Discrete-time Nonautonomous Dynamical Systems P.E. Kloeden, C. P¨ otzsche, and M. Rasmussen Abstract These notes present and discuss various aspects of the recent the- ory for time-dependent difference equations giving rise to nonautonomous dynamical systems on general metric spaces: First, basic concepts of autonomous difference equations and discrete-time (semi-) dynamical systems are reviewed for later contrast in the nonau- tonomous case. Then time-dependent difference equations or discrete-time nonautonomous dynamical systems are formulated as processes and as skew products. Their attractors including invariants sets, entire solutions, as well as the concepts of pullback attraction and pullback absorbing sets are in- troduced for both formulations. In particular, the limitations of pullback attractors for processes is highlighted. Beyond that Lyapunov functions for pullback attractors are discussed. Two bifurcation concepts for nonautonomous difference equations will be in- troduced, namely attractor and solution bifurcations. Finally, random difference equations and discrete-time random dynamical systems are investigated using random attractors and invariant measures. Key words: Nonautonomous dynamical system, process, two-parameter semigroup, discrete skew-product flow, difference equation, pullback attrac- tor, Lyapunov function, nonautonomous bifurcation, random dynamical sys- tem P.E. Kloeden Institut f¨ ur Mathematik, Goethe-Universit¨at, Postfach 11 19 32, 60054 Frankfurt a.M., Germany, e-mail: [email protected]C.P¨otzsche Institut f¨ ur Mathematik, Universit¨at Klagenfurt, Universit¨ atsstraße 65–67, 9020 Klagen- furt, Austria, e-mail: [email protected]M. Rasmussen Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom, e-mail: [email protected]1
Transcript
Discrete-timeNonautonomous Dynamical Systems
P.E. Kloeden, C. Potzsche, and M. Rasmussen
Abstract These notes present and discuss various aspects of the recent the-ory for time-dependent difference equations giving rise to nonautonomousdynamical systems on general metric spaces:First, basic concepts of autonomous difference equations and discrete-time(semi-) dynamical systems are reviewed for later contrast in the nonau-tonomous case. Then time-dependent difference equations or discrete-timenonautonomous dynamical systems are formulated as processes and as skewproducts. Their attractors including invariants sets, entire solutions, as wellas the concepts of pullback attraction and pullback absorbing sets are in-troduced for both formulations. In particular, the limitations of pullbackattractors for processes is highlighted. Beyond that Lyapunov functions forpullback attractors are discussed.Two bifurcation concepts for nonautonomous difference equations will be in-troduced, namely attractor and solution bifurcations.Finally, random difference equations and discrete-time random dynamicalsystems are investigated using random attractors and invariant measures.
The qualitative theory of dynamical systems has seen an enormous develop-ment since the groundbreaking contributions of Poincare and Lyapunov overa century ago. Meanwhile it provides a successful framework to describe andunderstand a large variety of phenomena in areas as diverse as physics, lifescience, engineering or sociology.
Such a success benefits, in part, from the fact that the law of evolutionin various problems from the above areas is static and does not change withtime (or chance). Thus a description with autonomous evolutionary equa-tions is appropriate. Nevertheless, many real world problems involve time-dependent parameters and, furthermore, one wants to understand control,modulation or other effects. In doing so, periodically or almost periodicallydriven systems are special cases, but, in principle, a theory for arbitrarytime-dependence is desirable. This led to the observation that many of themeanwhile well-established concepts, methods and results for autonomoussystems are not applicable and require an appropriate extension — the the-ory of nonautonomous dynamical systems.
The goal of these notes is to give a solid foundation to describe the long-term behaviour of nonautonomous evolutionary equations. Here we restrictto the discrete-time case in form of nonautonomous difference equations. Thishas the didactical advantage to feature many aspects of infinite-dimensionalcontinuous-time problems (namely nonexistence and uniqueness of backwardsolutions) without an involved theory to guarantee the existence of a semiflow.Moreover, even in low dimensions, discrete dynamics can be quite complex.
Beyond that a time-discrete theory is strongly motivated from applicationse.g., in population biology. In addition, it serves as a basic tool to understandnumerical temporal discretization and is often essential for the analysis ofcontinuous-time problems thorough concepts like time-1- or Poincare map-pings.
The focus of our presentation is on two formulations of time-discretenonautonomous dynamical systems, namely processes (two-parameter semi-groups) and skew-product systems. For both we construct, discuss and com-
5
6 1 Introduction
pare the so-called pullback attractor in Chapters 4–6. A pullback attractorserves as nonautonomous counterpart to the global attractor, i.e., the objectcapturing the essential dynamics of a system. Furthermore, in Chapter 7 wesketch two approaches to a bifurcation theory for time-dependent problemsto illuminate a current field of research. The final Chapter 8 on random dy-namical systems emphasises similarities to the corresponding nonautonomoustheory and provides results on random Markov chains and the approximationof invariant measures.
To conclude this introduction we point out that a significantly more com-prehensive approach is given in the up-coming monograph [32] (see also thelecture notes [52, 48]). In particular, we neglect various contributions to thediscrete-time nonautonomous theory: An appropriate spectral notion for lin-ear difference equations (cf. [55, 6, 43, 54]) substitutes the dynamical role ofeigenvalues from the autonomous special case. Gaps in this spectrum enableto construct nonautonomous invariant manifolds (so-called invariant fiberbundles, see [5, 50]). As special case they include centre fiber bundles andtherefore allow one to deduce a time-dependent version of Pliss’s reductionprinciple [41, 49]. The pullback attractors constructed in these notes are,generally, only upper semi-continuous in parameters. Thus, for approxima-tion purposes it might be advantageous to embedded them into a more robustdynamical object, namely a discrete counterpart to an inertial manifold [42].Topological linearization of nonautonomous difference equations has been ad-dressed in [7, 8], while a smooth linearization theory via normal forms wasdeveloped in [59].
Acknowledgements
This work was partially supported by the DFG grant KL 1203/7-1, theMinisterio de Ciencia e Innovacion project MTM2011-22411, the Consejerıade Innovacion, Ciencia y Empresa (Junta de Andalucıa) under the Ayuda2009/FQM314 and the Proyecto de Excelencia P07-FQM-02468 (Peter E.Kloeden). Martin Rasmussen was supported by an EPSRC Career Accelera-tion Fellowship.
The authors thank Dr. Thomas Lorenz for carefully reading parts of thearticle.
Chapter 2
Autonomous Difference Equations
A difference equation of the form
xn+1 = f (xn) , (1)
where f : Rd → Rd, is called a first-order autonomous difference equation on
the state space Rd. There is no loss of generality in the restriction to first-order difference equations (1), since higher-order difference equations can bereformulated as (1) by the use of an appropriate higher dimensional statespace.
Successive iteration of an autonomous difference equation (1) generatesthe forwards solution mapping π : Z+ × Rd → Rd defined by
xn = π(n, x0) = fn (x0) := f f · · · f︸ ︷︷ ︸n times
(x0),
which satisfies the initial condition π(0, x0) = x0 and the semigroup property
denote the nonnegative and nonpositive integers, respectively, and a discreteinterval is the intersection of a real interval with the set of integers Z.
Property (2) says that the solution mapping π forms a semigroup undercomposition; it is typically only a semigroup rather than a group since themapping f need not be invertible. It will be assumed here that the mapping fin the difference equation (1) is at least continuous, from which it follows thatthe mappings π(n, ·) are continuous for every n ∈ Z+. The solution mappingπ then generates a discrete-time semidynamical system on Rd.
7
8 2 Autonomous Difference Equations
More generally, the state space could be a metric space (X, d).
Definition 0.1. A mapping π : Z+ ×X → X satisfying
i) π(0, x0) = x0 for all x0 ∈ X,ii) π(m+ n, x0) = π(m,π(n, x0)) for all m, n ∈ Z+ and x0 ∈ X,
iii) the mapping x0 7→ π(n, x0) is continuous for each n ∈ Z+,
is called a (discrete-time) autonomous semidynamical system or asemigroup on the state space X.
The semigroup property ii) is illustrated in Figure 1 below. Note that suchan autonomous semidynamical system π on X is equivalent to a first orderautonomous difference equation on X with the right-hand side f defined byf(x) := π(1, x) for all x ∈ X.
If Z+ in Definition 0.1 is replaced by Z, then π is called a (discrete-time)autonomous dynamical system or group on the state space X.
X
x0
π(n, x0)
π(m+ n, ·)
π(m,π(n, x0))
Fig. 1 Semigroup property ii) of a discrete-time semidynamical system π : Z+ ×X → X
Autonomous dynamical systems need not be generated by autonomousdifference equations as above.
Example 0.1. Consider the space X = 1, · · · , rZ of bi-infinite sequencesx = knn∈Z with kn ∈ 1, · · · , r w.r.t. the group of left shift operatorsθn := θn for n ∈ Z, where the mapping θ : X → X is defined by θ(knn∈Z) =kn+1n∈Z. This forms an autonomous dynamical system on X, which is acompact metric space with the metric
d (x, x′) =∑n∈Z
(r + 1)−|n| |kn − k′n| .
The proximity and convergence of sets is given in terms of the Hausdorffseparation distX(A,B) of nonempty compact subsets A,B ⊆ X is defined as
1 Autonomous semidynamical systems 9
distX(A,B) := maxa∈A
dist(a,B) = maxa∈A
minb∈B
d(a, b)
and the Hausdorff metric HX(A,B) = max distX(A,B),distX(B,A) onthe space H(X) of nonempty compact subsets of X. In absence of possibleconfusion we simply write dist or H for the Hausdorff separation resp. metric.
1 Autonomous semidynamical systems
The dynamical behaviour of a semidynamical system π on a state space Xis characterised by its invariant sets and what happens in neighbourhoodsof such sets. A nonempty subset A of X is called invariant under π, or π-invariant, if
π(n,A) = A for all n ∈ Z+ (3)
or, equivalently, if f(A) = π(1, A) = A.Simple examples are equilibria (steady state solutions) and periodic so-
lutions; in the first case A consists of a single point, which must thus be afixed point of the mapping f , whereas for a solution with period r it consistsof a finite set of r distinct points p1, . . . , pr which are fixed point of thecomposite mapping fr (but not for an f j with j smaller than r).
Invariant sets can also be much more complicated, for example fractal sets.Many are the ω-limit sets of some trajectory, i.e., defined by
ω+(x0) = y ∈ X : ∃nj →∞, π(nj , x0)→ y ,
which is nonempty, compact and π-invariant when the forwards trajectoryπ(n, x0); n ∈ Z+ is a precompact subset of X and the metric space (X, d)is complete. However, ω(x0) needs not to be connected.
The asymptotic behaviour of a semidynamical system is characterised byits ω-limit sets, in general, and by its attractors and their associated absorbingsets, in particular. An attractor is a nonempty π-invariant compact set A∗
that attracts all trajectories starting in some neighbourhood U of A∗, that iswith ω+(x0) ⊂ A∗ for all x0 ∈ U or, equivalently, with
limn→∞
dist (π(n, x0), A∗) = 0 for all x0 ∈ U .
A∗ is called a maximal or global attractor when U is the entire state space X.Note that a global attractor, if it exists, must be unique. For later comparisonthe formal definition follows.
Definition 1.1. A nonempty compact subset A∗ of X is a globalattractor of the semidynamical system π on X if it is π-invariant and
10 2 Autonomous Difference Equations
attracts bounded sets, i.e.,
limn→∞
dist (π (n,D) , A∗) = 0 for any bounded subset D ⊂ X. (4)
As simple example consider the autonomous difference equation (1) onX = R with the map f(x) := max0, 4x(1− x) for x ∈ R. Then A∗ = [0, 1]is invariant and f(x0) ∈ A∗ for all x0 ∈ R, so A∗ is the maximal attractor.The dynamics are very simple outside of the attractor, but chaotic within it.
The existence and approximate location of a global attractor follows fromthat of more easily found absorbing sets, which typically have a convenientsimpler shape such as a ball or ellipsoid.
Definition 1.2. A nonempty compact subset B of X is called anabsorbing set of a semidynamical system π on X if for every boundedsubset D of X there exists a ND ∈ Z+ such that π(n,D) ⊂ B for alln ≥ ND in Z+.
Absorbing sets are often called attracting sets when they are also positivelyinvariant in the sense that π(n,B) ⊆ B holds for all n ∈ Z+, i.e., if one hasthe inclusion f(B) = π(1, B) ⊆ B. Attractors differ from attracting sets inthat they consist entirely of limit points of the system and are thus strictlyinvariant in the sense of (3).
Theorem 1.1 (Existence of global attractors). Suppose that asemidynamical system π on X has an absorbing set B. Then π hasa unique global attractor A∗ ⊂ B given by
A∗ =⋂m≥0
⋃n≥m
π(n,B), (5)
or simply by A∗ =⋂m≥0 π(n,B) when B is positively invariant.
For a proof we refer to the more general situation of Theorem 8.1.Similar results hold if the absorbing set is assumed to be only closed and
bounded and the mapping π to be compact or asymptotically compact.For later comparison note that, in view of the invariance of A∗, the attrac-
tion (4) can be written equivalently as the forwards convergence
2 Lyapunov functions for autonomous attractors 11
dist (π (n,D) , π (n,A∗))→ 0 as n→∞. (6)
A global attractor is, in fact, uniformly Lyapunov asymptotically stable.The asymptotic stability of attractors and that of attracting sets in generalcan be characterised by Lyapunov functions. Such Lyapunov functions canbe used to establish the existence of an absorbing set and hence that of anearby global attractor in a perturbed system.
2 Lyapunov functions for autonomous attractors
Consider an autonomous semidynamical system π on a compact metric space(X, d) which is generated by an autonomous difference equation
xn+1 = f(xn) , (7)
where f : X → X is globally Lipschitz continuous with Lipschitz constantL > 0, i.e.,
d(f(x), f(y)) ≤ Ld(x, y), for all x, y ∈ X .
Definition 2.1. A nonempty compact subset A ( X is called globallyuniformly asymptotically stable if it is both
i) Lyapunov stable , i.e., for all ε > 0, there exists a δ = δ(ε) > 0 with
dist(x,A) < δ ⇒ dist(fn(x), A) < ε for all n ∈ Z+ , (8)
ii) globally uniformly attracting , i.e., for all ε > 0, there exists an integerN = N(ε) > 1 such that
dist(fn(x), A) < ε for all x ∈ X, n ≥ N . (9)
Note that such a set A is the global attractor for the semidynamical sys-tem generated by an autonomous difference equation (7). In particular, it isinvariant, i.e., f(A) = A.
Global uniform asymptotical stability is characterized in terms of a Lya-punov function by the following necessary and sufficient conditions. The fol-lowing theorem is taken from Diamond & Kloeden [15].
Theorem 2.1. Let f : X → X be globally Lipschitz continuous, andlet A be a nonempty compact subset of X. Then A is globally uniformly
12 2 Autonomous Difference Equations
asymptotically stable w.r.t. the dynamical system generated by (7) if andonly if there exist
i) a Lyapunov function V : X → R+,ii) monotone increasing continuous functions α, β : R+ → R+ with
α(0) = β(0) = 0 and 0 < α(r) < β(r) for all r > 0, andiii) constants K > 0, 0 ≤ q < 1 such that for all x, y ∈ X, it holds that
1) |V (x)− V (y)| ≤ Kd(x, y),2) α(dist(x,A)) ≤ V (x) ≤ β(dist(x,A)) and3) V (f(x)) ≤ qV (x).
Proof. Sufficiency. Let V be a Lyapunov function as described in the theorem.
Choose ε > 0 arbitrarily and define δ := β−1(α(ε)/q), which means thatα(ε) = qβ(δ). This implies that
dist(fn(x), A) ≤ α−1 (qβ(dist(x,A))) ≤ α−1(α(ε)) ≤ ε for all n ∈ N,
when dist(x,A) < δ. Thus, A is Lyapunov stable. Now define
N := max
1, 1 +
⌊ln (α(ε)/V0)
ln q
⌋,
where V0 := maxx∈X V (x) is finite by continuity of V0 and compactness ofX. For n ≥ N , one has qn ≤ qN , since 0 ≤ q < 1. Since, from above,
α(dist(fn(x), A)) ≤ qnV (x) ≤ qnV0 ≤ qNV0 ≤ α(ε) for all n ≥ N ,
one has dist(fn(x), A) < ε for n ≥ N , x ∈ X. This means that A is globallyuniformly attracting and hence globally uniformly asymptotically stable.
Necessity. This will just be sketched here; the details can be found in [15].Let A be globally uniformly asymptotically stable, i.e., for given ε > 0, thereexists δ = δ(ε) such that (8) holds, and for given ε > 0, there exists N = N(ε)such that (9) holds. Define Gk : R+
0 → R+0 for k ∈ N by
Gk(r) :=
r − 1
k : r ≥ 1k ,
0 : 0 ≤ r < 1k ,
for all r ≥ 0 .
Then|Gk(r)−Gk(s)| ≤ |r − s| for all r, s ≥ 0 .
2 Lyapunov functions for autonomous attractors 13
Now choose q so that 0 < q < min1, L, where L is the Lipschitz constantof the mapping f , and define
gk :=( qL
)N(1/k)
for all k ∈ N
andVk(x) = gk sup
n∈Z+
q−nGk(dist(fn(x), A)) for all k ∈ N .
Then
i) Vk(x) = 0 if and only if dist(x,A) < δ(1/k), due to Lyapunov stability.ii) Since |dist(x,A)− dist(y,A)| ≤ d(x, y) and
d(fn(x), fn(y)) ≤ Ld(fn−1(x), fn−1(y)
)≤ · · · ≤ Lnd(x, y) ,
it follows that
|Vk(x)− Vk(y)|≤ gk sup
n≥0q−n |Gk(dist(fn(x), A))−Gk(dist(fn(y), A))|
≤ gk sup0≤n≤N(1/k)
q−n |Gk(dist(fn(x), A))−Gk(dist(fn(y), A))|
≤ gk sup0≤n≤N(1/k)
q−nd (fn(x), fn(y))
≤ gk sup0≤n≤N(1/k)
q−nLnd(x, y) = d(x, y) .
iii) From above, it holds that Vk(x) ≤ Vk(y) + d(x, y). For all y ∈ A, oneobtains that Vk(y) = 0 and Vk(x) ≤ d(x, y), and since A is compact, theminimum over all y ∈ A is attained and Vk(x) ≤ dist(x,A).
iv) Vk(f(x)) ≤ qVk(x), since
Vk(f(x)) ≤ gk supn≥0
q−nGk(dist(fn(f(x)), A))
= qgk supn≥0
q−n−1Gk(dist(fn+1(x), A))
= qgk supn≥1
q−nGk(dist(fn(x), A))
≤ qgk supn≥0
q−nGk(dist(fn(x), A)) = qVk(x)
Finally, define
V (x) =
∞∑k=1
2−kVk(x).
14 2 Autonomous Difference Equations
The main difficulty is to show the existence of the lower bound function α.This is systematically built up via the component functions Vk, which vanishsuccessively on a closed 1
k -neighbourhood of the set A. ut
Remarks
For a more comprehensive introduction to discrete dynamical systems andtheir attractors we refer to e.g. [40, 60]. In particular, for the case of infinite-dimensional state spaces see [18] and [57, Chapter 2], where also connected-ness issues of attractors or compactness properties for the semigroup π areaddressed.
Chapter 3
Nonautonomous Difference Equations
Difference equations on Rd of the form
xn+1 = fn (xn) , (∆)
in which continuous mappings fn : Rd → Rd on the right-hand side are al-lowed to vary with the time n, are called nonautonomous difference equations .
Such nonautonomous difference equations arise quite naturally in manydifferent ways. The mappings fn in (∆) may of course vary completely arbi-trarily, but often there is some relationship between them or some regularityin the way in which they are given.
For example, the mappings may all be the same as in the very specialautonomous subcase (1) or they may vary periodically within, or be cho-sen irregularly from, a finite family g1, · · · , gr, in which case (∆) can berewritten as
xn+1 = gkn (xn) , (10)
with the kn ∈ 1, . . . , r and fn = gkn .As another example, the difference equation (∆) may represent a variable
time-step discretization method for a differential equation x = f(x), thesimplest of which being the Euler method with a variable time-step hn > 0,
xn+1 = xn + hnf (xn) , (11)
in which case fn(x) = x+ hnf(x). More generally, a difference equation mayinvolve a parameter λ ∈ Λ which varies in time by choice or randomly, givingrise to the nonautonomous difference equation
xn+1 = g(xn, λn), (12)
so fn(x) = g(x, λn) here for the prescribed choice of λn ∈ Λ.The nonautonomous difference equation (∆) generates a solution mapping
φ : Z2≥ × Rd → Rd, where
15
16 3 Nonautonomous Difference Equations
Z2≥ := (n, n0) ∈ Z2 : n ≥ n0,
through iteration, i.e.,
φ(n0, n0, x0) := x0, φ(n, n0, x0) := fn−1 · · · fn0(x0) for all n > n0 ,
n0 ∈ Z, and each x0 ∈ Rd. This solution mapping satisfies the two-parametersemigroup property
φ(m,n0, x0) = φ(m,n, φ(n, n0, x0))
for all (n, n0) ∈ Z2≥, (m,n) ∈ Z2
≥ and x0 ∈ Rd. In this sense, φ is calledgeneral solution of (∆). In particular, as composition of continuous functions
the mapping x0 7→ φ(n, n0, x0) is continuous for (n, n0) ∈ Z2≥.
The general nonautonomous case differs crucially from the autonomous inthat the starting time n0 is just as important as the time that has elapsedsince starting, i.e., n − n0, and hence many of the concepts that have beendeveloped and extensively investigated for autonomous dynamical systemsin general and autonomous difference equations in particular are either toorestrictive or no longer valid or meaningful.
3 Processes
Solution mappings of nonautonomous difference equations (∆) are one of themain motivations for the process formulation of an abstract nonautonomousdynamical system on a metric state space (X, d) and time set Z.
The following definition originates from Dafermos [14] and Hale [18].
Definition 3.1. A (discrete-time) process on a state space X is a map-
ping φ : Z2≥ ×X → X, which satisfies the initial value, two-parameter
evolution and continuity properties:
i) φ(n0, n0, x0) = x0 for all n0 ∈ Z and x0 ∈ X,ii) φ(n2, n0, x0) = φ (n2, n1, φ(n1, n0, x0)) for all n0 ≤ n1 ≤ n2 in Z andx0 ∈ X,
iii) the mapping x0 7→ φ(n, n0, x0) of X into itself is continuous for alln0 ≤ n in Z.
The evolution property ii) is illustrated in Figure 2. Given a process φ onX there is an associated nonautonomous difference equation like (∆) on Xwith mappings defined by fn(x) := φ(n+ 1, n, x) for all x ∈ X and n ∈ Z.
4 Skew-product systems 17
Z
x0
φ(n1, n0, x0)φ(n2, n1, φ(n1, n0, x0))
X
φ(n2, n0, ·)
n0 n1 n2
Fig. 2 Property ii) of a discrete-time process φ : Z2≥ ×X → X
A process is often called a two-parameter semigroup on X in contrast withthe one-parameter semigroup of an autonomous semidynamical system sinceit depends on both the initial time n0 and the actual time n rather thanjust the elapsed time n − n0. This abstract formalism of a nonautonomousdynamical system is a natural and intuitive generalization of autonomoussystems to nonautonomous systems.
4 Skew-product systems
The skew-product formalism of a nonautonomous dynamical system is some-what less intuitive than the process formalism. It represents the nonau-tonomous system as an autonomous system on the cartesian product of theoriginal state space and some other space such as a function or sequencespace on which an autonomous dynamical systems called the driving systemacts. This driving system is the source of nonautonomity in the dynamics onthe original state space.
Let (P, dP ) be a metric space with metric dP and let θ = θn : n ∈ Zbe a group of continuous mappings from P onto itself. Essentially, θ is anautonomous dynamical system on P that models the driving mechanism forthe change in the mappings fn on the right-hand side of a nonautonomousdifference equation like (∆), that will now be written as
xn+1 = f (θn(p), xn) (13)
for n ∈ Z+, where f : P×Rd → Rd is continuous. The corresponding solutionmapping ϕ : Z+ × P × Rd → Rd is now defined by
ϕ(0, p, x) := x, ϕ(n, p, x) := f (θn−1(p), ·) · · · f (p, x) for all n ∈ N
18 3 Nonautonomous Difference Equations
and p ∈ P , x ∈ Rd. The mapping ϕ satisfies the cocycle property w.r.t. thedriving system θ on P , i.e.,
Consider now a state space X instead of Rd, where (X, d) is a metric spacewith metric d. The above considerations lead to the following definition ofa skew-product system, which is an alternative abstract formulation of adiscrete nonautonomous dynamical system on the state space X.
Definition 4.1. A (discrete-time) skew-product system (θ, φ) is de-fined in terms of a cocycle mapping ϕ on a state space X, driven by anautonomous dynamical system θ acting on a base space P .
Specifically, the driving system θ on P is a group of homeomorphismsθn : n ∈ Z under composition on P with the properties
i) θ0(p) = p for all p ∈ P ,ii) θm+n(p) = θm(θn(p)) for all m, n ∈ Z and p ∈ P ,
iii) the mapping p 7→ θn(p) is continuous for each n ∈ Z,
and the cocycle mapping φ : Z+ × P ×X → X satisfies
I) ϕ(0, p, x) = x for all p ∈ P and x ∈ X,II) ϕ(m + n, p, x) = ϕ(m, θn(p), ϕ(n, p, x)) for all m,n ∈ Z+, p ∈ P ,
x ∈ X,III) the mapping (p, x) 7→ φ(n, p, x) is continuous for each n ∈ Z.
For an illustration we refer to the subsequent Figure 3. A difference equa-tion of the form (13) can be obtained from a skew-product system by definingf(p, x) := ϕ(1, p, x) for all p ∈ P and x ∈ X.
A process φ admits a formulation as a skew-product system with P = Z,the time shift θn(n0) := n+ n0 and the cocycle mapping
ϕ(n, n0, x) := φ(n+ n0, n0, x) for all n ∈ Z+, x ∈ X.
The real advantage of the somewhat more complicated skew-product systemformulation of nonautonomous dynamical systems occurs when P is compact.This never happens for a process reformulated as a skew-product system as
4 Skew-product systems 19
P
p
θnp
θn+mp
p ×X
θnp ×X
θn+mp ×X
ϕ(n, p, ·)ϕ(m, θn+mp, ·)
ϕ(n+m, p, ·)
Fig. 3 A discrete-time skew-product system (θ, ϕ) over the base space P
above since the parameter space P then is Z, which is only locally compactand not compact.
4.2 Examples
The examples above can be reformulated as skew-product systems with ap-propriate choices of parameter space P and the driving system θ.
Example 4.1. A nonautonomous difference equation (∆) with continuousright-hand sides fn : Rd → Rd generates a cocycle mapping ϕ over theparameter set P = Z w.r.t. the group of left shift mappings θj := θj forj ∈ Z, where θ(n) := n+ 1 for n ∈ Z. Here ϕ is defined by
ϕ(0, n, x) := x and ϕ(j, n, x) := fn+j−1 · · · fn(x) for all j ∈ N
and n ∈ Z, x ∈ Rd. The mappings ϕ(j, n, ·) : Rd → Rd are all continuous.
Example 4.2. Let f : Rd → Rd be a continuous mapping used in an au-tonomous difference equation (1). The solution mapping ϕ defined by
ϕ(0, x) := x and ϕ(j, x) = f j(x) := f · · · f︸ ︷︷ ︸j times
(x) for all j ∈ N
and x ∈ Rd generates a semigroup on Rd. It can be considered as a cocyclemapping w.r.t. a singleton parameter set P = p0 and the singleton groupconsisting only of identity mapping θ := idP on P . Since the driving systemjust sits at p0, the dependence on the parameter in ϕ can be suppressed.
20 3 Nonautonomous Difference Equations
While the integers Z appears to be the natural choice for the parameter set inExample 4.1 and the choice is trivial in the autonomous case of Example 4.2,in the remaining examples the use of sequence spaces is more advantageousbecause such spaces are often compact.
Example 4.3. The nonautonomous difference equation (10) with continuousmappings gk : Rd → Rd for k ∈ 1, · · · , r generates a cocycle mapping overthe parameter set P = 1, · · · , rZ of bi-infinite sequences p = knn∈Z withkn ∈ 1, · · · , r w.r.t. the group of left shift operators θn := θn for n ∈ Z,where θ(knn∈Z) = kn+1n∈Z. The mapping ϕ is defined by
ϕ(0, p, x) := x and ϕ(j, p, x) := gkj−1 · · · gk0(x) for all j ∈ N
and x ∈ Rd, where p = knn∈Z, is a cocycle mapping. Note that the param-eter space 1, · · · , rZ here is a compact metric space with the metric
d (p, p′) =∑n∈Z
(r + 1)−|n| |kn − k′n| .
In addition, θn : P → P and ϕ(j, ·, ·) : P × Rd → Rd are all continuous.
We omit a reformulation of the numerical scheme (11) as it is similar tothe next example, but with a bi-infinite sequence p = hnn∈Z of stepsizessatisfying a constraint such as 1
2δ ≤ hn ≤ δ for n ∈ Z with appropriate δ > 0.
Example 4.4. As an example of a parametrically perturbed difference equa-tion (12), consider the mapping g : R1 ×
[12 , 1]7→ R1 defined by
g(x, λ) =|x|+ λ2
1 + λ,
which is continuous in x ∈ R1 and λ ∈[12 , 1]. Let P =
[12 , 1]Z
be the space of
bi-infinite sequences p = λnn∈Z taking values in[12 , 1], which is a compact
metric space with the metric
d (p, p′) =∑n∈Z
2−|n| |λn − λ′n| ,
and let θn, n ∈ Z be the group generated by the left shift operator θ onthis sequence space (analogously to Example 4.3). The mapping ϕ is definedby
ϕ(0, p, x) := x and ϕ(j, p, x) := g(qj−1, ·) · · · g(q0, x) for all j ∈ N
and x ∈ R1, where p = λnn∈Z, is a cocycle mapping on R1 with parameter
space[12 , 1]Z
and the above shift operators θn. The mappings θn : P → Pand ϕ(j, ·, ·) : P × Rd → Rd are all continuous here.
4 Skew-product systems 21
4.3 Skew-product systems as autonomoussemidynamical systems
A skew-product system (θ, ϕ) can be reformulated as autonomous semidy-namical system on the extended state space X := P ×X. Define a mappingπ : Z+ × X → X by
π (n, (p, x0)) :=(θn(p), φ(n, p, x0
)for all n ∈ Z+, (p, x0) ∈ X .
Note that the variable n in π (n, (p, x0)) is the time that has elapsed sincestarting at state (p, x0).
Theorem 4.1. π is an autonomous semidynamical system on X.
Proof. It is obvious that π(n, ·) is continuous in its variables (p, x0) for everyn ∈ Z+ and satisfies the initial condition
π(0, (p, x0)) = (p, ϕ(0, p, x0)) = (p, x0) for all p ∈ P, x0 ∈ X.
It also satisfies the one-parameter semigroup property
π(m+ n, (p, x0)) = π (m,π(n, (p, x0))) for all m,n ∈ Z+, p ∈ P, x0 ∈ X
since, by the group property of the driving system and the cocycle propertyof the skew-product,
π(m+ n, (p, x0)) = (θm+n(p), ϕ(m+ n, p, x0))
=(θm (θn(p)) , ϕ(m, θn(p), ϕ(n, p, x0))
)= π
(m, (θn(p), ϕ(n, p, x0))
)= π
(m,π(n, (p, x0))
).
ut
As seen in Example 4.1, a process φ on the state space X is also a skew-product on X with the shift operator θ on P := Z and thus generates anautonomous semidynamical system π on the extended state space X := Z×X.This semidynamical system has some unusual properties. In particular, πhas no nonempty ω-limit sets and, indeed, no compact subset of X can beπ-invariant. This is a direct consequence of the fact that the initial time is acomponent of the extended state space.
22 3 Nonautonomous Difference Equations
Remarks
An early reference to the description of nonautonomous discrete dynamicsvia processes or skew-product flows, is given in [40, pp. 45–56, Chapt. 4].
Chapter 4
Nonautonomous invariant sets andattractors of processes
Invariant sets and attractors are important regions of state space that char-acterize the long-term behaviour of a dynamical system.
Let φ : Z2≥ × X → X be a process on a metric state space (X, d). This
generates a solution xn = φ(n, n0, x0) to (∆) that depends on the start-ing time n0 as well as the current time n and not just on the time n − n0that has elapsed since starting as in an autonomous system. This has someprofound consequences in terms of definitions and the interpretation of dy-namical behaviour. As pointed out above, many concepts and results fromthe autonomous case are no longer valid or are too restrictive and excludemany interesting types of possible behaviour.
For example, it is too great a restriction of generality to consider a singlesubset A of X to be invariant under φ in the sense that
φ(n, n0, A) = A for all n ≥ n0, n0 ∈ Z,
which is equivalent to fn(A) = A for every n ∈ Z, where the fn are mappingsin the corresponding nonautonomous difference equation (∆). Then, in gen-eral, neither the trajectory χ∗n : n ∈ Z of a solution χ∗ that exists on all ofZ nor a nonautonomous ω-limit set defined by
ω+ (n0, x0) = y ∈ X : ∃nj →∞, φ (nj , n0, x0)→ y ,
will be invariant in such a sense.Moreover, such nonautonomous ω-limit sets exist in the infinite future in
absolute time rather than in current time like autonomous ω-limit sets, so itis not so clear how useful or even meaningful dynamically they are. Hence,the appropriate formulation of asymptotic behaviour of a nonautonomousdynamical system needs some careful consideration. Lyapunov asymptoti-cal stability of a solution of a nonautonomous system provides a clue. Thisrequires the definition of an entire solution.
23
24 4 Attractors of processes
Definition 4.2. An entire solution of a process φ on X is a sequenceχk : k ∈ Z in X such that
φ(n, n0, χn0) = χn for all n ≥ n0 and all n0 ∈ Z,
or equivalently, χn+1 = fn(χn) for all n ∈ Z in terms of the nonau-tonomous difference equation (∆) corresponding to the process φ.
Definition 4.3. An entire solution χ∗ of a process φ on X is said tobe (globally) Lyapunov asymptotically stable if it is Lyapunov stable ,i.e., for every ε > 0 and n0 ∈ Z there exists a δ = δ(ε, n0) > 0 such that
d (φ(n, n0, x0), χ∗n) < ε for all n ≥ n0 whenever d(x0, χ
∗n0
)< δ,
and attracting in the sense that
d (φ (n, n0, x0) , χ∗n)→ 0 as n→∞ (15)
for all x0 ∈ X and n0 ∈ Z.
Note, in particular, that the limiting “target” χ∗n exists for all time andis, in general, also changing in time as the limit is taken.
5 Nonautonomous invariant sets
Let χ∗ be an entire solution of a process φ on a metric space (X, d) andconsider the family A = An : n ∈ Z of singleton subsets An := χ∗n of X.Then by the definition of an entire solution it follows that
φ (n, n0, An0) = An for all n ≥ n0, n0 ∈ Z .
This suggests the following generalization of invariance for nonautonomousdynamical systems.
Definition 5.1. A family A = An : n ∈ Z of nonempty subsets ofX is invariant under a process φ on X, or φ-invariant , if
6 Forwards and pullback convergence 25
φ (n, n0, An0) = An for all n ≥ n0 and all n0 ∈ Z,
or, equivalently, if fn(An) = An+1 for all n ∈ Z in terms of the corre-sponding nonautonomous difference equation (∆).
A φ-invariant family consists of entire solutions. This is essentially due tothe fact that process is onto between the component subsets. The backwardsolutions, however, need not to be uniquely determined, since the mappingsfn are usually not assumed to be one-to-one.
Proposition 5.1 (Characterization of invariant sets). A familyA = An : n ∈ Z is φ-invariant if and only if for every pair n0 ∈ Zand x0 ∈ An0
there exists an entire solution χ such that χn0= x0 and
χn ∈ An for all n ∈ Z.Moreover, the entire solution χ is uniquely determined provided the
mapping fn(·) := φ(n+ 1, n, ·) : X → X is one-to-one for every n ∈ Z.
Proof. Sufficiency Let A be φ-invariant and pick an arbitrary x0 ∈ An0. For
n ≥ n0 define the sequence χn := φ(n, n0, x0). Then the φ-invariance of Ayields χn ∈ An. On the other hand, An0 = φ(n0, n,An) for n ≤ n0, so thereexists a sequence xn ∈ An with x0 = φ(n0, n, xn) and xn = φ(n, n− 1, xn−1)for all n < n0. Hence define χn := xn for n < n0 and χ becomes an entiresolution with the desired properties. If the mappings fn are all one-to-one,then the sequence xn is uniquely determined.
Necessity Suppose for an arbitrary n0 ∈ Z and x0 ∈ An0 that there isan entire solution χ with χn0 = x0 and χn ∈ An for all n ∈ Z. Henceφ(n, n0, x0) = φ(n, n0, χn0
) = χn ∈ An for n ≥ n0. From this it follows thatfn(An) ⊆ An+1. The remaining inclusion fn(An) ⊇ An+1 follows from thefact that x0 = φ(n0, n, χn) ∈ φ(n0, n,An) for n ≤ n0. ut
6 Forwards and pullback convergence
The convergence
d (φ (n, n0, x0) , χ∗n)→ 0 as n→∞ (n0 fixed)
in the attraction property (15) in the definition of a Lyapunov asymptoticallystable entire solution χ∗ of a process φ will be called forwards convergence
26 4 Attractors of processes
(cf. Figure 4) to distinguish it from another kind of convergence that is usefulfor nonautonomous systems.
Z
X
n0
χ∗
φ(·, n0, x0)
Fig. 4 Forward convergence n→∞
Forwards convergence does not, however, provide convergence to a par-ticular point χ∗n∗ for a fixed n∗ ∈ Z, which is important in many practicalsituations because the actual solution χ∗ may not be known and thus needsto be determined. To obtain such convergence one has to start progressivelyearlier. This leads to the concept of pullback convergence , defined by
d (φ (n, n0, x0) , χ∗n)→ 0 as n0 → −∞ (n fixed)
and illustrated in Figure 5.
Z
X
n0n0n0n0 n
χ∗x0
Fig. 5 Pullback convergence n0 → −∞
In terms of the elapsed time j, forwards convergence can be rewritten as
d(φ (n0 + j, n0, x0) , χ∗n0+j
)→ 0 as j →∞ (16)
for all x0 ∈ X and n0 ∈ Z, while pullback convergence becomes
d (φ (n, n− j, x0) , χ∗n)→ 0 as j →∞
7 Forwards and pullback attractors 27
for all x0 ∈ X and n ∈ Z.
Example 6.1. The nonautonomous difference equation xn+1 = 12xn + gn on
R has the solution mapping φ(j + n0, n0, x0) = 2−jx0 +∑jk=0 2−j+kgn0+n,
for which pullback convergence gives
φ(n0, n0 − j, x0) = 2−jx0 +
j∑k=0
2−kgn0−k →∞∑k=0
2−kgn0−k as j →∞,
provided the infinite series here converges. The limiting solution χ∗ is givenby χ∗n0
:=∑∞k=0 2−kgn0−k for each n0 ∈ Z. It is an entire solution of the
nonautonomous difference equation.
Pullback convergence makes use of information about the nonautonomousdynamical system from the past, while forwards convergence uses informationabout the future.
In autonomous dynamical systems, forwards and pullback convergence areequivalent since the elapsed time n− n0 →∞ if either n→∞ with n0 fixedor n0 → −∞ with n fixed. In nonautonomous dynamical systems pullbackconvergence and forwards convergence do not necessarily imply each other.
Example 6.2. Consider the process φ on R generated fn = g1 for n ≤ 0 andfn = g2 for n ≥ 1 where the mappings g1, g2 : R→ R are given by g1(x) := 1
2xand g2(x) := max0, 4x(1− x) for all x ∈ R. Then φ is pullback convergentto the entire solution χ∗ defined by χ∗n ≡ 0 for n ∈ Z, but is not forwardsconvergent to χ∗. In particular, χ∗ is not Lyapunov stable.
7 Forwards and pullback attractors
Forwards and pullback convergence can be used to define two distinct typesof nonautonomous attractors for a process φ on a state space X. Instead ofa family A = An : n ∈ Z of singleton subsets An := χ∗n for an entiresolution χ∗ of the process consider a φ-invariant family of A = An : n ∈ Zof nonempty subsets An of X.
In this context forwards convergence generalizes to
More generally, A is said to forwards (resp. pullback) attracts bounded sub-sets of X if x0 is replaced by an arbitrary bounded subset D of X in (17)(resp. (18)).
28 4 Attractors of processes
Definition 7.1. A φ–invariant family A = An : n ∈ Z of nonemptycompact subsets of X is called a forward attractor if it forward attractsbounded subsets of X and a pullback attractor if it pullback attractsbounded subsets of X.
As a φ-invariant family A of nonempty compact subsets of X, by Propo-sition 5.1, both pullback and forwards attractors consist of entire solutions.
In fact when the component subsets of a pullback attractor are uniformlybounded, i.e., if there exists a bounded subset B of X such that An ⊂ B forall n ∈ Z, then pullback attractors are characterized by the bounded entiresolutions of the process.
Proposition 7.1 (Dynamical characterization of pullback at-tractors). A uniformly bounded pullback attractor A = An : n ∈ Zadmits the dynamical characterization: for each n0 ∈ Z
x0 ∈ An0⇔ there exists a bounded entire solution χ with χn0
= x0.
Such a pullback attractor is therefore uniquely determined.
Proof. Sufficiency Pick n0 ∈ Z and x0 ∈ An0 arbitrarily. Then, due to theφ-invariance of the pullback attractor A, by Proposition 5.1 there exists anentire solution χ with χn0
= x0 and χn ∈ An for each n ∈ Z. Moreover, χis bounded since the component sets of the pullback attractor are uniformlybounded.
Necessity If there exists a bounded entire solution χ of the process φ, thenthe set of points Dχ := χn : n ∈ Z is bounded in X. Since A pullbackattracts bounded subsets of X, for each n ∈ Z,
0 ≤ dist (χn, An) ≤ limj→∞
dist (φ(n, n− j,Dχ), An) = 0,
so χn ∈ An. ut
8 Existence of pullback attractors 29
8 Existence of pullback attractors
Absorbing sets can also be defined for pullback attraction. Wider applicabilitycan be attained if they are also allowed to depend on time.
Definition 8.1. A family B = Bn : n ∈ Z of nonempty compactsubsets of X is called a pullback absorbing family for a process φ onX if for each n ∈ Z and every bounded subset D of X there exists anNn,D ∈ Z+ such that
φ (n, n− j,D) ⊆ Bn for all j ≥ Nn,D, n ∈ Z.
The existence of a pullback attractor follows from that of a pullback ab-sorbing family in the following generalization of Theorem 1.1 for autonomousglobal attractors. The proof is simpler if the pullback absorbing family is as-sumed to be φ-positive invariant.
Definition 8.2. A family B = Bn : n ∈ Z of nonempty compactsubsets of X is said to be φ-positive invariant if
φ (n, n0, Bn0) ⊆ Bn for all n ≥ n0.
Theorem 8.1 (Existence of pullback attractors). Suppose that aprocess φ on a complete metric space (X, d) has a φ-positive invariantpullback absorbing family B = Bn : n ∈ Z. Then there exists a globalpullback attractor A = An : n ∈ Z with component sets determinedby
An =⋂j≥0
φ (n, n− j, Bn−j) for all n ∈ Z. (19)
Moreover, if A is uniformly bounded then it is unique.
Proof. Let B be a pullback absorbing family and let An be defined as in (19).Clearly An ⊂ Bn for each n ∈ Z.
(i) First, it will be shown for any n ∈ Z that
30 4 Attractors of processes
limj→∞
dist (φ(n, n− j, Bn−j), An) = 0 . (20)
Assume to the contrary that there exist sequences xjk ∈ φ (n, n− jk, Bn−jk)⊂ Bn and jk → ∞ such that dist(xjk , An) > ε for all k ∈ N. The set xjk :k ∈ N ⊂ Bn is relatively compact, so there is a point x0 ∈ Bn and an indexsubsequence k′ →∞ such that xjk′ → x0. Now
xjk′ ∈ φ(n, n− jk′ , Bn−jk′
)⊂ φ (n, n− k,Bn−k)
for all kj′ ≥ k and each k ≥ 0 . This implies that
x0 ∈ φ (n, n− k,Bn−k) for all k ≥ 0 .
Hence, x0 ∈ An, which is a contradiction. This proves the assertion (20).(ii) By (20), for every ε > 0, n ∈ Z, there exists an N = Nε,n ≥ 0 such that
dist (φ(n, n−N,Bn−N ), An) < ε .
Let D be a bounded subset of X. The fact that B is a pullback absorbingfamily implies that φ (n, n− j,D) ⊂ Bn for all sufficiently large j. Hence, bythe cocycle property,
φ (n, n−N − j,D) = φ (n, n−N,φ(n−N,n−N − j,D))
⊂ φ (n, n−N,Bn−N ) .
(iii) The φ-invariance of the family A will now be shown. By (19), the setFm(n) := φ (n, n−m,Bn−m) is contained in Bn for every m ≥ 0, and bydefinition, An−j =
⋂m≥0 Fm (n− j). First, it will be shown that
φ
n, n− j, ⋂m≥0
Fm(n− j)
=⋂m≥0
φ (n, n− j, Fm(n− j)) . (21)
One sees directly that “⊂” holds. To prove “⊃”, let x be contained in the seton the right side. Then for any n ≥ 0, there exists an xm ∈ Fm(n−j) ⊂ Bn−jsuch that x = φ (n, n− j, xm). Since the sets Fm(n − j) are compact andmonotonically decreasing with increasing m, the set xm : m ≥ 0 has a limitpoint x ∈ ⋂m≥0 Fm(n−j) . By the continuity of φ (n, n− j, ·), it follows thatx = φ (n, n− j, x). Thus,
x ∈ φ
n, n− j, ⋂m≥0
Fm(n− j)
= φ (n, n− j, An−j) .
Hence, equation (21), the compactness of Fm(n − j) and the continuity ofφ (n, n− j, ·) imply that
8 Existence of pullback attractors 31
φ (n, n− j, An−j) =⋂m≥0
φ (n, n− j, Fm(n− j))
⊃⋂m≥0
φ (n, n− j, Fm(n− j))
=⋂m≥0
φ (n, n− j, φ (n− j, n− j −m,Bn−j−m))
=⋂m≥0
φ (n, n− j −m,Bn−j−m)
=⋂m≥j
φ (n, n−m,Bn−m) ⊃ An ,
which means that
An ⊂ φ (n, n− j, An−j) , j ∈ Z+ for all n ∈ Z . (22)
Replacing n by n−m in (22) and using the cocycle property gives
for all ε-neighborhoods Uε(An) of An, where ε > 0, provided that j = J(ε)is sufficiently large. Hence, φ (n, n−m,An−m) ⊂ An For all m ∈ Z+, n ∈ Z.With m replaced by j, this yields with (22) the φ-invariance of the familyAn : n ∈ Z.(iv) It remains to observe that if the sets in A = An : n ∈ Z are uniformlybounded, then the pullback attractor A is unique by Proposition 7.1. ut
Remark 8.1. There is no counterpart of Theorem 8.1 for nonautonomous for-wards attractors
If the pullback absorbing family B is not φ-positive invariant, then theproof is somewhat more complicated and the component subsets of the pull-back attractor of A are given by
An =⋂k≥0
⋃j≥k
φ (n, n− j, Bn−j).
32 4 Attractors of processes
However, the assumption in Theorem 8.1 that φ-positively invariant pullbackabsorbing systems is not a serious restriction.
Proposition 8.1. If B = Bn : n ∈ Z is a pullback absorbing systemfor a process φ fulfilling Bn ⊂ C for n ∈ Z, where C is a bounded subsetof X, then there exists a φ-positively invariant pullback absorbing systemB = Bn : n ∈ Z containing B = Bn : n ∈ Z component set-wise.
Proof. For each n ∈ Z define
Bn :=⋃j≥0
φ(n, n− j, Bn−j).
Obviously Bn ⊂ Bn for every n ∈ Z.To show positive invariance, the cocycle property is used in what follows.
φ(n+ 1, n, Bn) =⋃j≥0
φ(n+ 1, n, φ(n, n− j, Bn−j))
=⋃j≥0
φ(n+ 1, n− j, Bn−j)
=⋃i≥1
φ(n+ 1, n+ 1− i, Bn+1−i)
⊆⋃i≥0
φ(n+ 1, n+ 1− i, Bn+1−i) = Bn+1,
so φ(n+ 1, n, Bn) ⊆ Bn+1. By this and the cocycle property again
φ(n+ 2, n, Bn) = φ(n+ 2, n+ 1, φ(n+ 1, n, Bn)
)⊆ φ(n+ 2, n+ 1, Bn+1) ⊆ Bn+2.
The general positive invariance assertion then follows by induction.Now by the continuity of φ(n, n−j, ·) and the compactness of Bn−j , the set
φ(n, n− j, Bn−j) is compact for each j ≥ 0 and n ∈ Z. Moreover, Bn−j ⊂ Cfor each j ≥ 0 and n ∈ Z, so by the pullback absorbing property of B thereexists an N = Nn,C ∈ N such that
φ(n, n− j, Bn−j) ⊂ φ(n, n− j, C) ⊂ Bn
9 Limitations of pullback attractors 33
for all j ≥ N . Hence
Bn =⋃j≥0
φ(n, n− j, Bn−j)
⊆ Bn ∪⋃
0≤j<N
φ(n, n− j, Bn−j)
=⋃
0≤j<N
φ(n, n− j, Bn−j),
which is compact as a finite union of compact sets, so Bn is compact.To see that B so constructed is pullback absorbing, let D be a bounded
subset of X and fix n ∈ Z. Since B is pullback absorbing, there exists anNn,D ∈ N such that φ(n, n− j,D) ⊂ Bn for all j ≥ Nn,D. But Bn ⊂ Bn, so
φ(n, n− j,D) ⊂ Bn for all j ≥ Nn,D.
Hence B is pullback absorbing as required. ut
9 Limitations of pullback attractors
Pullback attractors are based on the behaviour of a nonautonomous systemin the past and may not capture the complete dynamics of a system when itis formulated in terms of a process. This was already indicated by Example6.2 and will be illustrated here through some simpler examples.
First consider the autonomous scalar difference equation
xn+1 =λxn
1 + |xn|(23)
depending on a real parameter λ > 0. Its zero solution x∗ = 0 exhibits apitchfork bifurcation at λ = 1. Its global dynamical behavior can be summa-rized as follows (see Figure 6):
• If λ ≤ 1, then x∗ = 0 is the only constant solution and is globally asymptot-ically stable. Thus 0 is the global attractor of the autonomous dynamicalsystem generated by the difference equation (23).
• If λ > 1, then there exist two additional nontrivial constant solutions givenby x± := ±(λ − 1). The zero solution x∗ = 0 is an unstable steady statesolution and the symmetric interval A = [x−, x+] is the global attractor.
These constant solutions are the fixed points of the mapping f(x) = λx1+|x| .
34 4 Attractors of processes
Fig. 6 Trajectories of the autonomous difference equation (23) with λ = 0.5 (left) andλ = 1.5 (right)
Piecewise autonomous difference equation: Consider now the piece-wise autonomous equation
xn+1 =λnxn
1 + |xn|, λn :=
λ, n ≥ 0,
λ−1, n < 0(24)
for some λ > 1, which corresponds to a switch between the two autonomousproblems (23) at n = 0.
The zero solution of the resulting nonautonomous system is the onlybounded entire solution, so by Proposition 7.1 the pullback attractor A hascomponent sets An ≡ 0 for all n ∈ Z. Note that the zero solution seems tobe “asymptotically stable” for n < 0 and then “unstable” for n ≥ 0. More-over the interval [x−, x+] is like a global attractor for the whole equation onZ, but it is not really one since it is not invariant or minimal for n < 0.
The nonautonomous difference equation (24) is asymptotically autonomousin both directions, but the pullback attractor does not reflect the full limitingdynamics (see Figure 7 (left)), in particular in the forwards time direction.Fully nonautonomous equation: If the parameters λn do not switch fromone constant to another as above, but increase monotonically, e.g., such asλn = 1 + 0.9n
1+|n| , then the dynamics is similar, although the limiting dynamics
is not so obvious from the equation. See Fig. 7 (left).Let λnn∈Z be a monotonically increasing sequence with limk→±∞ λn =
λ±1 for λ > 1. The nonautonomous problem
xn+1 = fn(xn) :=λnxn
1 + |xn|. (25)
is asymptotically autonomous in both directions with the limiting autonomoussystems given above.
9 Limitations of pullback attractors 35
Fig. 7 Trajectories of the piecewise autonomous equation (24) with λ = 1.5 (left) and theasymptotically autonomous equation (25) with λk = 1 + 0.9k
1+|k| (right)
Its pullback attractor A has component sets An ≡ 0 for all n ∈ Zcorresponding to the zero entire solution, which is the only bounded entiresolution. As above, the zero solution x∗ = 0 seems to be “asymptoticallystable” for n < 0 and then “unstable” for n ≥ 0. However, the forwardlimit points for nonzero solutions are ±(λ− 1), neither of which is a solutionat all. In particular, they are not entire solutions, so cannot belong to anattractor, forward or pullback, since these consist of entire solutions. SeeFigure 2 (right).
Remark 9.1. Pullback attraction alone does not characterize fully the boundedlimiting behaviour of a nonautonomous system formulated as a process.Something in addition like nonautonomous limit sets [32, 48], limiting equa-tions [22] or asymptotically invariant sets [23] and eventual asymptotic sta-bility [24] or a mixture of these ideas is needed to complete the picture.However, this varies from example to example and is somewhat ad hoc. Incontrast, this information is built into the skew-product system formulationof a nonautonomous dynamical system, especially when the state space P ofthe driving system is compact. Essentially, the skew-product system alreadyincludes the limiting dynamics and no further ad hoc methods are needed todetermine it.
Remarks
Pullback attractors for nonautonomous difference equations were introducedin [27, 28] and a comparison between different attractor types is given in [12](see also Section 11).
Without the assumption of being uniformly bounded, pullback attractorsof processes need not to be unique (see [48, p. 18, Example 1.3.5]). In ap-
36 4 Attractors of processes
plications, absorbing sets are frequently not compact and one has to assumeambient compactness properties of a process in order to establish the exis-tence of a pullback attractor (see [48, pp. 12ff]).
Chapter 5
Nonautonomous invariant sets andattractors: Skew-product systems
10 Existence of pullback attractors
Consider a discrete-time skew-product system (θ, ϕ) on P ×X, where (P, dP )and (X, d) are metric spaces. There are counterparts for skew-product sys-tems of the concepts of invariance, forwards and pullback convergence and for-wards and pullback attractors considered in the previous chapter for discrete-time processes.
Definition 10.1. A family A = Ap : p ∈ P of nonempty subsets ofX is called ϕ-invariant for a skew-product system (θ, ϕ) on P ×X if
ϕ(n, p,Ap) = Aθn(p) for all n ∈ Z+, p ∈ P.
It is called ϕ-positively invariant if
ϕ(n, p,Ap) ⊆ Aθn(p) for all n ∈ Z+, p ∈ P.
Definition 10.2. A family A = Ap : p ∈ P of nonempty compactsubsets of X is called pullback attractor of a skew-product system (θ, ϕ)on P ×X if it is ϕ-invariant and pullback attracts bounded sets, i.e.,
dist (ϕ(j, θ−j(p), D), Ap)→ 0 for j →∞ (26)
for all p ∈ P and all bounded subsets D of X. It is called a forwardsattractor if it is ϕ-invariant and forward attracts bounded sets, i.e.,
37
38 5 Attractors of skew-product systems
dist(ϕ(j, p,D), Aθj(p)
)→ 0 for j →∞. (27)
As with processes, the existence of a pullback attractor for skew-productsystems is ensured by that of a pullback absorbing system.
Definition 10.3. A family B = Bp : p ∈ P of nonempty compactsubsets of X is called a pullback absorbing family for a skew-productsystem (θ, ϕ) on P ×X if for each p ∈ P and every bounded subset Dof X there exists an Np,D ∈ Z+ such that
ϕ (j, θ−j(p), D) ⊆ Bp for all j ≥ Np,D, p ∈ P.
The following result generalizes Theorem 1.1 for autonomous semidynami-cal systems and the first half is the counterpart of Theorem 8.1 for processes.The proof is similar in the latter case, essentially with j and θ−j(p) changedto n0 and n0 − j, respectively, but additional complications due to the factthat the pullback absorbing family is no longer assumed to be ϕ-positivelyinvariant. See [33] for details.
Theorem 10.1 (Existence of pullback attractors). Let (X, d) and(P, dP ) be complete metric spaces and suppose that a skew-product sys-tem (θ, ϕ) has a pullback absorbing set family B = Bp : p ∈ P. Thenthere exists a pullback attractor A = Ap : p ∈ P with component setsdetermined by
Ap =⋂n≥0
⋃j≥n
ϕ(j, θ−j(p), Bθ−j(p)
); (28)
it is unique if its component sets are uniformly bounded.
The pullback attractor of a skew-product system (θ, ϕ) has some niceproperties when its component subsets are contained in a common compactsubset or if the state space P of the driving system is compact.
Proposition 10.1 (Upper semi-continuity of pullback attrac-tors). Suppose that A(P ) :=
⋃p∈P Ap is compact for a pullback at-
10 Existence of pullback attractors 39
tractor A = Ap : p ∈ P. Then the set-valued mapping p 7→ Ap isupper semi-continuous in the sense that
dist (Aq, Ap)→ 0 as q → p.
On the other hand, if P is compact and the set-valued mapping p 7→ Apis upper semi-continuous, then A(P ) is compact.
Proof. First note that, since A(P ) is compact, the pullback attractor is uni-formly bounded by a compact set and hence is uniquely determined.
Assume that the set-valued mapping p 7→ Ap is not upper semi-continuous.Then there exist an ε0 > 0 and a sequence pn → p0 in P such thatdist (Apn , Ap0) ≥ 3ε0 for all n ∈ N. Since the sets Apn are compact, thereexists an an ∈ Apn such that
dist (an, Ap0) = dist (Apn , Ap0) ≥ 3ε0 for each n ∈ N . (29)
By pullback attraction, dist (ϕ (m, θ−m(p0), B) , Ap0) ≤ ε0 for m ≥ MB,ε0 forany bounded subset B of X; in particular, below A(P ) will be used for the setB. By the ϕ-invariance of the pullback attractor, there exist bn ∈ Aθ−m(pn) ⊂A(P ) for n ∈ N such that ϕ (m, θ−m(pn), bn) = an. Since A(P ) is compact,there is a convergent subsequence bn′ → b ∈ A(P ). Finally, by the continuityof θ−m(·) and of the cocycle mapping ϕ(n, ·, ·),
which contradicts (29). Hence, p 7→ Ap must be upper semi-continuous.The remaining assertion follows since the image of a compact subset under
an upper semi-continuous set-valued mapping is compact (cf. [4]). utPullback attractors are in general not forwards attractors. When, however,
the state space P of the driving system is compact, then one has the followingpartial forwards convergence result for the pullback attractor.
Theorem 10.2. In addition to the assumptions of Theorem 10.1, sup-pose that P is compact and suppose that the pullback absorbing family
40 5 Attractors of skew-product systems
B is uniformly bounded by a compact subset C of X. Then
limn→∞
supp∈P
dist (ϕ(n, p,D), A(P )) = 0 (30)
for every bounded subset D of X, where A(P ) :=⋃p∈P Ap.
Proof. First note that A(P ) is compact since the component subsets Ap areall contained in the common compact set C. This means also that the pullbackattractor is unique.
Suppose to the contrary that the convergence (30) does not hold. Thenthere exist an ε0 > 0 and sequences nj →∞, pj ∈ P and xj ∈ C such that
dist (ϕ(nj , pj , xj), A(P )) > ε0 . (31)
Set pj = θnj (pj). By the compactness of P , there exists a convergent subse-quence pj′ → p0 ∈ P . From the pullback attraction, there exists an n > 0such that
dist (ϕ(n, θ−n(p0), C), Ap0) <ε02.
The cocycle property then gives
ϕ(nj , θ−nj (pj), xj
)= ϕ
(n, θ−n(pj), ϕ
(nj − n, θ−nj (pj), xj
))for any nj > n. By the pullback absorption of B, it follows that
ϕ(nj − n, θ−nj (pj), xj
)⊂ Bθ−n(pj) ⊂ C ,
and since C is compact, there is a further index subsequence j′′ of j′ (de-pending on n) such that
znj′′ := ϕ(nj′′ − n, θ−nj′′ (pj′′), xj′′
)→ z0 ∈ C.
The continuity of the skew-product mappings in the p and x variables implies
Proposition 11.1 (Uniform and global attractors). Suppose thatA is a uniform attractor (i.e., uniformly attracting in both the forwardand pullback senses) of a skew-product system (θ, ϕ) and that
⋃p∈P Ap
is precompact in X. Then the union A :=⋃p∈P p × Ap is the global
attractor of the autonomous semidynamical system π.
Proof. The π-invariance of A follows from the ϕ-invariance of A, and theθ-invariance of P via
π(n,A) =⋃p∈Pθn(p)×ϕ(n, p,Ap) =
⋃p∈Pθn(p)×Aθn(p) =
⋃q∈Pq×Aq = A .
Since A is also a pullback attractor and⋃p∈P Ap is precompact in X (and P
is compact too), the set-valued mapping p 7→ Ap is upper semi-continuous,which means that p 7→ F (p) := p × Ap is also upper semi-continuous.Hence, F (P ) = A is a compact subset of X. Moreover, the definition of themetric distX on X implies that
where π(n, (p, x)) = (θn(p), ϕ(n, p, x)). The desired attraction to A w.r.t. πthen follows from the forward attraction of A w.r.t. ϕ. ut
Without uniform attraction as in Proposition 11.1 a pullback attractorneed not give a global attractor, but the following result does hold.
42 5 Attractors of skew-product systems
Proposition 11.2. If A is a pullback attractor for a skew-product sys-tem (θ, ϕ) and
⋃p∈P Ap is precompact in X, then A :=
⋃p∈P p ×Ap
is the maximal invariant compact set of the autonomous semidynamicalsystem π.
Proof. The compactness and π-invariance of A are proved in the same wayas in first part of the proof of Proposition 11.1. To prove that the compactinvariant set A is maximal, let C be any other compact invariant set of theautonomous semidynamical system π. Then A is a compact and ϕ-invariantfamily of compact sets, and by pullback attraction,
dist(ϕ(n, θ−n(p), Cθ−n(p)
), Ap
)≤ dist (ϕ (n, θ−n(p),K) , Ap)→ 0
as n → ∞, where K :=⋃p∈P Cp is compact. Hence, Cp ⊆ Ap for all p ∈ P ,
i.e., C :=⋃p∈P p × Cp ⊆ A, which finally means that A is a maximal
π-invariant set. ut
The set A here need not be the global attractor of π. In the oppositedirection, the global attractor of the associated autonomous semidynamicalsystem always forms a pullback attractor of the skew-product system.
Proposition 11.3 (Global and pullback attractors). If an auton-omous semidynamical system π has a global attractor
A =⋃p∈Pp ×Ap,
then A = Ap : p ∈ P is a pullback attractor for the skew-productsystem (θ, ϕ).
Proof. The sets P and K :=⋃p∈P Ap are compact by the compactness of A.
12 Limitations of pullback attractors revisited 43
for all (p, x) ∈ P ×D and every arbitrary bounded subset D of X, since A isthe global attractor of π.
Hence, replacing p by θ−n(p) implies
limn→∞
dist (ϕ(n, θ−n(p), D),K) = 0 .
Then the system is pullback asymptotic compact (see the definition in Chap-ter 12 of [32]) and by Theorem 12.12 in [32] this is a sufficient condition forthe existence of a pullback attractor A′ = A′p : p ∈ P with
⋃p∈P A
′p ⊂ K.
From Proposition 11.2, A′ :=⋃p∈P p×A′p is the maximal π-invariant sub-
set of X, but so is the global attractor A. This means that A′ = A. Thus, Ais a pullback attractor of the skew-product system (θ, ϕ). ut
12 Limitations of pullback attractors revisited
The limitations of pullback attraction for processes were illustrated in Sec-tion 9 through the scalar nonautonomous difference equation
xn+1 = fn(xn) :=λnxn
1 + |xn|, (32)
where λnn∈Z is an increasing sequence with limn→±∞ λn = λ±1 for λ > 1.The pullback attractor A of the corresponding process has component
sets An ≡ 0 for all n ∈ Z corresponding to the zero entire solution, whichis the only bounded entire solution. The zero solution x∗ = 0 seems to be“asymptotically stable” for n < 0 and then “unstable” for n ≥ 0. Howeverthe forward limit points for nonzero solutions are ±(λ − 1), which both arenot solutions at all. In particular, they are not entire solutions.
An elegant way to resolve the problem is to consider the skew-productsystem formulation of a nonautonomous dynamical system. This includes anautonomous dynamical system as a driving mechanism, which is responsiblefor the temporal change in the dynamics of the nonautonomous differenceequation. It also includes the dynamics of the asymptotically autonomousdifference equations above and their limiting autonomous systems.
The nonautonomous difference equation (32) can be formulated as a skew-product system with the diving system defined in terms of the shift operatorθ on the space of bi-infinite sequences
ΛL = λ = λnn∈Z : λn ∈ [0, L] , n ∈ Z
for some L > λ > 1. It yields a compact metric space with the metric
dΛL (λ, λ′) :=∑n∈Z
2−|n| |λn − λ′n| .
44 5 Attractors of skew-product systems
This is coupled with a cocycle mapping with values xn = ϕ(n, λ, x0) on Rgenerated by the difference equation (32) with a given coefficient sequence λ.
For the sequence λ from (32), the limit of the shifted sequences θn(λ) inthe above metric as n → ∞ is the constant sequence λ∗+ equal to λ, whilethe limit as n→ −∞ is the sequence λ∗− with all components equal to λ−1.
The pullback attractor of the corresponding skew-product system (θ, ϕ)on Λ × R consists of compact subsets Aλ of R for each λ ∈ ΛL. It is easyto see that Aλ = 0 for any λ with components λn < 1 for n ≤ 0, whichincludes the constant sequence λ∗− as well as the switched sequence in (32).On the other hand, Aλ∗+ = [−λ, λ]. Here ∪λ∈ΛLAλ is precompact, so containsall future limiting dynamics.
The pullback attractor of the skew-product system includes that of the pro-cess for a given bi-infinite coefficient sequence, but also includes its forwardasymptotic limits and much more. The coefficient sequence set ΛL includes allpossibilities, in fact, far more than may be of interest in particular situation.
If one is interested in the dynamics of a process corresponding to a specificλ ∈ ΛL, then it would suffice to consider the skew-product system w.r.t. thedriving system on the smaller space Λλ defined as the hull of this sequence,
i.e., the set of accumulation points of the set θn(λ) : n ∈ Z in the metric
space (ΛL, dΛL). In particular, if λ is the specific sequence in (32), then theunion ∪λ∈ΛλAλ = Aλ∗+ = [−λ, λ] contains all future limiting dynamics, i.e.,
limn→∞
dist(ϕ(n, λ, x), [−λ, λ]
)= 0 for all x ∈ R.
The example described by nonautonomous difference equation (32) isasymptotically autonomous with Λλ = λ∗± ∪ θn(λ) : n ∈ Z. The for-ward limit points ±(λ − 1) of the process generated by (25), which werenot steady states of the process, are now locally asymptotic steady states ofthe skew product flow with base space P = Λ consisting of the single con-stant sequence λk ≡ λ, when the skew product system is interpreted as anautonomous semidynamical system on the product space P ×X. More gen-erally, unlike the process formulation, the skew-product system formulationand its pullback attractor include the forwards limiting dynamics.
13 Local pullback attractors
Less uniform behaviour such as parameter dependent domains of definitionand local pullback attractors can be handled by introducing the concept of abasin of attraction system.
13 Local pullback attractors 45
Let Domp ⊂ X be the domain of definition of f(p, ·) in the nonautonomousequation (13), which requires f(p,Domp) ⊂ Domθ(p). Then the correspondingcocycle mapping ϕ has the domain of definition Z+ × ⋃p∈P (p ×Domp).Consequently one needs to restrict the admissible families of bounded sets inthe pullback convergence to subsets of Domp for each p ∈ P .
Definition 13.1. An ensemble Dad of families D = Dp : p ∈ P ofnonempty subsets X is called admissible ifi) Dp is bounded and Dp ⊂ Domp for each p ∈ P and every D = Dp :p ∈ P ∈ Dad ; and
ii) D(1) = D(1)p : p ∈ P ∈ Dad whenever D(2) = D(2)
p : p ∈ P ∈Dad and D
(1)p ⊆ D(2)
p for all p ∈ P .
Further restrictions will allow one to consider local or otherwise restrictedform of pullback attraction.
Definition 13.2. A ϕ-invariant family A = Ap : p ∈ P of nonemptycompact subsets of X with Ap ⊂ Domp for each p ∈ P is called apullback attractor w.r.t. the basin of attraction system Datt if Datt isan admissible ensemble of families of subsets such that
limj→∞
dist(ϕ(j, θ−j(p), Dθ−j(p)), Ap
)= 0 (33)
for every D = Dp : p ∈ P ∈ Datt.
In this case a pullback absorbing set system B = Bp : p ∈ P shouldalso satisfy B ∈ Datt and the pullback absorbing property be modified to
ϕ(j, θ−j(p), Dθ−j(p))
)⊆ Bp
for all j ≥ Np,D, p ∈ P and D = Dp; p ∈ P ∈ Datt.A counterpart of Theorem 10.1 then holds here. In this case the pull-
back attractor is unique within the basin of attraction system, but the skew-product system may have other pullback attractors within other basin of at-traction systems, which may be either disjoint from or a proper sub-ensembleof the original basin of attraction system.
Example 13.1. Consider the scalar nonautonomous difference equation
xn+1 = fn (xn) := xn + γnxn(1− x2n
)(34)
46 5 Attractors of skew-product systems
for given parameters γn > 0, n ∈ Z.First let γn ≡ γ for all n ∈ Z, so the system is autonomous. It has the at-
tractor A∗ = [−1, 1] for the maximal basin of attraction(−1− γ−1, 1 + γ−1
),
but if one restricts attention further to the basin of attraction(0, 1 + γ−1
)then the attractor is only A∗∗ = 1.
Now let γn be variable with γn ∈[12 γ, γ
]for each n ∈ Z, so the system
is now nonautonomous and representable as a skew-product on the statespace X = Z × R with the parameter set P = Z. Then A∗ = A∗n : n ∈ Zwith A∗n = [−1, 1] for all n ∈ Z is the pullback attractor for the basin ofattraction system Datt consisting of all families D = Dn : n ∈ Z satisfyingDn ⊂
(−1− γ−1, 1 + γ−1
), whereas A∗∗ = A∗∗n : n ∈ Z with A∗∗n = 1
for all n ∈ Z is the pullback attractor for the basin of attraction system Datt
consisting of all families D = Dn : n ∈ Z with Dn ⊂(0, 1 + γ−1
).
Chapter 6
Lyapunov functions for pullbackattractors
A Lyapunov function characterizing pullback attraction and pullback attrac-tors for a discrete-time process in Rd will be constructed here. Consider anonautonomous difference equation
xn+1 = fn(xn) (∆)
on Rd, where the fn : Rd → Rd are Lipschitz continuous mappings. Thisgenerates a process φ : Z2
≥ × Rd → Rd through iteration by
φ(n, n0, x0) = fn−1 · · · fn0(x0) for all n ≥ n0
and each x0 ∈ Rd, which in particular satisfies the continuity property
x0 7→ φ(n, n0, x0) is Lipschitz continuous for all n ≥ n0.
The pullback attraction is taken w.r.t. a basin of attraction system, whichis defined as follows for a process.
Definition 13.3. A basin of attraction system Datt consists of families
D = Dn : n ∈ Z of nonempty bounded subsets of Rd with the property
that D(1) = D(1)n : n ∈ Z ∈ Datt if D(2) = D(2)
n : n ∈ Z ∈ Datt and
D(1)n ⊆ D
(2)n for all n ∈ Z.
Although somewhat complicated, the use of such a basin of attractionsystem allows both nonuniform and local attraction regions, which are typicalin nonautonomous systems, to be handled.
47
48 6 Lyapunov functions
Definition 13.4. A φ-invariant family of nonempty compact subsets A= An : n ∈ Z is called a pullback attractor w.r.t. a basin of attractionsystem Datt if it is pullback attracting
limj→∞
dist (φ(n, n− j,Dn−j), An) = 0 (35)
for all n ∈ Z and all D = Dn : n ∈ Z ∈ Datt.
Obviously A ∈ Datt.
The construction of the Lyapunov function requires the existence of apullback absorbing neighbourhood family.
14 Existence of a pullback absorbing neighbourhoodsystem
The following lemma shows that there always exists such a pullback absorbingneighbourhood system for any given pullback attractor. This will be requiredfor the construction of the Lyapunov function for the proof of Theorem 15.1.The proof is very similar to that of Proposition 8.1.
Lemma 14.1. If A is a pullback attractor with a basin of attraction systemDatt for a process φ, then there exists a pullback absorbing neighbourhoodsystem B ⊂ Datt of A w.r.t. φ. Moreover, B is φ-positive invariant.
Proof. For each n0 ∈ Z pick δn0 > 0 such that
B[An0; δn0
] := x ∈ Rd : dist(x,An0) ≤ δn0
such that B[An0; δn0
] : n0 ∈ Z ∈ Datt and define
Bn0:=⋃j≥0
φ(n0, n0 − j, B[An0−j ; δn0−j ]).
Obviously An0⊂ intB[An0
; δn0] ⊂ Bn0
. To show positive invariance thetwo-parameter semigroup property will be used in what follows.
14 Existence of a pullback absorbing neighbourhood system 49
The general positive invariance assertion then follows by induction.Now referring to the continuity of φ(n0, n0 − j, ·) and the compactness of
B[An0−j ; δn0−j ], the set φ(n0, n0 − j, B[An0−j ; δn0−j ]) is compact for eachj ≥ 0 and n0 ∈ Z. Moreover, by pullback convergence, there exists an N =N(n0, δn0) ∈ N such that
which is compact, so Bn0 is compact.To see that B so constructed is pullback absorbing w.r.t. Datt, letD ∈ Datt.
Fix n0 ∈ Z. Since A is pullback attracting, there exists an N(D, δn0, n0) ∈ N
such thatdist (φ(n0, n0 − j,Dn0−j), An0
) < δn0
for all j ≥ N(D, δn0, n0). But (φ(n0, n0 − j,Dn0−j) ⊂ intB[An0
; δn0] and
B[An0; δn0
] ⊂ Bn0, so
50 6 Lyapunov functions
φ(n0, n0 − j,Dn0−j) ⊂ intBn0
for all j ≥ N(D, δn0 , n0). Hence B is pullback absorbing as required. ut
15 Necessary and sufficient conditions
The main result is the construction of a Lyapunov function that characterizesthis pullback attraction.
Theorem 15.1. Let the fn be uniformly Lipschitz continuous on Rdfor each n ∈ Z and let φ be the process that they generate. In addition,let A be aφ-invariant family of nonempty compact sets that is pullbackattracting with respect to φ with a basin of attraction system Datt. Thenthere exists a Lipschitz continuous function V : Z×Rd → R such that
Property 1 (upper bound): For all n0 ∈ Z and x0 ∈ Rd
V (n0, x0) ≤ dist(x0, An0); (36)
Property 2 (lower bound): For each n0 ∈ Z there exists a functiona(n0, ·) : R+ → R+ with a(n0, 0) = 0 and a(n0, r) > 0 for all r > 0which is monotonically increasing in r such that
a(n0,dist(x0, An0)) ≤ V (n0, x0) for all x0 ∈ Rd; (37)
Property 3 (Lipschitz condition): For all n0 ∈ Z and x0, y0 ∈ Rd
|V (n0, x0)− V (n0, y0)| ≤ ‖x0 − y0‖; (38)
Property 4 (pullback convergence): For all n0 ∈ Z and any D ∈ Datt
limsupn→∞ supzn0−n∈Dn0−n
V (n0, φ(n0, n0 − n, zn0−n)) = 0. (39)
In addition,Property 5 (forwards convergence): There exists N ∈ Datt. which ispositively invariant under φ and consists of nonempty compact sets Nn0
V (n0 + j, φ(j, n0, x0)) ≤ e−jV (n0, x0) for all x0 ∈ Nn0, j ∈ N. (41)
Proof. The aim is to construct a Lyapunov function V (n0, x0) that charac-terizes a pullback attractor A and satisfies properties 1–5 of Theorem 15.1.For this define
V (n0, x0) := supn∈N
e−Tn0,n dist (x0, φ(n0, n0 − n,Bn0−n))
for all n0 ∈ Z and x0 ∈ Rd, where
Tn0,n = n+
n∑j=1
α+n0−j
with Tn0,0 = 0. Here αn = logLn, where Ln is the uniform Lipschitz constantof fn on Rd, and a+ = (a+ |a|)/2, i.e., the positive part of a real number a.
Note 4: Tn0,n ≥ n and Tn0,n+m = Tn0,n + Tn0−n,m for n, m ∈ N, n0 ∈ Z.
Proof of property 1
Since e−Tn0,n ≤ 1 for all n ∈ N and dist (x0, φ(n0, n0 − n,Bn0−n)) is monoton-ically increasing from 0 ≤ dist (x0, φ(n0, n0, Bn0
If x0 ∈ An0 , then V (n0, x0) = 0 by Property 1, so assume that x0 ∈ Rd \An0 .Now in
V (n0, x0) = supn≥0
e−Tn0,ndist (x0, φ(n0, n0 − n,Bn0−n))
the supremum involves the product of an exponentially decreasing quantitybounded below by zero and a bounded increasing function, since the setsφ(n0, n0−n,Bn0−n) are a nested family of compact sets decreasing to An0
withincreasing n. In particular,
dist (x0, An0) ≥ dist (x0, φ(n0, n0 − n,Bn0−n)) for all n ∈ N.
Hence there exists an N∗ = N∗(n0, x0) ∈ N such that
for any x0, y0 ∈ Rd and nonempty compact subset C of Rd.
Proof of property 4
Assume the opposite. Then there exists an ε0 > 0, a sequence nj →∞ in Nand points xj ∈ φ(n0, n0−nj , Dn0−nj ) such that V (n0, xj) ≥ ε0 for all j ∈ N.Since D ∈ Datt and B is pullback absorbing, there exists an N = N(D, n0)∈ N such that
φ(n0, n0 − nj , Dn0−nj ) ⊂ Bn0 for all nj ≥ N.
Hence, for all j such that nj ≥ N , it holds xj ∈ Bn0, which is a compact set,
so there exists a convergent subsequence xj′ → x∗ ∈ Bn0 . But also
xj′ ∈⋃
n≥nj′
φ(n0, n0 − n,Dn0−n)
and ⋂nj′
⋃n≥nj′
φ(n0, n0 − n,Dn0−n) ⊆ An0
by the definition of a pullback attractor. Hence x∗ ∈ An0and V (n0, x
∗) = 0.But V is Lipschitz continuous in its second variable by property 3, so
ε0 ≤ V (n0, xj′) = ‖V (n0, xj′)− V (n0, x∗)‖ ≤ ‖xj′ − x∗‖,
which contradicts the convergence xj′ → x∗. Hence property 4 must hold.
Proof of property 5
DefineNn0
:= x0 ∈ B[Bn0; 1] : φ(n0 + 1, n0, x0) ∈ Bn0+1 ,
54 6 Lyapunov functions
where B[Bn0; 1] = x0 : dist(x0, Bn0
) ≤ 1 is bounded because Bn0is
compact and Rd is locally compact, so Nn0is bounded. It is also closed,
hence compact, since φ(n0 + 1, n0, ·) is continuous and Bn0+1 is compact.Now An0 ⊂ intBn0 and Bn0 ⊂ Nn0 , so An0 ⊂ intNn0 . In addition,
φ(n0 + 1, n0, Nn0) ⊂ Bn0+1 ⊂ Nn0+1,
so N is positive invariant.It remains to establish the exponential decay inequality (40). This needs
the following Lipschitz condition on φ(n0 + 1, n0, ·) ≡ fn0(·):
which is the desired inequality. Moreover, since φ(1, n0, x0) ∈ Bn0+1 ⊂ Nn0+1,the proof continues inductively to give
V (n0 + j, φ(n0 + j, n0, x0)) ≤ e−jV (n0, x0) for all j ∈ N.
This completes the proof of Theorem 15.1. ut
15.1 Comments on Theorem 15.1
Note 1: It would be nice to use φ(n0, n0 − n, x0) for a fixed x0 in the pull-back convergence property (39), but this may not always be possible due tononuniformity of the attraction region, i.e., there may not be a D ∈ Datt
with x0 ∈ Dn0−n for all n ∈ N.
Note 2: The forwards convergence inequality (41) does not imply forwardsLyapunov stability or Lyapunov asymptotical stability. Although
there is no guarantee (without additional assumptions) that
infj≥0
a(n0 + j, r) > 0
for r > 0, so dist(φ(n0 + j, n0, x0), An0+j) need not become small as j →∞.As a counterexample consider Example 6.2 of the process φ on R generated
by (∆) with fn = g1 for n ≤ 0 and fn = g2 for n ≥ 1 where the mappingsg1, g2 : R→ R are given by g1(x) := 1
2x and g2(x) := max0, 4x(1−x) for allx ∈ R. Then A with An0 = 0 for all n0 ∈ Z is pullback attracting for φ butis not forwards Lyapunov asymptotically stable. (Note one can restrict g1, g2to [−R,R] → [−R,R] for any fixed R > 1 to ensure the required uniformLipschitz continuity of the fn).
Note 3: The forwards convergence inequality (41) can be rewritten as
since An0 ⊆ φ(n0, n0−n−j, Bn0−n−j) and φ(n0, n0−m, zn0−m) ∈ Bn0 . Thus
V (n0, φ(n0, n0 −m, zn0−m)) ≤ e−Tn0,ndist (Bn0, An0
)
for all zn0−m ∈ Dn0−m, m ≥ n + N(D, n0, n) and n ≥ 0.It can be assumed that the mapping n 7→ n + N(D, n0, n) is monotonic in-
creasing in n (by taking a larger N(D, n0, n) if necessary), and is hence invert-ible. Let the inverse of m = n + N(D, n0, n) be n = M(m) = M(D, n0,m).Then
V (n0, φ(n0, n0 −m, zn0−m)) ≤ e−Tn0,M(m)dist (Bn0, An0
)
58 6 Lyapunov functions
for all m ≥ N(D, n0, 0) ≥ 0. Usually N(D, n0, 0) > 0. This expression can bemodified to hold for all m ≥ 0 by replacing M(m) by M∗(m) defined for allm ≥ 0 and introducing a constant KD,n0
≥ 1 to account for the behaviourover the finite time set 0 ≤ m < N(D, n0, 0). For all m ≥ 0 this gives
V (n0, φ(n0, n0 −m, zn0−m)) ≤ KD,n0e−Tn0,M
∗(m)dist (Bn0, An0
) .
Chapter 7
Bifurcations
The classical theory of dynamical bifurcation focusses on autonomous differ-ence equations
xn+1 = g(xn, λ) (43)
with a right-hand side g : Rd × Λ → Rd depending on a parameter λ fromsome parameter space Λ, which is typically a subset of Rn (cf., e.g., [37]or [21]). A central question is how stability and multiplicity of invariant setsfor (43) changes when the parameter λ is varied. In the simplest, and mostoften considered situation, these invariant sets are fixed points or periodicsolutions to (43).
Given some parameter value λ∗, a fixed point x∗ = g(x∗, λ∗) of (43) iscalled hyperbolic , if the derivative D1g(x∗, λ∗) has no eigenvalue on the
complex unit circle S1. Then it is an easy consequence of the implicit func-tion theorem (cf. [38, p. 365, Theorem 2.1]) that x∗ allows a unique con-tinuation x(λ) ≡ g(x(λ), λ) in a neighborhood of λ∗. In particular, hyper-bolicity rules out bifurcations understood as topological changes in the setx ∈ Rd : g(x, λ) = x near (x∗, λ∗) or a stability change of x∗.
On the other hand, eigenvalues on the complex unit circle give rise tovarious well-understood autonomous bifurcation scenarios. Examples includefold, transcritical or pitchfork bifurcations (eigenvalue 1), flip bifurcations(eigenvalue −1) or the Sacker–Neimark bifurcation (a pair of complex conju-gate eigenvalues for d ≥ 2).
16 Hyperbolicity and simple examples
Even in the autonomous set-up of (43) one easily encounters intrinsicallynonautonomous problems, where the classical methods of, for instance, [37,21] do not apply:
59
60 7 Bifurcations
1. Investigate the behaviour of (43) along an entire reference solution (χn)n∈Z,which is not constant or periodic. This is typically done using the (obvi-ously nonautonomous) equation of perturbed motion
xn+1 = g(xn + χn, λ)− g(χn, λ).
2. Replace the constant parameter λ in (43) by a sequence (λn)n∈Z in Λ,which varies in time. Also the resulting parametrically perturbed equation
xn+1 = g(xn, λn)
becomes nonautonomous. This situation is highly relevant from an appliedpoint of view, since parameters in real world problems are typically subjectto random perturbations or an intrinsic background noise.
Both the above problems fit into the framework of general nonautonomousdifference equations
xn+1 = fn(xn, λ) (∆λ)
with a sufficiently smooth right-hand side fn : Rd × Λ → Rd, n ∈ Z. Inaddition, suppose that fn and its derivatives map bounded subsets of Rd×Λinto bounded sets uniformly in n ∈ Z.
Generically, nonautonomous equations (∆λ) do not have constant solu-tions, and the fixed point sequences x∗n = fn(x∗n, λ
∗) are usually not solutionsto (∆λ). This gives rise to the following question:
If there are no equilibria, what should bifurcate in a nonautonomous set-up?
Before suggesting an answer, a criterion to exclude bifurcations is pro-posed. For motivational purposes consider again the autonomous case (43)and the problem of parametric perturbations.
Example 16.1. The autonomous difference equation xn+1 = 12xn + λ has the
unique fixed point x∗(λ) = 2λ for all λ ∈ R. Replace λ by a bounded sequence(λn)n∈Z and observe as in Example 6.1 that the nonautonomous counterpart
xn+1 = 12xn + λn
has a unique bounded entire solution χ∗n :=∑n−1k=−∞
(12
)n−k−1λk. For the
special case λn ≡ λ, this solution reduces to the known fixed point χ∗n ≡ 2λ.
This simple example yields the conjecture that equilibria of autonomousequations persist as bounded entire solutions under parametric perturbations.It will be shown below in Theorem 16.1 (or in [45, Theorem 3.4]) that thisconjecture is generically true in the sense that the fixed point of (43) has tobe hyperbolic in order to persist under parametric perturbations.
Example 16.2. The linear difference equation xn+1 = xn+λn has the forwardsolution xn = x0+
∑n−1k=0 λn, whose boundedness requires the assumption that
16 Hyperbolicity and simple examples 61
the real sequence (λn)n≥0 is summable. Thus, the nonhyperbolic equilibriax∗ of xn+1 = xn do not necessarily persist as bounded entire solutions underarbitrary bounded parametric perturbations.
Typical examples of nonautonomous equations having an equilibrium,given by the trivial solution are equations of perturbed motion. Their vari-ational equation along (χn)n∈Z is given by xn+1 = D1g(χn, λ)xn and inves-tigating the behaviour of its trivial solution under variation of λ requires anappropriate nonautonomous notion of hyperbolicity.
Suppose that An ∈ Rd×d, n ∈ Z, is a sequence of invertible matrices, andconsider a linear difference equation
xn+1 = Anxn (44)
with the transition matrix
Φ(n, l) :=
An−1 · · ·Al, l < n,
I, n = l,
A−1n · · ·A−1l−1, n < l.
Let I be a discrete interval and define I′ := k ∈ I : k+1 ∈ I. An invariant
projector for (44) is a sequence Pn ∈ Rd×d, n ∈ I, of projections Pn = P 2n
such thatAn+1Pn = PnAn for all n ∈ I′.
Definition 16.1. A linear difference equation (44) is said to admit anexponential dichotomy on I, if there exist an invariant projector Pn andreal numbers K ≥ 0, α ∈ (0, 1) such that for all n, l ∈ I one has
‖Φ(n, l)Pl‖ ≤ Kαn−l if l ≤ n,‖Φ(n, l)[id−Pl]‖ ≤ Kαl−n if n ≤ l.
Remark 16.1. An autonomous difference equation xn+1 = Axn has an ex-ponential dichotomy, if and only if the coefficient matrix A ∈ Rd×d has noeigenvalues on the complex unit circle.
In terms of this terminology an entire solution (χn)n∈Z of (∆λ) is calledhyperbolic , if the variational equation
xn+1 = D1fn(χn, λ)xn (Vλ)
has an exponential dichotomy on Z.Let `∞ denote the space of bounded sequences in Rd.
62 7 Bifurcations
Theorem 16.1 (Continuation of bounded entire solutions). Ifχ∗ = (χ∗n)n∈Z is an entire bounded and hyperbolic solution of (∆λ∗),then there exists an open neighborhood Λ0 ⊆ Λ of λ∗ and a uniquefunction χ : Λ0 → `∞ such that
i) χ(λ∗) = χ∗,ii) each χ(λ) is a bounded entire and hyperbolic solution of (∆λ),
iii) χ : Λ0 → `∞ is as smooth as the functions fn.
Proof. The proof is based on the idea to formulate a nonautonomous differ-ence equation (∆λ) as an abstract equation F (χ, λ) = 0 in the space `∞.This is solved using the implicit mapping theorem, where the invertibilityof the Frechet derivative D1F (χ∗, λ∗) is characterised by the hyperbolicityassumption on χ∗. For details, see [47, Theorem 2.11]. ut
Consequently, in order to deduce sufficient conditions for bifurcations, onemust violate the hyperbolicity of χ∗. For this purpose, the following charac-terisation of an exponential dichotomy is useful.
Theorem 16.2 (Characterization of exponential dichotomies).A variational equation (Vλ) has an exponential dichotomy on Z, if andonly if the following conditions are fulfilled:
i) (Vλ) has an exponential dichotomy on Z+ with projector P+n , as well
as an exponential dichotomy on Z− with projector P−n ,ii) R(P+
0 )⊕N(P−0 ) = Rd.
Proof. See [10, Lemma 2.4]. ut
The subsequent examples illustrate various scenarios that can arise, if acondition stated in Theorem 16.2 is violated.
Example 16.3 (Pitchfork bifurcation). Consider the difference equation
xn+1 = fn(xn, λ), fn(x, λ) :=λx
1 + |x|
from Section 9. It is a prototypical example of a supercritical autonomouspitchfork bifurcation (cf., e.g. [37, pp. 119ff, Sect. 4.4]), where the uniqueasymptotically stable equilibrium x∗ = 0 for λ ∈ (0, 1) bifurcates into twoasymptotically stable equilibria x± := ±(λ− 1) for λ > 1.
16 Hyperbolicity and simple examples 63
Along the trivial solution the variational equation xn+1 = λxn becomesnonhyperbolic for λ = 1. Indeed, criterion i) of Theorem 16.2 is violated,since the variational equation does not admit a dichotomy on Z+ or on Z−.This loss of hyperbolicity causes an attractor bifurcation, since for• λ ∈ (0, 1), the set x∗ = 0 is the global attractor• λ > 1, the trivial equilibrium x∗ = 0 becomes unstable and the symmetricinterval A = [x−, x+] is the global attractor.
Bifurcations of pullback attractors can be observed as nonautonomousversions of pitchfork bifurcations.
Example 16.4 (Pullback attractor bifurcation). Consider for parameter valuesλ > 0 the difference equation
xn+1 = λxn −
minanx
3n,
λ2xn
, xn ≥ 0,
maxanx
3n,
λ2xn
, xn < 0,
where (an)n∈Z is a sequence which is both bounded and bounded away fromzero. Note that in a neighborhood U of 0, the difference equation is given byxn+1 = λxn − anx3n, and outside of a set V ⊃ U , the difference equation isgiven by xn+1 = λ
2xn. Both U and V here can be chosen independently of λnear λ = 1. Moreover, for fixed n ∈ Z, the right-hand side of this equationlies between the functions x 7→ λ
2x and x 7→ λx.It is clear that for λ ∈ (0, 1), the global pullback attractor is given by
the trivial solution, which follows from the fact that points are contracted ateach time step by the factor λ. For λ > 1, the trivial solution is no longerattractive, but there exists a (nontrivial) pullback attractor for λ ∈ (1, 2).This follows from Theorem 10.1, because the family B = V : n ∈ Z ispullback absorbing (the right-hand is given by x 7→ λ
2x outside of V ).At the parameter value λ = 1, the global pullback attractor changes its
dimension. Thus, this difference equation provides an example of a nonau-tonomous pitchfork bifurcation, which will be treated below in Section 17.
While these two examples show how (autonomous) bifurcations can beunderstood as attractor bifurcations, the following scenario is intrinsicallynonautonomous (see [44] for a deeper analysis).
Example 16.5 (Shovel bifurcation). Consider a scalar difference equation
xn+1 = an(λ)xn, an(λ) :=
12 + λ, n < 0,
λ, n ≥ 0,(45)
with parameters λ > 0. In order to understand the dynamics of (45), distin-guish three cases:
64 7 Bifurcations
i) λ ∈ (0, 12 ): The equation (45) has an exponential dichotomy on Z withprojector Pn ≡ 1. The uniquely determined bounded entire solution is thetrivial one, which is uniformly asymptotically stable.
ii) λ > 1: The equation (45) has an exponential dichotomy on Z with pro-jector Pk ≡ 0. Again, 0 is the unique bounded entire solution, but is nowunstable.
iii) λ ∈ ( 12 , 1): In this situation, (45) has an exponential dichotomy on Z+
with projector P+n ≡ 1, as well as an exponential dichotomy on Z− with
projector P−n = 0. Thus condition ii) in Theorem 16.2 is violated and 0is a nonhyperbolic solution. For this parameter regime, every solution of(45) is bounded. Moreover, (45) is asymptotically stable, but not uniformlyasymptotically stable on the whole time axis Z.
The parameter values λ ∈ 12 , 1 are critical. In both situations, the num-ber of bounded entire solutions to the linear difference equation (45) changesdrastically. Furthermore, there is a loss of stability in two steps: From uni-formly asymptotically stable to asymptotically stable, and finally to unstable,as λ increases through the values 1
2 and 1. Hence, both values can be con-sidered as bifurcation values, since the number of bounded entire solutionschanges as well as their stability properties.
The next example requires the state space to be at least two-dimensional.
Example 16.6 (Fold solution bifurcation). Consider the planar equation
xn+1 = fn(xn, λ) :=
(bn 00 cn
)xn +
(0
(x1n)2
)− λ
(01
)(46)
with components xn = (x1n, x2n), depending on a parameter λ ∈ R and asymp-
totically constant sequences
bn :=
2, n < 0,12 , n ≥ 0,
cn :=
12 , n < 0,
2, n ≥ 0.(47)
The variational equation for (46) corresponding to the trivial solution andthe parameter λ∗ = 0 reads as
xn+1 = D1fn(0, 0)xn :=
(bn 00 cn
)xn.
It admits an exponential dichotomy on Z+, as well as on Z− with correspond-ing invariant projectors P+
n ≡(1 00 0
)and P−n ≡
(0 00 1
). This yields
R(P+0 ) ∩N(P−0 ) = R
(10
), R(P+
0 ) +N(P−0 ) = R(
10
)and therefore condition ii) of Theorem 16.2 is violated. Hence, the trivialsolution to (46) for λ = 0 is not hyperbolic.
16 Hyperbolicity and simple examples 65
Λ
λ < λ∗
λ = λ∗
λ > λ∗
R2
R2
R2
Λ
λ < λ∗
λ = λ∗
λ > λ∗
R2
R2
R2
Fig. 8 Left (supercritical fold): Initial values η ∈ R2 yielding a bounded solution φλ(·, 0, η)
of (46) for different parameter values λ.Right (cusp): Initial values η ∈ R2 yielding a bounded solution φλ(·, 0, η) of (49) for
different parameter values λ
Let φλ(·, 0, η) be the general solution to (46). Its first component φ1λ is
φ1λ(n, 0, η) = 2−|n|η1 for all n ∈ Z, (48)
while the variation of constants formula (cf. [1, p. 59]) can be used to deducethe asymptotic representation
φ2λ(n, 0, η) =
2n(η2 + 4
7η21 − λ
)+O(1), n→∞,
12n
(η2 − 1
2η21 + 2λ
)+O(1), n→ −∞.
Therefore, the sequence φλ(·, 0, η) is bounded if and only if η2 = − 47η
21 + λ
and η2 = 12η
21 − 2λ holds, i.e., η21 = 7
2λ, η2 = −λ. From the first relation, onesees that there exist two bounded solutions if λ > 0, the trivial solution isthe unique bounded solution for λ = 0 and there are no bounded solutionsfor λ < 0; see Figure 8 (left) for an illustration. For this reason, λ = 0 can beinterpreted as bifurcation value, since the number of bounded entire solutionsincreases from 0 to 2 as λ increases through 0.
The method of explicit solutions can also be applied to the nonlinear equa-tion
xn+1 = fn(xn, λ) :=
(bn 00 cn
)xn +
(0
(x1n)3
)− λ
(01
). (49)
However, using the variation of constants formula (cf. [1, p. 59]), it is possibleto show that the crucial second component of the general solution φλ(·, 0, η)for (49) fulfills
φ2λ(n, 0, η) =
2n(η2 + 8
15η31 − λ
)+O(1), n→∞,
12n
(η2 − 2
15η31 + 2λ
)+O(1), n→ −∞.
66 7 Bifurcations
Since the first component is given in (48), φλ(·, 0, η) is bounded if and onlyif η2 = − 8
15η31 + λ and η2 = 2
15η31 − 2λ, which in turn is equivalent to
η1 =3
√9
2λ, η2 = −7
5λ.
Hence, these particular initial values η ∈ R2 given by the cusp shaped curvedepicted in Figure 8 (right) lead to bounded entire solutions of (49).
17 Attractor bifurcation
An easy example for a bifurcation of a pullback attractor was discussed al-ready in Example 16.4. Now a general bifurcation pattern will be derived,which ensures, under certain conditions on Taylor coefficients, that a pull-back attractor changes qualitatively under variation of the parameter. Thisgeneralizes the autonomous pitchfork bifurcation pattern. Although the pull-back attractor discussed in Example 16.4 is a global attractor, the pitchforkbifurcation only yields results for a local pullback attractor.
Definition 17.1. Consider a process φ on a metric state space X. Aφ-invariant family A = An : n ∈ Z of nonempty compact subsets ofX is called a local pullback attractor if there exists an η > 0 such that
limk→∞
dist(φ(n, n− k,Bη(An−k)), An
)= 0 for all n ∈ Z .
A local pullback attractor is a special case of a pullback attractor w.r.t. acertain basin of attraction, which was introduced in Definition 13.2. Here, thebasin of attraction has to be chosen as a neighborhood of the local pullbackattractor.
Suppose now that (∆λ) is a scalar equation (d = 1) with the trivial solutionfor all parameters λ from an interval Λ ⊆ R. The transition matrix of thecorresponding variational equation
xn+1 = D1fn(0, λ)xn
is denoted by Φλ(n, l) ∈ R.The hyperbolicity condition i) in Theorem 16.2 will be violated when deal-
ing with attractor bifurcations. This yields a nonautonomous counterpart tothe classical pitchfork bifurcation pattern.
17 Attractor bifurcation 67
Theorem 17.1 (Nonautonomous pitchfork bifurcation). Supposethat fn(·, λ) : R→ R is invertible and of class C4 with
D21fn(0, λ) = 0 for all n ∈ Z and λ ∈ Λ.
Suppose there exists a λ∗ ∈ R such that the following hypotheses hold.
• Hypothesis on linear part : There exists a K ≥ 1 and functionsβ1, β2 : Λ → (0,∞) which are either both increasing or decreasingwith limλ→λ∗ b1(λ) = limλ→λ∗ b2(λ) = 1 and
Φλ(n, l) ≤ Kβ1(λ)n−l for all l ≤ n,Φλ(n, l) ≤ Kβ2(λ)n−l for all n ≤ l
and all λ ∈ Λ.• Hypothesis on nonlinearity : Assume that if the functions β1 and β2
are increasing, then
−∞ < lim infλ→λ∗
infn∈Z
D31fn(0, λ) ≤ lim sup
λ→λ∗supn∈Z
D31fn(0, λ) < 0,
and otherwise (i.e., if the functions β1 and β2 are decreasing), then
0 < lim infλ→λ∗
infn∈Z
D31fn(0, λ) ≤ lim sup
λ→λ∗supn∈Z
D31fn(0, λ) <∞.
In addition, suppose that the remainder satisfies
limx→0
supλ∈(λ∗−x2,λ∗+x2)
supn≤0
x
∫ 1
0
(1− t)3D4fn(tx, λ) dt = 0,
lim supλ→λ∗
lim supx→0
supn≤0
Kx3
1−minβ1(λ), β2(λ)−1
∫ 1
0
(1− t)3D4fn(tx, λ) dt < 3.
Then there exist λ− < λ∗ < λ+ so that the following statements hold:
1) If the functions β1, β2 are increasing, the trivial solution is a localpullback attractor for λ ∈ (λ−, λ
∗), which bifurcates to a nontriviallocal pullback attractor Aλn : n ∈ Z, λ ∈ (λ∗, λ+), satisfying thelimit
limλ→λ∗
supn≤0
dist(Aλn, 0) = 0.
2) If the functions β1, β2 are decreasing, the trivial solution is a localpullback attractor for λ ∈ (λ∗, λ+), which bifurcates to a nontriviallocal pullback attractor Aλn : n ∈ Z, λ ∈ (λ−, λ
∗), satisfying thelimit
68 7 Bifurcations
limλ→λ∗
supn≤0
dist(Aλn, 0) = 0.
For a proof of this theorem and extensions (to both different time domainsand repellers), see [51, 53].
The next example, taken from [19], illustrates the above theorem.
Example 17.1. Consider the nonautonomous difference equation
xn+1 =λxn
1 + bnqλ xqn
, (50)
where q ∈ N and the sequence (bn)n∈N is positive and both bounded andbounded away from zero. For q = 1, this difference equation can be trans-formed into the well-known Beverton–Holt equation, which describes the den-sity of a population in a fluctuating environment. It was shown in [19] that inthis case, the system admits a nonautonomous transcritical bifurcation (thebifurcation pattern of which was derived in [51]).
For q = 2, a nonautonomous pitchfork bifurcation occurs. The above the-orem can be applied, because the Taylor expansion of the right-hand side of(50) reads as λxn + bnx
q+1n + O(x2q+1), and the remainder fulfills the con-
ditions of the theorem (see [19] for details). This means that for λ ∈ (0, 1),the trivial solution is a local pullback attractor, which undergoes a transitionto a nontrivial local pullback attractor when λ > 1. Note that the extremesolutions of the nontrivial local pullback attractor for λ > 1 are also localpullback attractors, which gives the interpretation of this bifurcation as abifurcation of locally pullback attractive solutions.
18 Solution bifurcation
In the previous section on attractor bifurcations, the first hyperbolicity condi-tion i) in Theorem 16.2, given by exponential dichotomies on both semiaxes,was violated.
The present concept of solution bifurcation is based on the assumption thatmerely condition ii) of Theorem 16.2 does not hold. This requires the vari-ational difference equation (Vλ) to be intrinsically nonautonomous. Indeed,if (Vλ) is almost periodic, then an exponential dichotomy on a semiaxis ex-tends to the whole integer axis (cf. [61, Theorem 2]) and the reference solutionχ = (χn)n∈Z becomes hyperbolic. For this reason the following bifurcationscenarios cannot occur for periodic or autonomous difference equations.
The crucial and standing assumption is the following:
18 Solution bifurcation 69
Hypothesis: The variational equation (Vλ) admits an ED both on Z+ (withprojector P+
n ) and on Z− (with projector P−n ) such that there exists nonzerovectors ξ1 ∈ Rd, ξ′1 ∈ Rd satisfying
R(P+0 ) ∩N(P−0 ) = Rξ1, (R(P+
0 ) +N(P−0 ))⊥ = Rξ′1. (51)
Then a solution bifurcation is understood as follows: Suppose that for afixed parameter λ∗ ∈ Λ, the difference equation (∆λ∗) has an entire boundedreference solution χ∗ = χ(λ∗). One says that (∆λ) undergoes a bifurcation atλ = λ∗ along χ∗, or χ∗ bifurcates at λ∗, if there exists a convergent parametersequence (λn)n∈N in Λ with limit λ∗ so that (∆λn) has two distinct entiresolutions χ1
λn, χ2
λn∈ `∞ both satisfying
limn→∞
χ1λn = lim
n→∞χ2λn = χ∗.
The above Hypothesis allows a geometrical insight into the following abstractbifurcation results using invariant fiber bundles , i.e., nonautonomous coun-terparts to invariant manifolds: Because (Vλ) has an exponential dichotomyon Z+, there exists a stable fiber bundle χ∗ +W+
λ consisting of all solutionsto (∆λ) approaching χ∗ in forward time. Here,W+
λ is locally a graph over thestable vector bundle R(P+
n ) : n ∈ Z+. Analogously, the dichotomy on Z−guarantees an unstable fiber bundle χ∗+W−λ consisting of solutions decayingto χ∗ in backward time (cf. [47, Corollary 2.23]). Then the bounded entiresolutions to (∆λ) are contained in the intersection (χ∗ +W+
λ ) ∩ (χ∗ +W−λ ).In particular, the intersection of the fibers
Sλ :=(χ∗0 +W+
λ,0
)∩(χ∗0 +W−λ,0
)⊆ Rd
yields initial values for bounded entire solutions (see Figure 9).It can be assumed without loss of generality, using the equation of per-
turbed motion, that χ∗ = 0. In addition suppose that
fn(0, λ) ≡ 0 on Z,
which means that (∆λ) has the trivial solution for all λ ∈ Λ. The correspond-ing variational equation is
xn+1 = D1fn(0, λ)xn
with transition matrix Φλ(n, l) ∈ Rd×d.
Theorem 18.1 (Bifurcation from known solutions). Let Λ ⊆ Rand suppose fn is of class Cm, m ≥ 2. If the transversality condition
70 7 Bifurcations
φ∗ +W+λ
φ∗ +W−λ
φ1
φ2Rd
ZSλ
k = 0
Fig. 9 Intersection Sλ ⊆ Rd of the stable fiber bundle φ∗ +W+λ ⊆ Z+ × Rd with the
unstable fiber bundle φ∗ + W−λ ⊆ Z− × Rd at time k = 0 yields two bounded entire
solutions φ1, φ2 to (∆λ) indicated as dotted dashed lines
is satisfied, then the trivial solution of a difference equation (∆λ)bifurcates at λ∗. In particular, there exists a ρ > 0, open convexneighborhoods U ⊆ `∞(Ω) of 0, Λ0 ⊆ Λ of λ∗ and Cm−1-functionsψ : (−ρ, ρ)→ U , λ : (−ρ, ρ)→ Λ0 with
1) ψ(0) = 0, λ(0) = λ∗ and ψ(0) = Φλ∗(·, 0)ξ1,2) each ψ(s) is a nontrivial solution of (∆)λ(s) homoclinic to 0, i.e.,
limn→±∞
ψ(s)n = 0.
Proof. See [46, Theorem 2.14]. ut
Corollary 18.1 (Transcritical bifurcation). Under the additionalassumption
g20 :=∑n∈Z〈Φλ∗(0, n+ 1)′ξ′1, D
21fn(0, λ∗)[Φλ∗(n, 0)ξ1]2〉 6= 0
one has λ(0) = − g202g11
and the following holds locally in U × Λ0: The
difference equation (∆λ) has a unique nontrivial entire bounded solutionψ(λ) for λ 6= λ∗ and 0 is the unique entire bounded solution of (∆)λ∗ ;moreover, ψ(λ) is homoclinic to 0.
18 Solution bifurcation 71
Λ
λ < λ∗
λ = λ∗
λ > λ∗
R2
R2
R2
Λ
λ < λ∗
λ = λ∗
λ > λ∗
R2
R2
R2
Fig. 10 Left (transcritical): Initial values η ∈ R2 yielding a homoclinic solution φλ(·, 0, η)
of (53) for different parameter values λ.Right (supercritical pitchfork): Initial values η ∈ R2 yielding a homoclinic solution
φλ(·, 0, η) of (54) for different parameter values λ
Proof. See [46, Corollary 2.16]. ut
Example 18.1. Consider the nonlinear difference equation
xn+1 = fn(xn, λ) :=
(bn 0λ cn
)xn +
(0
(x1n)2
)(53)
depending on a bifurcation parameter λ ∈ R and sequences bn, cn defined in(47). As in Example 16.6, the assumptions hold with λ∗ = 0 and
g11 =4
36= 0, g20 =
12
76= 0.
Hence, Corollary 18.1 can be applied in order to see that the trivial solutionof (53) has a transcritical bifurcation at λ = 0. Again, this bifurcation will bedescribed quantitatively. While the first component of the general solutionφλ(·, 0, η) given by (48) is homoclinic, the second component satisfies
φ2λ(n, 0, η) =
2n(η2 + 4
7η21 + 2λ
3 η1)
+ o(1), n→∞,
2−n(η2 − 2
7η21 − 2λ
3 η1)
+ o(1), n→ −∞.
In conclusion, one sees that φλ(·, 0, η) is bounded if and only if η = (0, 0) or
η1 = −14
9λ, η2 =
28
81λ2.
Hence, besides the zero solution, there is a unique nontrivial entire solutionpassing through the initial point η = (η1, η2) at time n = 0 for λ 6= 0. Thismeans the solution bifurcation pattern sketched in Figure 10 (left) holds.
72 7 Bifurcations
Corollary 18.2 (Pitchfork bifurcation). For m ≥ 3 and under theadditional assumptions∑
n∈Z〈Φλ∗(0, n+ 1)′ξ′1, D
21fn(0, λ∗)[Φλ∗(n, 0)ξ1]2〉 = 0,
g30 :=∑n∈Z〈Φλ∗(0, n+ 1)′ξ′1, D
31fn(0, λ∗)[Φλ∗(n, 0)ξ1]3〉 6= 0
one has λ(0) = 0, λ(0) = − g303g11
and the following holds locally in U×Λ0:
3) Subcritical case : If g30/g11 > 0, then the unique entire bounded solu-tion of (∆λ) is the trivial one for λ ≥ λ∗ and (∆λ) has exactly twonontrivial entire solutions for λ < λ∗; both are homoclinic to 0.
4) Supercritical case : If g30/g11 < 0, then the unique entire boundedsolution of (∆λ) is the trivial one for λ ≤ λ∗ and (∆λ) has exactlytwo nontrivial entire solutions for λ > λ∗; both are homoclinic to 0.
Proof. See [46, Corollary 2.16]. ut
Example 18.2. Let δ be a fixed nonzero real number and consider the nonlin-ear difference equation
xn+1 = fn(xn, λ) :=
(bn 0λ cn
)xn + δ
(0
(x1n)3
)(54)
depending on a bifurcation parameter λ ∈ R and the bn, cn defined in (47).As in our above Example 18.1, the assumptions of Corollary 18.2 are fulfilledwith λ∗ = 0. The transversality condition here reads g11 = 4
3 6= 0. Moreover,D2
1fn(0, 0) ≡ 0 on Z implies g20 = 0, whereas the relation D31fn(0, 0)ζ3 =( 0
6δζ31
)for all n ∈ Z, ζ ∈ R2 leads to g30 = 4δ 6= 0. This gives the crucial
quotient g30g11
= 3δ. By Corollary 18.2, the trivial solution to (54) undergoes a
subcritical (supercritical) pitchfork bifurcation at λ = 0 provided δ > 0 (resp.δ < 0). As before one can illustrate this result using the general solutionφλ(·, 0, η) to (54). The first component is given by (48) and helps to show forthe second component that
φ2λ(n, 0, η) =
2n(η2 + 8δ
15η31 + 2λ
3 η1)
+ o(1), n→∞,
2−n(η2 − 2δ
15η31 − 4λ
3 η1)
+ o(1), n→ −∞.
This asymptotic representation shows that φλ(·, 0, η) is homoclinic to 0 if
and only if η = 0 or η21 = − 2δλ and η2 = 4
15(5δ+16λ)
δ2 λ2. Hence, there is a
18 Solution bifurcation 73
correspondence to the pitchfork solution bifurcation from in Corollary 18.2.See Figure 10 (right) for an illustration.
Remarks
In [51, Theorem 5.1] one finds a nonautonomous generalization for transcrit-ical bifurcations.
Chapter 8
Random dynamical systems
Random dynamical systems on a state space X are nonautonomous by thevery nature of the driving noise. They can be formulated as skew-productsystems with the driving system acting a probability sample space Ω ratherthan on a topological or metric parameter space P . A major difference is thatonly measurability and not continuity w.r.t. the parameter can be assumed,which changes the types of results that can be proved. In particular, the skew-product system does not form an autonomous semidynamical system on theproduct space Ω×X. Nevertheless, there are many interesting parallels withthe theory of deterministic nonautonomous dynamical systems.
For further details see Arnold [2] and, for example, also [29], where thetemporal discretization of random differential equations is also considered.
19 Random difference equations
Let (Ω,F ,P) be a probability space and let ξn, n ∈ Z be a discrete-timestochastic process taking values in some space Ξ, i.e., a sequence of randomvariables or, equivalently, F-measurable mappings ξn : Ω → Ξ for n ∈ Z. Let(X, d) be a complete metric space and consider a mapping g : Ξ ×X → X.
Thenxn+1(ω) = g (ξn(ω), xn(ω)) for all n ∈ Z, ω ∈ Ω, (55)
is a random difference equation on X driven by the stochastic process ξn.Greater generality can be achieved by representing the driving noise pro-
cess by a metrical (i.e., measure theoretic) dynamical system θ on some canon-ical sample space Ω, i.e., the group of F-measurable mappings θn, n ∈ Zunder composition formed by iterating a measurable mapping θ : Ω → Ωand its measurable inverse mapping θ−1 : Ω → Ω, i.e., with θ0 = idΩ and
θn+1 := θ θn, θ−n−1 := θ−1 θ−n for all n ∈ N,
75
76 8 Random dynamical systems
where θ−1 := θ−1. It is usually assumed that θ generates an ergodic processon Ω.
Let f : Ω×X → X be an F ×B(X)-measurable mapping, where B(X) isthe Borel σ-algebra on X. Then, in this context, a random difference equationhas the form
xn+1(ω) = f (θn(ω), xn(ω)) for all n ∈ Z, ω ∈ Ω. (56)
Define recursively a solution mapping ϕ : Z+×Ω× X → X for the randomdifference equation (56) by ϕ(ω, 0, x) := x and
ϕ(ω, n+ 1, x) = f(θn(ω), φ(θn(ω), n, x)) for all n ∈ N, x ∈ X
and ω ∈ Ω. Then, ϕ satisfies the discrete-time cocycle property w.r.t. θ, i.e.,
ϕ(n+m,ω, x) = ϕ (n, θm(ω), ϕ(m,ω, x0)) for all m,n ∈ Z+,
x ∈ X and ω ∈ Ω. The mapping ϕ is called a cocycle mapping.In terms of Arnold [2], the random difference equation (56) generates a
discrete-time random dynamical system (θ, φ) on Ω × X with the metricdynamical system θ on the probability space (Ω,F ,P) and the cocycle map-ping ϕ on the state space X.
Definition 19.1. A (discrete-time) random dynamical system (θ, ϕ)on Ω ×X consists of a metrical dynamical system θ on Ω, i.e., a groupof measure preserving mappings θn : Ω → Ω, n ∈ Z, such that
i) θ0 = idΩ and θn θm = θn+m for all n, m ∈ Z,ii) the map ω 7→ θn(ω) is measurable and invariant w.r.t. P in the sense
that θn(P) = P for each n ∈ Z,
and a cocycle mapping ϕ : Z+ ×Ω ×X → X such that
a) ϕ(0, ω, x0) = x0 for all x0 ∈ X and ω ∈ Ω,b) ϕ(n+m,ω, x0) = ϕ (n, θm(ω), ϕ(m,ω, x0)) for all n,m ∈ Z+, x0 ∈ X
and ω ∈ Ω,c) x0 7→ ϕ(n, ω, x0) is continuous for each (n, ω) ∈ Z+ ×Ω,d) ω 7→ ϕ(n, ω, x0) is F-measurable for all (n, x0) ∈ Z+ ×X.
The notation θn(P) = P for the measure preserving property of θn w.r.t.P is just a compact way of writing
P(θn(A)) = P(A) for all n ∈ Z, A ∈ F .
20 Random attractors 77
A systematic treatment of the random dynamical system theory, bothcontinuous and discrete time, is propounded in Arnold [2]. Note that π =(θ, φ) has a skew-product structure on Ω ×X, but it is not an autonomoussemidynamical system on Ω × X since no topological structure is assumedon Ω.
20 Random attractors
Unlike a deterministic skew-product system, a random dynamical system(θ, ϕ) on Ω×X is not an autonomous semidynamical system on Ω×X. Nev-ertheless, skew-product deterministic systems and random dynamical systemshave many analogous properties, and concepts and results for one can oftenbe used with appropriate modifications for the other. The most significantmodification concerns measurability and the nonautonomous sets under con-sideration are random sets.
Let (X, d) be a complete and separable metric space (i.e., a Polish space)
Definition 20.1. A family D = Dω, ω ∈ Ω of nonempty subsets of Xis called a random set if the mapping ω 7→ dist(x,Dω) is F-measurablefor all x ∈ X. A random set D is called a random closed set if Dω
is closed for each ω ∈ Ω and is called a random compact set if Dω iscompact for each ω ∈ Ω.
Random sets are called tempered if their growth w.r.t. the driving systemθ is sub-exponential (cf. Definition 15.1).
Definition 20.2. A random set D = Dω, ω ∈ Ω in X is said to betempered if there exists a x0 ∈ X such that
Dω ⊂ x ∈ X : d(x, x0) ≤ r(ω) for all ω ∈ Ω ,
where the random variable r(ω) > 0 is tempered, i.e.,
supn∈Zr(θn(ω))e−γ|n| <∞ for all ω ∈ Ω, γ > 0.
The collection of all tempered random sets in X will be denoted by D.
78 8 Random dynamical systems
A random attractor of a random dynamical system is a random set whichis a pullback attractor in the pathwise sense w.r.t. the attracting basin oftempered random sets.
Definition 20.3. A random compact setA = (Aω)ω∈Ω from D is calleda random attractor of a random dynamical system (θ, ϕ) on Ω ×X inD if A is a ϕ-invariant set, i.e.,
ϕ(n, ω,Aω) = Aθn(ω) for all n ∈ Z+, ω ∈ Ω ,
and pathwise pullback attracting in D, i.e.,
limn→∞
dist(ϕ(n, θ−n(ω), D(θ−n(ω))
), Aω
)= 0 for all ω ∈ Ω, D ∈ D.
If the random attractor consists of singleton sets, i.e., Aω = Z∗(ω) forsome random variable Z∗ with Z∗(ω) ∈ X, then Zn(ω) := Z∗(θn(ω)) is astationary stochastic process on X.
The existence of a random attractor is ensured by that of a pullback ab-sorbing set. The tempered random set B = Bω, ω ∈ Ω in the followingtheorem is called a pullback absorbing random set.
Theorem 20.1 (Existence of random attractors). Let (θ, ϕ) be arandom dynamical system on Ω × X such that ϕ(n, ω, ·) : X → X isa compact operator for each fixed n > 0 and ω ∈ Ω. If there exista tempered random set B = Bω, ω ∈ Ω with closed and boundedcomponent sets and an ND,ω ≥ 0 such that
ϕ(n, θ−n(ω), D(θ−n(ω))
)⊂ Bω for all n ≥ ND,ω, (57)
and every tempered random set D = Dω, ω ∈ Ω, then the ran-dom dynamical system (θ, ϕ) possesses a random pullback attractorA = Aω : ω ∈ Ω with component sets defined by
Aω =⋂m>0
⋃n≥m
ϕ(n, θ−n(ω), B(θ−n(ω)) for all ω ∈ Ω. (58)
The proof of Theorem 20.1 is essentially the same as its counterparts fordeterministic skew-product systems. The only new feature is that of measur-ability, i.e., to show that A = Aω), ω ∈ Ω is a random set. This follows
21 Random Markov chains 79
from the fact that the set-valued mappings ω 7→ ϕ(n, θ−n(ω), B(θ−n(ω))
)are
measurable for each n ∈ Z+.Arnold & Schmalfuß [3] showed that a random attractor is also a forward
attractor in the weaker sense of convergence in probability, i.e.,
limn→∞
∫Ω
dist(ϕ(n, ω,Dω), Aθn(ω)
)P(dω) = 0
for all D ∈ D. This allows individual sample paths to have large deviationsfrom the attractor, but for all to converge in this probabilistic sense.
21 Random Markov chains
Discrete-time finite state Markov chains with a tridiagonal structure are com-mon in biological applications. They have a transition matrix [IN +∆Q],where IN is the N × N identity matrix and Q is the tridiagonal N × N -matrix
where the qj are positive constants.Such a Markov chain is a first order linear difference equation
p(n+1) = [IN +∆Q] p(n) (60)
on the probability simplex ΣN in RN defined by
ΣN =p = (p1, · · · , pN )T :
N∑j=1
pj = 1, p1, . . . , pN ∈ [0, 1].
The Perron-Frobenius theorem applies to the matrix L∆ := IN+∆Q when∆ > 0 is chosen sufficiently small. In particular, it has eigenvalue λ = 1 andthere is a positive eigenvector x, which can be normalized (in the ‖ ·‖1 norm)to give a probability vector p, i.e., [IN +∆Q]p = p, so Qp = 0. Specifically,the probability vector
p1 =1
‖x‖1, pj+1 =
1
‖x‖1
j∏i=1
q2i−1q2i
for all j = 1, . . . , N − 1,
80 8 Random dynamical systems
where
‖x‖1 =
N∑j=1
xj = 1 +
N−1∑j=1
j∏i=1
q2i−1q2i
.
The following result is well known.
Theorem 21.1. The probability eigenvector p is an asymptotically sta-ble steady state of the difference equation (60) on the simplex ΣN .
In a random environment, e.g., with randomly varying food supply, thetransition probabilities may be random, i.e., the band entries qi of the matricQ may depend on the sample space parameter ω ∈ Ω. Thus, qi = qi(ω) fori = 1, 2, . . ., 2N − 2, and these may vary in turn according to some metricdynamical system θ on the probability space (Ω,F ,P). The following basicassumption will be used.
Assumption 1 There exist numbers 0 < α ≤ β < ∞ such that the uniformestimates hold
α ≤ qi(ω) ≤ β for all ω ∈ Ω, i = 1, 2, . . . , 2N − 2. (61)
Let L be a set of linear operators Lω : RN → RN parametrized by theparameter ω taking values in some set Ω and let θn, n ∈ Z be a groupof maps of Ω onto itself. The maps Lωx serve as the generator of a linearcocycle FL(n, ω). Then (θ, FL) is a random dynamical system on Ω ×ΣN .
Theorem 21.2. Let FL(n, ω)x be the linear cocycle
FL(n, ω)x = Lθn−1ω · · ·Lθ1ωLθ0ωx.
with matrices Lω := IN +∆Q(ω), where the tridiagonal matrices Q(ω)are of the form (59) with the entries qi = qi(ω) satisfying the uniformestimates (61) in Assumption 1. In addition, suppose that 0 < ∆ < 1
2β .
Then, the simplex ΣN is positively invariant under FL(n, ω), i.e.,
FL(n, ω)ΣN ⊆ ΣN for all ω ∈ Ω.
Moreover, for n large enough, the restriction of FL(n, ω)x to the set ΣNis a uniformly dissipative and uniformly contractive cocycle (w.r.t. theHilbert metric), which has a random attractor A = Aω, ω ∈ Ω suchthat each set Aω, ω ∈ Ω, consists of a single point.
22 Approximating invariant measures 81
The proof can be found in [30]. It involves positive matrices and the Hilbertprojective metric on positive cones in RN .
Henceforth write Aω = aω for the singleton component subsets of therandom attractor A. Then the random attractor is an entire random sequence
aθnω, n ∈ Z in ΣN (γ) ⊂ΣN , where
ΣN (γ) =
x = (x1, x2, . . . , xN ) :
N∑i=1
xi = 1, x1, x2, . . . , xN ≥ γN−1.
with γ := min∆α, 1 − 2∆β > 0. It attracts other iterates of the randomMarkov chain in the pullback sense. Pullback convergence involves startingat earlier initial times with a fixed end time. It is, generally, not the sameas forward convergence in the sense usually understood in dynamical sys-tems, but in this case it is the same due to the uniform boundedness of thecontractive rate w.r.t. ω.
Corollary 21.1. For any norm ‖ · ‖ on RN , p(0) ∈ ΣN and ω ∈ Ω∥∥∥p(n)(ω)− aθnω∥∥∥→ 0 as n→∞.
The random attractor is, in fact, asymptotically Lyapunov stable in theconventional forward sense.
22 Approximating invariant measures
Consider now a compact metric space (X, d). A random difference equation(56) on X driven by the noise process θ generates a random dynamical system(θ, ϕ). It can be reformulated as a difference equation with a triangular orskew-product structure
(ω, x) 7→ F (ω, x) :=
(θ(ω)f(ω, x)
)An invariant measure µ of F = (θ, ϕ) on Ω ×X defined by µ = F ∗µ (whichis shorthand for an integral expression) can be decomposed as
µ(ω,B) = µω(B)P(dω) for all B ∈ B(X),
where the measures µω on X are θ-invariant w.r.t. f , i.e.,
82 8 Random dynamical systems
µθ(ω)(B) = µω(f−1(ω,B)
)for all B ∈ B(X), ω ∈ Ω.
This decomposition is very important since only the state space X, but notthe sample space Ω, can be discretized.
To compute a given invariant measure µ consider a sequence of finite sub-sets XN of X given by
XN = x(N)1 , · · · , x(N)
N ⊂ X,
for N ∈ N with maximal step size
hN = supx∈X
dist(x,XN )
such that hN → 0 as N →∞.Then the invariant µ will be approximated by a sequence of invariant
stochastic vectors associated with random Markov chains describing transi-tions between the states of the discretized state spaces XN . These involverandom N ×N matrices, i.e., measurable mappings
PN : Ω → SN ,
where SN denotes the set of N ×N (nonrandom) stochastic matrices, satis-fying the property
PnN (θm(ω))PmN (ω) = Pm+nN (ω) for all m,n ∈ Z+. (62)
Recall that a stochastic matrix has non-negative entries with the columnssumming to 1.
Consider a random Markov chain PN (ω), ω ∈ Ω and a random proba-bility vector pN (ω), ω ∈ Ω on the deterministic grid XN . Then
pN,n+1(θn+1(ω)) = pN,n(θn(ω))PN (θn(ω))
and an equilibrium probability vector is defined by
pN (θ(ω)) = pN (ω)PN (ω) for all ω ∈ Ω.
It can be represented trivially as a random measure µN,ω on X.The distance between random probability measures will be given with the
Prokhorov metric ρ and the distance of a random Markov chain P : Ω → SNand the generating mapping f of the random dynamical system is defined by
D(P (ω), f) =
N∑i,j=1
(pi,j(ω) distX×X((x
(N)i , x
(N)j ),Grf(ω, ·)
), (63)
where the distance to the random graph is given by
22 Approximating invariant measures 83
distX×X((x, y),Grf(ω, ·)) = infz∈X
maxd(x, z), d(y, f(ω, z)) for all x, y ∈ X.
The following necessary and sufficient result holds if θ-semi-invariantrather than θ-invariant families of decomposed probability measures are used.
Definition 22.1. A family of probability measures µω on X is calledθ-semi-invariant w.r.t. f , if
µθ(ω)(B) ≤ µω(f−1(ω,B)
)for all B ∈ B(X), ω ∈ Ω.
Such θ-semi-invariant families are, in fact, θ-invariant when the mappingsx 7→ f(ω, x) are continuous.
Theorem 22.1. A random probability measure µω, ω ∈ Ω is θ-semi-invariant w.r.t. f on X if and only if it is randomly stochasticallyapproachable , i.e., for each N there exist
i) a grid XN with fineness hN → 0 as N →∞ii) a random Markov chain PN (ω), ω ∈ Ω on XN
iii) random probability measure µN,ω, ω ∈ Ω on X correspondingto a random equilibrium probability vector pN (ω), ω ∈ Ω ofPN (ω), ω ∈ Ω on XN
with the expected convergences
ED (PN (ω), f(ω, ·))→ 0, Eρ (µN,ω, µω)→ 0 as n→∞.
Proof. See Imkeller & Kloeden [20]. ut
The double terminology “random stochastic” seems to be an overkill, butjust think of a Markov chain for which the transition probabilities are notfixed, but can vary randomly in time.
84 8 Random dynamical systems
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