An Introduction to the Qualitative Theory of …...Qualitative Theory of Nonautonomous Dynamical...

An Introduction to theQualitative Theory

of Nonautonomous Dynamical Systems

Martin Rasmussen

Imperial College London

9th Meeting of the

European Study Group on Cardiovascular Oscillations

Lancaster, UK, 10–14 April 2016

11 April 2016

G.D. Birkhoff (1884–1944) Dynamical Systems (1927)

Dynamical systemsas abstractions and generalisations of

ordinary differential equations.

G.D. Birkhoff (1884–1944) Dynamical Systems (1927)

Dynamical systemsas abstractions and generalisations of

ordinary differential equations.

Birkhoff was strongly influenced by the two fathers of theQualitative Theory of Dynamical Systems

A.M. Lyapunov (1857–1918) H. Poincare (1854–1912)

Theory of ordinary differential equations before the era ofLyapunov/Poincare/Birkhoff:

Concentration on finding analytical expressions forsolutions of specific differential equations

Explicit expressions of solutions are unavailablefor the most interesting differential equations!

Numerical integration

Approximate solutions vianumerical schemes such as Euler

and Runge–Kutta method.

Qualitative theory

Study of qualitative propertieswithout having the knowledge of

solutions







Qualitative theory


solutions







Qualitative theory


solutions







Qualitative theory


solutions

Differential Equations Random Systems Control Systems

Nonautonomous Dynamical Systems

Natural Sciences Medicine Financial Markets

Basic theoryAttractor theory

Linear theoryNonlinear theory

Bifurcation theory

Contents

1 Basic theory

2 Attractor theory

3 Linear theory

4 Nonlinear theory

5 Bifurcation theory

Martin Rasmussen Qualitative Theory of Nonautonomous Dynamical Systems



Bifurcation theory

Process formulationTranslation invarianceInvarianceSkew product flow formulation

Nonautonomous differential equations

An initial value problem is the combination of

• a nonautonomous ordinary differential equation (ODE)

x :=dx

dt= f (t, x) ,

where f : R× Rd → Rd is assumed to be continuous, and

• an initial condition x(t0) = x0 for some fixed t0 ∈ R and x0 ∈ Rd .

Process formulation

Assuming global existence and uniqueness of solutions, the solutions forma mapping

(t, t0, x0) 7→ ϕ(t, t0, x0) ∈ Rd for all (t, t0, x0) ∈ R× R× Rd .

This mapping is called the process induced by the above ODE.




Bifurcation theory


The initial value and cocycle property

The initial value and cocycle property

A process is a continuous mapping (t, t0, x0) 7→ ϕ(t, t0, x0) ∈ Rd whichsatisfies

initial value condition. ϕ(t0, t0, x0) = x0,

cocycle property. ϕ(t2, t0, x0) = ϕ(t2, t1, ϕ(t1, t0, x0)

).

t0t1 t2

x0

ϕ(t1,t0,x0)

ϕ(t2,t0,x0)

=ϕ(t2,t1,ϕ(t1,t0,x0))




Bifurcation theory


The autonomous case

The situation is much simpler in the autonomous case, where x = f (x).

Translation invariance

The process ϕ(t, t0, x0) induced by an autonomous differential equationdoes not depend separately on

initial time t0 and actual time t,

but only on the elapsed time t − t0.

This induces a flow φ(t − t0, x0) := ϕ(t, t0, x0), fulfilling

• initial value condition: φ(0, x0) = x0,

• group property: φ(t + s, x0) = φ(t, φ(s, x0)).




Bifurcation theory


The autonomous case

The situation is much simpler in the autonomous case, where x = f (x).

Translation invariance

The process ϕ(t, t0, x0) induced by an autonomous differential equationdoes not depend separately on

initial time t0 and actual time t,

but only on the elapsed time t − t0.

This induces a flow φ(t − t0, x0) := ϕ(t, t0, x0), fulfilling

• initial value condition: φ(0, x0) = x0,

• group property: φ(t + s, x0) = φ(t, φ(s, x0)).




Bifurcation theory


Flow versus process

Autonomous ODEx = f (x)

Nonautonomous ODEx = f (t, x)

Flow φ(t, x0)

depends on one parameter(initial state x0)

Process ϕ(t, t0, x0)

depends on two parameters(initial time t0 and initial state x0)

All interesting objects aresubsets of thephase space(x0-space)

All interesting objects aresubsets of the

extended phase space((t0, x0)-space)

We call an arbitrary subset of theextended phase space a

nonautonomous set




Bifurcation theory


Invariant sets

The long-term behaviour of a flow is characterised by its invariant sets.

Definition (Invariance)

A subset M ⊂ Rd is called invariant w.r.t. the flow φ : R× Rd → Rd if

φ(t,M) = M for all t ∈ R .

Examples are given by

equilibria periodic orbits or fractals.




Bifurcation theory


Invariant nonautonomous sets

For a nonautonomous set M ⊂ R× Rd and some time t ∈ R, the t-fiberof M is given by

M(t) :={x ∈ Rd : (t, x) ∈ M

}.

Definition (Invariant nonautonomous set)

A nonautonomous set M ⊂ R× Rd for a process ϕ : R× R× Rd → Rd

is called invariant if

M(t) = ϕ(t, t0,M(t0)

)for all t, t0 ∈ R .

R

Rd

t

M(t)

M




Bifurcation theory


A simple example

Consider the nonautonomous differential equation

x = 2tx

which induces the process

φ(t, t0, x0) = x0et2−t20 .

Then the set

M :={

(t, x) : −et2

≤ x ≤ et2}

is an invariant nonautonomous set. t

x




Bifurcation theory


A simple example


x = 2tx


φ(t, t0, x0) = x0et2−t20 .

Then the set

M :={

(t, x) : −et2

≤ x ≤ et2}

is an invariant nonautonomous set.

M

t

x

It is exactly given by all solution curves with initial values x0 ∈ [−1, 1]starting at time t0 = 0.




Bifurcation theory


An interesting observation

If ϕ : R× R× Rd → Rd is the process generated by the nonautonomousdifferential equation

x = f (t, x) ,

then the mapping(t, t0, x0) 7→ (t, ϕ(t, t0, x0))

is a flow!

A process is also a flow

This flow is induced by the (d + 1)-dimensional autonomous differentialequation

t = 1 ,

x = f (t, x) .

Why do we need a nonautonomous theory?




Bifurcation theory


An interesting observation

If ϕ : R× R× Rd → Rd is the process generated by the nonautonomousdifferential equation

x = f (t, x) ,

then the mapping(t, t0, x0) 7→ (t, ϕ(t, t0, x0))

is a flow!

A process is also a flow

This flow is induced by the (d + 1)-dimensional autonomous differentialequation

t = 1 ,

x = f (t, x) .

Why do we need a nonautonomous theory?




Bifurcation theory


Skew product flows

But we can learn something from the example

t = 1 , x = f (t, x) .

nonautonomous influence

t-system on Rx-system on Rd

The drawback: All solutions of the t-system are unbounded!




Bifurcation theory


Skew product flows

But we can learn something from the example

t = 1 , x = f (t, x) .


t-system on Rx-system on Rd

The drawback: All solutions of the t-system are unbounded!




Bifurcation theory


Skew product flows


flow θ on a compact set P

x-system on Rd

Definition (Skew product flow)

A skew product flow (θ, ϕ) consists of a flow θ : R× P → P on a baseset P and a cocycle ϕ : R× P × Rd → Rd over θ, i.e., one has

initial value condition: ϕ(0, p, x) = x ,

cocycle property: ϕ(t + s, p, x) = ϕ(t, θ(s, p), ϕ(s, p, x)

).




Bifurcation theory


A short summary

Nonautonomous differential equations give rise to processes, whichdepend on two parameters (initial time t0 and initial value x0) incontrast to flows (which only depend on the initial value x0).

The interesting sets in the nonautonomous case are nonautonomoussets, which are subsets of the extended phase space.

An invariant nonautonomous set can be seen as union of solutioncurves.




Bifurcation theory

Autonomous attractorsNonautonomous attractorsForward attractorsPullback attractorsExistence of attractors

Attractor theory




Bifurcation theory


The autonomous case

An attractor describes attraction in forward time.

Definition (Attractor)

Let φ : R× Rd → Rd be a flow. A compactinvariant set A is called global attractor if

dist(φ(t,B),A)→ 0 as t →∞

for all bounded sets B ⊂ Rd .




Bifurcation theory


The nonautonomous case

Let ϕ : R× R× Rd → Rd be a process, and let A ⊂ R× Rd be aninvariant nonautonomous set with compact fibers. There are essentiallytwo possibilities of attraction in the nonautonomous case.

Forward and pullback attraction

For all bounded sets B ⊂ Rd , one has

dist(ϕ(t, t0,B),A(t)

)→ 0

as t →∞ with t0 fixed

(forward attraction)as t0 → −∞ with t fixed

(pullback attraction).

Definition (Forward and pullback attractor)

In case of forward attraction, the nonautonomous set A is called forwardattractor; and it is called pullback attractor in case of pullback attraction.




Bifurcation theory


The nonautonomous case

Let ϕ : R× R× Rd → Rd be a process, and let A ⊂ R× Rd be aninvariant nonautonomous set with compact fibers. There are essentiallytwo possibilities of attraction in the nonautonomous case.

Forward and pullback attraction

For all bounded sets B ⊂ Rd , one has

dist(ϕ(t, t0,B),A(t)

)→ 0

as t →∞ with t0 fixed

(forward attraction)as t0 → −∞ with t fixed

(pullback attraction).

Definition (Forward and pullback attractor)

In case of forward attraction, the nonautonomous set A is called forwardattractor; and it is called pullback attractor in case of pullback attraction.




Bifurcation theory


Forward attractor

Forward attractor

A forward attractor A fulfills for all x0 ∈ Rd that

dist(ϕ(t, t0, x0),A(t)

)→ 0 as t →∞ with t0 fixed.

t0

x0

t

x ϕ(·, t0, x0)

A




Bifurcation theory


Pullback attractor

Pullback attractor

A pullback attractor A fulfills for all x0 ∈ Rd that

dist(ϕ(t, t0, x0),A(t)

)→ 0 as t0 → −∞ with t fixed.

t0t0t0 t

x0

ϕ(·, t0, x0)

A(t)

A




Bifurcation theory


An example


x = 2tx


φ(t, t0, x0) = x0et20−t

2

.

Then

the nonautonomous setR× {0} is a pullbackattractor,

there exists no forwardattractor.

t

x




Bifurcation theory


An example


x = −2tx


φ(t, t0, x0) = x0et2−t20 .

Then

there exists no pullbackattractor,

every invariantnonautonomous set withcompact fibers is a forwardattractor.

t

x




Bifurcation theory


Pullback versus forward attraction

The following general statements about pullback and forward attractorshold:

The concepts of pullback and forward attraction are independent ofeach other.

Pullback attraction describes attraction for the past of the system,whereas forward attraction concerns the future of a system.

If a pullback attractor exists, it is unique.

Forward attractors are intrinsically nonunique.




Bifurcation theory


Existence of attractors

The existence of autonomous attractors follows from absorbing sets.

Definition (Absorbing set)

A nonempty compact set D ⊂ Rd is called an absorbing set for a flow φif for every bounded set B ⊂ Rd , there exists a T > 0 such thatφ(t,B) ⊂ D for all t ≥ T .

Theorem (Existence of global attractors)

Suppose that a flow φ has an absorbing set D. Then φ has a uniqueglobal attractor A, given by

A =⋂s≥0

⋃t≥s

φ(t,D) .




Bifurcation theory


Existence of pullback attractors

Definition (Pullback absorbing set)

Let ϕ be a process. A nonempty compact set D ⊂ Rd is called pullbackabsorbing if for each t ∈ R and every bounded set B ⊂ Rd , there exists aT > 0 such that

ϕ(t, t0,B) ⊂ D for all t0 ≤ t − T .

Theorem (Existence of pullback attractors)

Let ϕ be a process with a pullback absorbing set D. Then there exists apullback attractor A uniquely determined by

A(t) =⋂τ≥0

⋃t0≤−τ

ϕ(t, t0,D) for all t ∈ R .




Bifurcation theory

Linearisations in the autonomous caseLinearisations in the nonautonomous caseExponential dichotomySacker–Sell spectrum

Linear theory




Bifurcation theory


Linearisations in the autonomous case

Consider the autonomous differential equation

x = f (x) ,

with a smooth function f : Rd → Rd , having an equilibrium x∗ ∈ Rd ,i.e., f (x∗) = 0.

Spectrum

The linearisation in this equilibrium is then given by the linear system

x = Df (x∗) x ,

the spectrum of which is defined by

σ :={

Reλ : λ is an eigenvalue of Df (x∗)}

= {a1, . . . , an} .




Bifurcation theory


Spectral decomposition

The spectrum describes the exponential growth behaviour of the system.

Spectral decomposition

Suppose that σ = {a1, . . . , an} is the spectrum of x = Df (x∗) x . Thenfor all elements ai ∈ σ, there is a invariant linear subspace

Wi :=⊕

λ∈C,Reλ=ai

ker(A− λ1)d

such thatsolutions in Wi grow as fast as eai t .

The subspaces Wi form a linear decomposition of Rd , i.e.,

W1 ⊕ · · · ⊕Wn = Rd .




Bifurcation theory


Hierarchy of invariant subspaces

Assume that σ = {a1, a2, a3} with a1 < a2 = 0 < a3 andW1 ⊕W2 ⊕W3 = Rd . This gives a hierarchy of invariant subspaces.


W1 ⊂ W1 ⊕W2 ⊂ Rd

∪ ∪W2 ⊂ W2 ⊕W3

∪W3

Stable subspace W1

Contains all trajectories which converge exponentially to 0 in forwardtime.




Bifurcation theory





W1 ⊂ W1 ⊕W2 ⊂ Rd

∪ ∪W2 ⊂ W2 ⊕W3

∪W3

Center-stable subspace W1 ⊕W2

Contains all trajectories which grow not too fast in forward time.




Bifurcation theory





W1 ⊂ W1 ⊕W2 ⊂ Rd

∪ ∪W2 ⊂ W2 ⊕W3

∪W3

Unstable subspace W3

Contains all trajectories which converge exponentially to 0 in backwardtime.




Bifurcation theory





W1 ⊂ W1 ⊕W2 ⊂ Rd

∪ ∪W2 ⊂ W2 ⊕W3

∪W3

Center-unstable subspace W2 ⊕W3

Contains all trajectories which grow not too fast in backward time.




Bifurcation theory





W1 ⊂ W1 ⊕W2 ⊂ Rd

∪ ∪W2 ⊂ W2 ⊕W3

∪W3

Center subspace W2

Contains trajectories which are both in W1 ⊕W2 and W2 ⊕W3 (inparticular, bounded trajectories).




Bifurcation theory


Linearisations in the nonautonomous case

We have linearised an autonomous differential equation

x = f (x)

in an equilibrium x∗.

In the nonautonomous context,it is possible to linearise along an arbitrary solution.




Bifurcation theory



We have linearised an autonomous differential equation

x = f (x)

in an equilibrium x∗.

In the nonautonomous context,it is possible to linearise along an arbitrary solution.




Bifurcation theory




x = f (t, x) ,

with a smooth function f : R× Rd → Rd . Let µ : R→ Rd be a solutionof this differential equation.

1. Transformation to the trivial equilibrium. Introduce a new variabley = x − µ(t). This yields the differential equation

y = f (t, y + µ(t))− f (t, µ(t)) ,

the so-called equation of perturbed motion.

2. Identifying linear and nonlinear part. The y -equation reads as

y = D2f (t, µ(t)) y︸︷︷︸linear part

+ f (t, y + µ(t)) + f (t, µ(t))− D2f (t, µ(t)) y︸︷︷︸nonlinear part




Bifurcation theory



This means that the solution µ corresponds to the zero solution of adifferential equation of the form

y = A(t)y + r(t, y) .

We now study the linear part of this equation, which is the so-calledvariational equation

y = D2f (t, µ(t))y .

Remark

In the autonomous context, such a transformation is only possible forequilibria (without leaving the class of autonomous systems).




Bifurcation theory



Consider the variational equation

x = D2f (t, µ(t)))︸︷︷︸=:A(t)

x .

with an induced process ϕ.

Warning

The eigenvalues of A(t) do not give information about the stability of thesystem. This can be see by the example

A(t) :=

(−2 cos2 t −1− 2 cos t sin t

1− 2 cos t sin t −2 sin2 t

).

For each t ∈ R, the matrix A(t) has double eigenvalue −1, but thereexist solutions which converge to ∞.




Bifurcation theory


Exponential dichotomy

Definition (Exponential dichotomy)

Let α < β. ϕ admits an exponential dichotomy with gap (α, β) if thereare K ≥ 1 and invariant nonautonomous sets S+ and S− with

S+(t)⊕ S−(t) =Rd for all t ∈ R ,

‖ϕ(t, t0, x0)‖ ≤Keα(t−t0)‖x0‖ for all t ≥ t0 and x0 ∈ S+(t0) ,

‖ϕ(t, t0, x0)‖ ≥ 1

Keβ(t−t0)‖x0‖ for all t ≥ t0 and x0 ∈ S−(t0) .

R

RdS+

S−




Bifurcation theory


Sacker–Sell spectrum

Definition (Sacker–Sell spectrum)

The Sacker–Sell spectrum of the above linear system is defined by

Σ :=

R \⋃

ED with gap (α,β)

(α, β)

∪{−∞} ∪ {∞}︸︷︷︸possibly

.

Remark

All Lyapunov exponents are included in the Sacker–Sell spectrum.




Bifurcation theory


Spectral Theorem

Theorem (Spectral Theorem)

The Sacker–Sell spectrum is given by

Σ = [a1, b1] ∪ · · · ∪ [an, bn] ,

where −∞ ≤ a1 ≤ b1 < a2 ≤ · · · < an ≤ bn ≤ ∞ and 1 ≤ n ≤ d , and foreach spectral interval [ai , bi ], there exists a linear and invariantnonautonomous set Si , a so-called spectral manifold with

S1(t)⊕ · · · ⊕ Sn(t) = Rd for all t ∈ R .

Remark

Solutions lying in the spectral manifolds admit an exponential growthbehaviour, which is determined by the corresponding spectral interval.




Bifurcation theory

Invariant manifoldsAn exampleThe importance of invariant manifolds

Nonlinear theory




Bifurcation theory


Invariant manifolds

Consider the nonlinear autonomous differential equation

x = Ax + r(x) .

We assume the following:

Hypothesis on linear part.σ(A) = {a1, a2, a3}.

Hypothesis on nonlinearity.Dr(0) = 0.

Existence of invariant manifolds

Then, in a neighbourhood of 0, there exist invariants manifolds, whichinherit the exponential growth behaviour from the linear subspaces.




Bifurcation theory


Invariant manifolds

Consider the nonlinear autonomous differential equation

x = Ax + r(x) .

We assume the following:

Hypothesis on linear part.σ(A) = {a1, a2, a3}.Hypothesis on nonlinearity.Dr(0) = 0.

Existence of invariant manifolds

Then, in a neighbourhood of 0, there exist invariants manifolds, whichinherit the exponential growth behaviour from the linear subspaces.




Bifurcation theory


Nonlinear systems

We have seen that an arbitrary solution µ of a nonautonomousdifferential equation

x = f (t, x)

can be transformed to the trivial solution of a differential equation of theform

x = A(t)x + r(t, x) ,

where r(t, 0) = 0 for all t ∈ R. We assume that r is a C 1-function.




Bifurcation theory


Invariant manifolds

Theorem (Existence of local invariant manifolds)

We assume:

x = A(t)x admits an exponential dichotomy with gap (α, β).

The nonlinearity fulfills limx→0 supt∈R∥∥ ∂r∂x (t, x)

∥∥ = 0.

Then, locally, there exist invariant nonautonomous sets T+ and T−,so-called invariants manifolds, which inherit the growth behaviour fromlinear manifolds S+ and S−.

R

RNS+

S−




Bifurcation theory


Invariant manifolds


We assume:



∥∥ = 0.


R

RNS+

S−




Bifurcation theory


Invariant manifolds


We assume:



∥∥ = 0.


R

RNT+

T−




Bifurcation theory


Example

We consider the system

x =− x + 0.4 cos(t)(sin(x) + sin(y)) ,

y = y + 0.4 sin(t) sin(x) .

The system fulfills the conditions of the invariant manifold theorem, andwe obtain a stable (green) and an unstable (yellow) manifold.




Bifurcation theory


The importance of invariant manifolds

Invariant manifolds are important for the study of bothlocal and global

dynamical behavior.

Locally

• Stable and unstable manifolds describe the saddle point structurearound hyperbolic equilibria.

• Center manifolds capture the essential dynamics, which makes thema main object in bifurcation and stability theory.

Globally

• Invariant manifolds serve as separatrix between different domains ofthe space.

• Attractors consist of unstable manifolds.

• Inertial manifolds allow a reduction to finite-dimensional dynamics.




Bifurcation theory

Autonomous pitchfork bifurcationNonautonomous pitchfork bifurcation

Bifurcation theory




Bifurcation theory


Autonomous pitchfork bifurcation

αx We consider for α ∈ R the differential equa-

tionx = αx − x3 ,

which is a prototype for an autonomouspitchfork bifurcation.

• For α < 0, we have only oneequilibrium (x = 0), which isattractive.

• For α > 0, x = 0 becomes repulsive,and two other equilibria are created(x = ±

√α), which are attractive.




Bifurcation theory



αx

A(α)




Alternative interpretation

A(α) := [−√α,√α] is an attractor, which

shrinks down in the limit α↘ 0.




Bifurcation theory



αx

A(α)










Bifurcation theory



αx

A(α)










Bifurcation theory



Recall the prototype of an autonomous pitchfork bifurcation:

x = αx − x3 .


We obtain a pitchfork bifurcation of an autonomous differential equation

x = a(α)x + b(α)x3 + r(α, x)

under the following conditions:

• Transversal exchange of stability: a(0)=0 , dadα (0) > 0,

• Non-vanishing third Taylor coefficient: b(0) < 0,

• Conditions on the remainder: r(α, x) = O(x4, x2α, xα2).




Bifurcation theory



Recall the prototype of an autonomous pitchfork bifurcation:

x = αx − x3 .


We obtain a pitchfork bifurcation of an autonomous differential equation

x = a(α)x + b(α)x3 + r(α, x)

under the following conditions:

• Transversal exchange of stability: a(0)=0 , dadα (0) > 0,

• Non-vanishing third Taylor coefficient: b(0) < 0,

• Conditions on the remainder: r(α, x) = O(x4, x2α, xα2).




Bifurcation theory


Nonautonomous pitchfork bifurcation


x = a(t, α)x + b(t, α)x3 + r(t, α, x) ,

for which we want to formulate nonautonomous conditions for

• transversal exchange of stability,

• non-vanishing third Taylor coefficient,

• conditions on the remainder.




Bifurcation theory




x = a(t, α)x + b(t, α)x3 + r(t, α, x) ,

Transversal exchange of stability

The autonomous case:spec

α




Bifurcation theory




x = a(t, α)x + b(t, α)x3 + r(t, α, x) ,


The nonautonomous case:spec

α

β1

β2




Bifurcation theory




x = a(t, α)x + b(t, α)x3 + r(t, α, x) ,


There exist functions β1, β2 and K ≥ 1 such that for τ ≤ t ∈ I , we have∫ t

τ

a(s, α)ds ≤ β1(α)(t − τ) logK ,

∫ τ

t

a(s, α)ds ≤ β2(α)(t − τ) logK

and

lim supα→0

lim supx→0

supt∈R

2K |r(t, x , α)||x |max

{− β1(α), β2(α)

} < 1 .




Bifurcation theory




x = a(t, α)x + b(t, α)x3 + r(t, α, x) ,

Non-vanishing third Taylor coefficient

The autonomous condition b(0) < 0 is generalised by

−∞ < lim infα→0

inft∈R

b(t, α) ≤ lim supα→0

supt∈R

b(t, α) < 0 .




Bifurcation theory




x = a(t, α)x + b(t, α)x3 + r(t, α, x) ,

Conditions on the remainder

The autonomous condition r(α, x) = O(x4, x2α, xα2) is generalised by

limx→0

supα∈(−x2,x2)

supt∈R

|r(t, x , α)||x |3

= 0 .




Bifurcation theory



Nonautonomous pitchfork bifurcation (supercritical case)

Under the above hypotheses, the nonautonomous differential equation

x = a(t, α)x + b(t, α)x3 + r(t, α, x)

admits a bifurcation of pullback attractors; more precisely, for negative αnear 0, the trivial solution is a local pullback attractor, which bifurcatesto a nontrivial pullback attractor Aα for positive α near 0. One has thelimit relation

limα↘0

dist(Aα(t), {0}

)= 0 for all t ∈ R .




Bifurcation theory


Summary

We have seen that there are two different types of attraction in thenonautonomous case: forward and pullback attraction.

The existence of pullback attractors follows via absorbing sets.

Hyperbolicity for nonautonomous systems is described by theconcept of an exponential dichotomy.

The Sacker–Sell spectrum gives information about the exponentialgrowth behaviour of linear systems.

The growth rates are attained in the spectral manifolds.

The spectral manifolds persist under nonlinear perturbation and givethe invariant manifolds.

Nonautonomous bifurcations can be described as attractorbifurcations; we have seen how the autonomous pitchfork bifurcationpattern translates to the nonautonomous case.




Bifurcation theory


Questions?

If you have questions...




Bifurcation theory


Questions?

If you have questions...




Bifurcation theory


Thanks!

Thanks for your attention!And thanks to:

L. Arnold, A. Berger, B. Aulbach, T. Caraballo, A. Carvalho, D. Cheban,C. Chicone, I. Chueshov, F. Colonius, W. Coppel, H. Crauel, T.S. Doan,

R. Fabbri, L. Grune, T. Huls, P. Imkeller, R. Johnson, J. Langa,C. Nunez, W. Kliemann, P.E. Kloeden, Y. Latushkin, S. Novo, R. Obaya,

R. Ortega, K. Palmer, C. Potzsche, J. Robinson, R. Sacker,B. Schmalfuß, G. Sell, S. Siegmund, T. Wanner, Y. Yi.


Date post:	06-Jul-2020
Category:	Documents
Upload:	others
View:	5 times
Download:	0 times

An Introduction to the Qualitative Theory of …...Qualitative Theory of Nonautonomous Dynamical...

Documents