Chapter 8 Structural Stability and Bifurcations 8.1 Smooth and Sudden Changes So far, we discussed the behavior of solutions of a dynamical system ˙ x = G(x), x(0) = x 0 , (8.1) under changes of the initial point x 0 . Since practical problems modelled by (8.1) can be expected to involve uncertainties in the mapping G, we should address also the question of the sensitivity of the solutions of (8.1) to perturbations of G. Then, however, we may see a very different behavior, than in the case of changes of x 0 . By ODE-theory, the solution of (8.1) depends continuously on the initial point, as is obvious for the solution x(t) = exp(tA)x 0 of the linear problem ˙ x = Ax, x(0) = x 0 . On the other hand, small changes of the mapping G may well produce abrupt changes in the solution behavior. We saw this already in the case of the discrete logistics problem x k+1 = g(x k ), k =0, 1, 2,..., g(x) := μx(1 - x), μ> 0. (8.2) Here, the fixed points are x * = 0 and x ** =(μ - 1)/μ, where x * = 0 is asymptotically stable for 0 <μ< 1 and unstable for μ> 1, while x ** is unstable for 0 <μ< 1 and asymptotically stable for 1 ≤ μ< 3. But, at μ = 3 the solution behavior changes significantly: In fact, for any μ> 3 the iterates tend to a 2-cycle consisting of the points z ± := 1 2μ (μ + 1) ± p (μ + 1)(μ - 3) ; 130
Transcript
Chapter 8
Structural Stability and
Bifurcations
8.1 Smooth and Sudden Changes
So far, we discussed the behavior of solutions of a dynamical system
x = G(x), x(0) = x0, (8.1)
under changes of the initial point x0. Since practical problems modelled by (8.1) can beexpected to involve uncertainties in the mapping G, we should address also the questionof the sensitivity of the solutions of (8.1) to perturbations of G. Then, however, we maysee a very different behavior, than in the case of changes of x0.
By ODE-theory, the solution of (8.1) depends continuously on the initial point, as isobvious for the solution x(t) = exp(tA)x0 of the linear problem x = Ax, x(0) = x0. Onthe other hand, small changes of the mapping G may well produce abrupt changes in thesolution behavior. We saw this already in the case of the discrete logistics problem
Here, the fixed points are x∗ = 0 and x∗∗ = (µ − 1)/µ, where x∗ = 0 is asymptoticallystable for 0 < µ < 1 and unstable for µ > 1, while x∗∗ is unstable for 0 < µ < 1and asymptotically stable for 1 ≤ µ < 3. But, at µ = 3 the solution behavior changessignificantly: In fact, for any µ > 3 the iterates tend to a 2-cycle consisting of the points
z± :=1
2µ[(µ+ 1)±
√(µ+ 1)(µ− 3)
];
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that is, the attractor consisting of the single point x∗∗ has become an attractor-setz+, z− of cardinality two.
Such sudden changes in the dynamics caused by smooth alterations of a system occur inmany problems. Bridges bend continuously under an increasing load, but then suddenlybreak. As the saying goes, ’there is one straw, that breaks the camels back’. Similarly,forces in a rock-structure can build up until friction no longer holds and the structuresuddenly ruptures in an earthquake. A living cell suddenly changes its reproductive rythmand doubles and redoubles cancerously. In each example, continuous changes suddenly leadto an abrupt change in the structure of the solution field. Accordingly, these situationsare also called structural instabilities.
8.2 Structural Stability
In this section we outline – without proofs – the basic ideas behind the concept of structuralstability. In essence, a dynamical systems (8.1) is structurally stable if its phase portraitremains topologically invariant under small perturbations of the mapping G. This isillustrated by the two pairs of pictures in Figures 8.1 and 8.2. The two phase portraits8.1 can be continuously deformed into each other, while, in the second pair, the connectionbetween the two saddles is severed and thus there can no longer be a such a deformation.
Figure 8.1: Conjugate Figure 8.2: Not conjugate
A precise definition of structural stability builds on the concept of topological equivalenceof mappings. If x 7→ Xx is a coordinate transformation on Rn defined by some nonsingularX ∈ Rn×n, then any matrix A ∈ Rn×n transforms into X−1AX. Analogously, supposethat Rn is mapped onto itself by a homeomorphism h : Rn −→ Rn; i.e., a bijective,continuous mapping with a continuous inverse, then a nonlinear mapping f : Rn −→ Rn
is transformed into the mapping g = h−1fh. Accordingly, the mappings f and g are
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called topologically equivalent, if for some homeomorphism h on Rn the following diagramis commutative:
Rn
h
f // Rn
h
Rn
g // Rn
For the discussion of the topological equivalence of phase portraits of dynamical systems itis customary to use vector fields. Recall, that for a C1-mapping G : Rn −→ Rn, the uniquesolution of (8.1) passing through x ∈ Rn has at x the tangent vector G(x). Accordingly,we will view G as defining a C1 vector field on Rn to be denoted by (Rn, G). The solutionsof (8.1) are the orbits of this vector field.
Two vector fields (Rn, G1), (Rn, G2) are topologically equivalent near x1, x2 ∈ Rn, if thereexist neighborhoods U1,U2 ⊂ Rn of x1, x2 and a homeomorphism h : U1 −→ U2 withh(x1) = x2, which maps each orbit of the first vector field in U1 onto an orbit of thesecond vector field in U2, preserving the direction of time.
A linear vector field (Rn, A) is hyperbolic, if A ∈ Rn×n has no eigenvalues with zeroreal part. Here, the next result shows, that a characteristic quantity is the number ofeigenvalues with negative real part, called the index of A:
8.2.1. The hyperbolic (linear) vector fields (Rn, A) and (Rn, B) are topologically equivalentif and only if A and B have the same index.
A proof will not be given; it involves consideration of certain bases of generalized eigen-vectors corresponding to the two matrices. As a simple example, let
A =
(−1 −3−3 −1
), B =
(2 00 −4
).
Then we have
B = XAX−1, X =1√2
(1 −11 1
)and hence X exp(tA) = exp(tB)X. Since x 7−→ Xx is a homeomorphism on R2, thevector fields (R2, A) and (R2, B) are indeed topologically equivalent.
As indicated, structural stability of a vector field (Rn, G) requires, that all sufficiently smallperturbation ofG produce topologically equivalent vector fields. For this we have to specifywhat perturbations are allowed. In the following definition we work with the normed linearspace (C1(Ω), ‖ · ‖1) of all continuously differentiable mappings G : Ω ⊂ Rn −→ Rn on agiven open set Ω, with the norm
‖G‖1 := supx∈Ω‖G(x)‖2 + sup
x∈Ω‖DG(x)‖2, ∀G ∈ C1(Ω).
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8.2.2 Definition. For G0 ∈ C1(Ω) the vector field (Ω, G0) is structurally stable in(C1(Ω), ‖ · ‖1), if there is an ε > 0, such that for all G ∈ C1(Ω) with ‖G−G0‖1 < ε,the vector fields (Ω, G0) and (Ω, G) are topologically equivalent.
We will not enter into a further study of this concept. The theory quickly reaches aconsiderable mathematical depth.
For a linear vector field (Rn, A) defined by some matrix A it is natural to restrict theperturbations to the normed linear space (Rn×n, ‖ · ‖2). Recall that the eigenvalues ofA ∈ Rn×n depend continuously on the elements of the matrix. Hence, if the vectorfield (Rn, A) is hyperbolic, then there exists ε > 0, such that for each B ∈ Rn×n with‖B −A‖2 < ε, the vector field (Rn, B) is also hyperbolic. Moreover, it can be shown,that, for sufficiently small ε, the index of any such B is the same as that of A. Hence8.2.1 implies, that the vector fields of A and B are topologically equivalent. This leadsto the following result:
8.2.3. A linear vector field (Rn, A) is structurally stable in (Rn×n, ‖ · ‖2) if and only if Ais hyperbolic.
For nonlinear mappings the Hartman-Grobman theorem provides a connection with thislinear result:
8.2.4. Let Ω ⊂ Rn be an open set and for given G ∈ C1(Ω) denote by Φ the flow of thedynamical system (8.1). Suppose, that x∗ ∈ Ω is a hyperbolic equilibrium of (8.1). Thenthere exists a homeomorphism h of an open neighborhood U1 ⊂ Ω of x∗ onto an openneighborhood U2 ⊂ Rn of the origin, such that h(x∗) = 0 and
where the open interval J , 0 ∈ J , is locally determined.
Under the conditions of the theorem, (8.3) implies that the homeomorphism h maps eachsolution of (8.1) in the neighborhood U1 of x∗ onto a solution of the linearized system
y = DG(x∗)y, y(0) = h(x),
in the neighborhood U2 of the origin and preserves the t–orientation. In other words, locallynear the hyperbolic equilbrium x∗, the vector field (Rn, G) is topologically equivalent withthe hyperbolic linear vector field (Rn, DG(x∗)) near the origin.
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8.3 Introduction to Bifurcations
Bifurcation theory is the mathematical study of changes in the topological properties ofthe phase portraits of a family of dynamical systems. We sketch only some elementaryideas underlying this large topic.
Many physical, chemical, and biological problems lead to parameter-dependent dynamicalsystems
x = G(x, µ), x(0) = x0, G : Ω ⊂ Rn := Rm × Rd −→ Rm, n = m+ d, (8.4)
where G ∈ C1(Ω) on an open set Ω. Then, in essence, a bifurcation occurs at a parametervalue µ0, if there are values of µ arbitrarily close to µ0, for which the phase portraits of (8.4)are no longer topologically equivalent with the one for µ0. In other words, bifurcationsidentify the failure of the structural stability of the vector field (Ω, G(·, µ)) for specificparameter values.
For instance, the planar system
x = A(µ)x, A(µ) :=
(µ −11 µ
), (8.5)
has the origin as an equilibrium for each value of µ. For µ 6= 0 this equilibrium ishyperbolic and hence, by 8.2.3, the linear vector field (R2, A(µ)) is structurally stable.More specifically, for µ < 0 the phase portraits are asymptotically stable spirals and forµ > 0 they are unstable spirals (see Figure 8.3). But for µ = 0 the origin is a center pointand no homeomorphism can map the periodic orbits for µ = 0 onto the spiral trajectoriesfor any nonzero µ, no matter how small. In other words, we have a bifurcation at µ = 0.
Figure 8.3: Structural instability
Bifurcation theory develops strategies for investigating the types of bifurcations occurringwithin a family of dynamical systems, and for classifying and naming the different patterns.One aim is to produce subdivisions of the parameter space into regions induced by thetopological equivalence of the corresponding phase portraits. Then bifurcations occur atpoints, that do not lie in the interior of one of these regions. A bifurcation diagram is amap of these regions, together with representative phase portraits for each of them.
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Suppose that the system (8.4) involves a scalar parameter, i.e., that d = 1. If, say,G(x∗, µ∗) = 0, and x∗ is a hyperbolic equilibrium, then the vector field (Rn, G(, ·, µ∗)) isstructurally stable near x∗. Generically, there are only two ways in which variation of theparameter can lead to a violation of the hyperbolicity condition, namely
• for some value of the parameter, a simple real eigenvalue λ of DxG(x, µ) becomeszero, or
• a pair of simple complex eigenvalues passes the imaginary axis and on the axis wehave λ1,2 = ±iγ, γ > 0.
It can be shown, that a higher dimensional parameter space is needed to allocate additionaleigenvalues on the imaginary axis.
The two cases correspond to some basic types of bifurcations. A typical approach for char-acterizing such types is the use of certain normal forms for model systems, that exemplifythe topological pattern of the phase portraits near the bifurcation.
Limit Points: These are also called turning points or saddle-node bifurcations. Theyare characterized by the collision and annihilation of two equilibria. A normal form is
x = x2 + µ, (x, µ)> ∈ R2. (8.6)
Hence equilibria exist only for µ ≤ 0 and they lie on a smooth curve in R2, namely, theparabola of Figure 8.4.
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
unstable
stable
Figure 8.4: Limit point
For µ < 0 the equilibria are hyperbolic and the upper branch x+(µ) :=√−µ, µ < 0, of the
parabola is unstable, while the lower branch x−(µ) := −√−µ, µ < 0, is asymptotically
stable. When µ crosses zero from negative to positive values, the two equilibria collideand then disappear for µ > 0. The term ”collision” is here appropriate, since the velocity(d/dµ) x±(µ) of the approach tends to infinity as µ → 0. The limit point (0, 0)> ∈ R2 is
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characterized by the property, that the tangent of the equilibrium curve at this point isorthogonal to the parameter axis.
The bifurcation can also occur in reverse. The system
x = x2 − µ, (x, µ)> ∈ R2,
has no equilibria for µ < 0 and two of them for µ > 0.
The indicated name ’saddle-node bifurcation’ becomes clearer, if we consider (8.6) in theplanar form
x =
(x2
1 + µ
−x2
), (x, µ)> ∈ R3. (8.7)
For µ < 0 the two equilibria are now a stable node and a saddle, and they are annihilatedat µ = 0, as shown in Figure 8.5 (where β stand for µ).
Figure 8.5: Saddle-node bifurcation
Hopf Bifurcations: The characteristic feature of these bifurcations is the emergenceof a periodic orbit from a stable equilibrium or vice versa. A normal form is
x = G(x, µ) := A(µ)x− (x>x) x, A(µ) :=
(µ −11 µ
), (x, µ)> ∈ R3. (8.8)
Evidently, x∗ = 0 is an equilibrium for all µ, and DxG(x∗, µ) = A(µ) has the eigenvaluesλ1,2 := µ± i. The system can be written as the complex equation
z = (µ+ i)z − |z|2z, z = x1 + ix2 = ρeiθ,
or in the decoupled form
ρ = ρ(µ− ρ2)
θ = 1.(8.9)
For µ < 0 the phase portrait consists of asymptotically stable spirals, as shown in Figure8.6 with µ = −1. For µ = 0 both eigenvalues pass through the imaginary axis with thevelocity dλ1,2/dt = 1. For µ > 0 the first equation of (8.9) has the unstable equilibrium
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−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 8.6: Equ (8.8) µ = −1
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 8.7: Equ (8.8) µ = 1
ρ1 = 0 and the asymptotically stable equilibrium ρ2(µ) =√µ. For the full system (8.9)
the latter defines a limit cycle with radius√µ, as shown in Figure 8.7 with µ = 1:
Note that, generally, a limit cycle is an isolated closed orbit for which neighboring trajec-tories are not closed. In other words, the orbits of any center point are not limit cycles.
As before, the bifurcation can also be reversed. The system
x = A(µ)x+ (x>x)x, (x, µ)> ∈ R3, (8.10)
has for µ < 0 an unstable limit cycle, which disappears when µ crosses zero from negativeto positive values and then turns into unstable spirals when µ > 0. The bifurcationsin (8.8) and (8.10) are sometimes called supercritical and subcritical Hopf bifurcations,respectively. This terminology is somewhat misleading, since ”super” means here ”after”the bifurcation point, and ”sub” ”before” the point, which, however, depends on the chosendirection of the parameter variation.
We followed here the wide-spread practice of using the term ”Hopf-bifurcation” for theemergence of a periodic orbit from a stable equilibrium as a parameter crosses a criticalvalue. But the subject has its origins in the work of Poincare around 1892, followed byextensive studies by Andronov and coworkers around 1930, while the fundamental paperof E. Hopf appeared in 1942. Accordingly, the term Poincare-Andronov-Hopf bifurcationis thought to be more justified by many.
Transcritical Bifurcation: A normal form is
x = x(µ− x), (x, µ)> ∈ R2. (8.11)
Thus the equilibria form two intersecting straight lines in R2, as shown in Figure 8.8(where λ stands for µ). At the crossing point (0, 0)> these equilibria exchange their
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dx/dt = λ − x) x
stable
stable
instable
instable
λ
x(
Figure 8.8: Transcritical bifurcation
stability properties; that is, the unstable equilibria become asymptotically stable and viceversa.
Note, that beyond the bifurcation point the number of equilibria has not changed, whilein the case of limit points two equilibria appear or disappear.
Cusp Bifurcation: This is also called a pitchfork bifurcation or the occurrence of abranch point. A normal form corresponds with the first equation of (8.9); i.e.,
x = x(µ− x2), (x, µ)> ∈ R2. (8.12)
Evidently, x∗ = 0 is an equilibrium for all µ, which is asymptotically stable for µ < 0and unstable for µ > 0. Then, for µ > 0 two additional, asymptotically stable equilibriax±(µ) = ±√µ appear, as shown in Figure 8.9.
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
stable
stable
stable
instable
Figure 8.9: Pitchfork bifurcation
Thus, in this case, when µ passes through zero from negative to positive values, the asymp-totically stable equilibrium x∗ = 0 becomes unstable and two curves of asymptoticallystable equilibria are branching off,
As before, this bifurcation can also occur in reverse. The system
x = x(µ+ x2), (x, µ)> ∈ R2,
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has for µ < 0 the stable equilibrium x∗ = 0 and the two unstable equilibria x± = ±√−µ.
Then, for µ > 0 only the unstable equilibrium x∗ = 0 remains.
8.4 Two Examples
8.4.1 A Simple Mechanical Example
As an illustration of the appearance of limit points and branch points, we consider thesimple framework of Figure 8.10, where the two straight, rigid, and pin-jointed rods aresubjected to two (dead)-loads µ and ν. The deformation is specified by the angle y and,without loads, the (linear) spring forces the rods into the horizontal reference configuration,where y = 0.
y
1 1ν
µ
Figure 8.10: A simple framework
The angle y corresponds to a lengthening of the spring by 2y, which therefore has theelastic energy (1/2)k(2y)2. Here k denotes the spring constant, which we set to one. Upto constants, the potential energy due to the forces µ and ν equals µ(2 cos y) and ν sin y,respectively. Thus at a given state x := (y, µ, ν)> in the open set
Ω :=
(y, µ, ν)> : |y| < π
2, µ, ν ∈ R1
, (8.13)
the system has the total potential energy
V = V (y, µ, ν) :=12
(2y)2 + 2µ cos y − ν sin y, (y, µ, ν)> ∈ Ω, (8.14)
and the equilibria are the solutions of the parameterized nonlinear equation
G(y, µ, ν) := DyV (y, µ, ν) = 2y − 2µ sin y − ν cos y, (y, µ, ν)> ∈ Ω. (8.15)
is the 2-dimensional surface in R3 shown in Figure 8.11. For further insight, Figure 8.12depicts some contour lines corresponding to constant loads ν in the plane of the variablesy, µ. Evidently, M has a saddle-point at (0, 1, 0)>.
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−1−0.5
00.5
10
0.51
1.5
−8
−6
−4
−2
0
2
4
6
8
Figure 8.11: Equilibrium surface
0 0.5 1 1.5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
turning points, bifurcation point
ν = 0
ν = 0
ν = 0.2
ν = −0.2
ν = 0.1
ν = −0.1
ν = 0.2
ν = −0.2ν = −0.1
ν = 0.1
µ
y
stabilityboundary
Figure 8.12: Contours
Theoretical mechanics provides, that a point on the surface is stable or unstable if thesecond derivative of the potential energy V ; i.e.,
DyyV (y, µ, ν) := 2− 2µ cos y + ν sin y, (8.17)
is positive or negative, respectively. Hence, the solutions of the system of equations(G(y, µ, ν)DyG(y, µ, ν)
)= 0, (y, µ, ν) ∈ Ω, (8.18)
define the stability boundary of the system.
We fix the load ν = ν0 and allow only µ to vary. In other words, we consider the reducedequation
Gν0(y, µ) := G(y, µ, ν0) = 0. (8.19)
As Figure 8.12 indicates, for ν0 6= 0 the solution set of (8.19) consists of two smoothcurves on M corresponding to positive and negative values of y, respectively. Let t 7−→(y(t), µ(t)) be a (smooth) parametric representation of either one of these curves, then, bydifferentiation, it follows that
This implies that µ′(t) = 0 and, therefore, that, at a point on the intersection of the curvewith the stability boundary, the tangent must be perpendicular to the µ–axis. Accordingly,these points are limit points with respect to µ. Figure 8.12 indicates, that that there exists
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exactly one limit point for each ν0 6= 0. The line of the limit points defines the stabilityboundary sketched in Figure 8.12.
Suppose that we start at a point on the curve for ν0 = −0.1 with y = 1 and someµ > 1. Then a gradual decrease of the load µ reduces the state-angle y until we reachthe limit point of the particular curve and hence the stability boundary. Beyond thispoint the equilibria on the curve are unstable, and physical considerations suggest, thatthe frame will snap through to a point with y < 0 on the corresponding stable branchfor the load ν0 = −0.1. This is sketched by the dotted line in Figure 8.12. Of course,this jump cannot be derived from our static equilibrium model. After the snap-through,a continuing decrease of the load µ causes the system to follow the stable load path.
In the case ν0 = 0, the solutions of the reduced equation (8.19) form the four branches
which meet at the saddle point y∗ = (0, 1, 0)> of the surface. Evidently, this is the onlypoint where the stability boundary intersects the four curves, and, as Figure 8.12 shows,at this point we have a pitchfork bifurcation. If we are, say, at the origin of Figure 8.12,then the rods are in a straight line, and there are no applied loads. With increasing µ
the system follows branch (a); i.e., the state y remains zero until we reach µ = 1. Thecontinuing branch (b) is unstable and hence an increase of µ beyond 1 will cause thesystem to depart from the straight position and to follow either branch (c) or (d). Inpractice, the specific choice is determined by small, inevitable perturbing forces.
8.4.2 Hopf Bifurcation in the Lorenz System
The well known ODE-system developed by E. N. Lorenz is a simplified model of certainconvection phenomena in the earth’s atmosphere and has the form
x = G(x, µ), G(x, µ) :=
a(x2 − x1)x1(µ− x3)− x2
x1x2 − bx3
, (x, µ)> ∈ R4.
Here, the parameter µ denotes a (relative) Raleigh number and the constants are oftenset to a = 10 and b = 8/3. Evidently, the origin x∗ = 0 of R3 is an equilibrium for all µand, it is easily checked, that for µ ≥ 1 there are the two additional equilibria
Thus x∗ is asymptotically stable for µ < 1. At µ = 1 one eigenvalue becomes zeroand (0, 0, 1)> is a cusp point, where the origin becomes unstable and for µ > 1 the twoequilibrium curves (8.22) branch off. Their projections from R3 onto the (x1, µ)–plane areshown in Figure 8.13:
0 5 10 15 20 25 30
−8
−6
−4
−2
0
2
4
6
8
Hopf bifurcation
Hopf bifurcation
unstable
stable
stable
x1
µ
Figure 8.13: Equilibria of the Lorenz equations
A short calculation shows, that the negative characteristic polynomial of DxG(x±(µ), µ)equals
A method for computing Hopf bifurcations by J. Guckenheimer, M. Myers, and B. Sturm-fels (SIAM J. Num. Anal 34, 1997, 1-21) is based on the observation, that any (monic)polynomial q has a nonzero pair of roots λ and −λ if and only if λ is a common root ofthe polynomials q(λ) + q(−λ) and q(λ)− q(−λ). In our case, this implies that (8.24) hasa nonzero root-pair exactly if there exists a solution z = λ2 of the two equations
c0 + c2z = 0, c1 + z = 0
This requires that 0 = c0 − c1c2 = 2ab(µ− 1)− b(a+ b+ 1)(a+ µ), and hence that
µ = µh :=a(3 + a+ b)a− b− 1
.
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Indeed, the eigenvalues of DxG(x±(µh), µh) are
λ1 = −(a+ b+ 1), λ2,3 = ±i√
2ab(a+ 1)a− b− 1
.
A closer study of the characteristic polynomial (8.24) shows, that the equilibria x±(µ) areasymptotically stable for 1 < µ < µh and unstable for µ > µh. For µ = µh we expect Hopfbifurcations at the points (x±(µh), µh)> on the two branches. The details of the situationare more complicated and are discussed in a monograph by C. Sparrow, (Springer, 1982).
8.5 Some Further Aspects of Bifurcation Theory
The bifurcations discussed in Section 8.3 involved systems with a scalar parameter and,in each case, the bifurcation type was solely determined by properties of the system atthe particular equilibria. This need not always be the so; in fact, one has to distinguishbetween two principal classes of bifurcations:
• Local bifurcations, which can be analysed entirely in terms of the local stabilityproperties of equilibria as parameters cross through critical thresholds.
• Global bifurcations, which involve topological changes in the behavior of the trajec-tories in phase space extending over larger distances and involving, e.g., interrelatedchanges at several equilibria.
For an example of a global bifurcation consider the system
x = G(x, µ) :=
(x2
x1 + µx2 − x21
), (x, µ)> ∈ R2 × R1. (8.25)
For each µ, the system has the same two equilibria, x∗ := (0, 0)> and x∗∗ := (1, 0)>.Because of
DxG(x, µ) =
(0 1
1− 2x1 µ
),
x∗ is always a saddle, while x∗∗ changes from a spiral sink for µ < 0 to a spiral source forµ > 0. The phase portraits for the cases µ = 0, µ = −1 and µ = 1 are shown in Figures8.14, 8.15, and 8.16.
For µ = 0 there exists a trajectory, that both emanates and terminates at x∗. This is a socalled homoclinic orbit and x∗ is said to be a homoclinic point. The region bounded by thehomoclinic orbit forms a neighborhood of the equilibrium x∗∗ and contains periodic orbitsaround that point. Thus, locally, x∗∗ is a stable center point. Evidently, the eigenvalues
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−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 8.14: Equ (8.25) µ = 0
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 8.15: Equ (8.25) µ = −1
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 8.16: Equ (8.25) µ = 1
of DxG(x∗∗, 0) are ±i and hence we might expect a Hopf bifurcation for µ = 0. But thehomoclinic orbit is formed at the other equilibrium x∗ and hence is not a local feature ofx∗∗. In other words, the solution behavior is only partially determined by the eigenvaluesat that equilibrium and depends on the entire ODE.
So far we considered only problems with a scalar parameter. For higher dimensional pa-rameter spaces the phenomena quickly become more complicated and even an introductorydiscussion would exceed our framework. We expand only on one aspect, which arose inconnection with the mechanical example in subsection 8.4.1. This example involved atwo-dimensional parameter vector and, as Figure 8.11 showed, the equilibria formed atwo-dimensional surface in R3.
In essence, the existence of such equilibrium manifolds depends on the number of param-eters in the equations.
As before, consider a parameter-dependent dynamical system
x = G(x, µ), x(0) = x0, G : Ω ⊂ Rn := Rm × Rd, n = m+ d, (8.26)
144
defined by a C1 mapping G on an open set Ω. Then our interest centers on conditionsunder which the set of all equilibria
M := (x, µ) ∈ Rn : G(x, µ) = 0 (8.27)
is a differentiable manifold.
We will not enter into the theory of manifolds, but use – in place of a definition – thefollowing sufficient characterization of submanifolds of Rn:
8.5.1. Let F : Ω ⊂ Rn −→ Rm, n > m, be continuously differentiable on the open set Ωand assume that the zero set M = x ∈ Ω : F (x) = 0 is not empty. If rankDF (x) = m
for all x ∈M , then M is a C1-submanifold of Rn of dimension d = n−m.
Generally, the mapping F is a submersion at x ∈ Ω if DF (x) ∈ Rm×n has maximal rankm and hence maps onto Rm. Thus, M is a submanifold of Rn, if F is a submersion ateach point of this set. Accordingly, 8.5.1 is also called the submersion theorem.
A simple example is the Euclidean unit sphere
Sn−1 :=x ∈ Rn : x>x = 1
, (8.28)
centered at the origin. It is the zero set of x ∈ Rn 7→ F (x) := x>x − 1, and, sinceDF (x) = rank(2x1, . . . , 2xn) = 1 for x ∈ Sn−1, it follows from 8.5.1 that Sn−1 is an(n− 1)-submanifold of Rn.
For the normal form (8.6) of a limit-point the derivative DG(x, µ) = (2x, 1) has rankone for all (x, µ)> ∈ R2. Thus, by 8.5.1, the equilibrium set M is a one-dimensionalsubmanifold of R2, namely the parabola of Figure 8.4.
of all equilibria of the cusp bifurcation (8.12) shown in Figure 8.9. Here, we haveDG(x, µ) = (µ − 3x2, x) and hence rankDG(0, 0) = 0; i.e., G is not a submersion atthe bifurcation point (0, 0)> ∈M . A d-dimensional manifold is a topological space, whereeach point necessarily has an open neighborhood, which is homeomorphic with an opensubset of Rd. But, clearly, in (8.29) the bifurcation point (0, 0)> ∈M has no neighborhoodin M , which is homeomorphic with an open interval of R1. In other words, the entire set(8.29) is not a one-dimensional submanifold of R2. In a sense, the problem is here, thatthe equation (8.12) does not include enough parameters. If, instead, the two-parametersystem
x = G(x, µ, ν) := ν + µx− x3, x ∈ R1 µ, ν ∈ R1 (8.30)
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is used, then rankDG(x, µ, ν) = 1 on all of R3 and the equilibrium–set is a two–dimensionalsubmanifold of R3. One calls (8.30) an unfolding of the original system (8.12).
The construction of suitable unfoldings of a given parameterized system (8.26) is an im-portant topic of bifurcation theory. We consider only another simple example involving anaturally unfolded system with a cusp bifurcation.
As a simplified, two-dimensional model of the roll-behavior of ships, consider a thin, flatlamina, that stands on edge on a plane and consists of a material, which allows the centerof gravity to shift. More specifically, the lamina is assumed to have the shape of a parabolacut off by a line perpendicular to the symmetry axis as shown in Figure 8.17.
η−axis
ξ−axis
Figure 8.17: Parabolic lamina
−3 −2 −1 0 1 2 30
0.5
1
−1
−0.5
0
0.5
1
x
y
ξ
Figure 8.18: Equilibrium surface
In the indicated (ξ, η) coordinate system the parabola is given by η = ξ2 and (x, y)>
denotes the center of gravity. At equilibrium the line between the point of contact and thecenter of gravity must be perpendicular to the resting plane. In other words, if (ξ, ξ2)> isthe point of contact, then the tangent vector (1, 2ξ)> of the parabola at that point is inthe resting plane and must be orthogonal to the vector (x− ξ, y − ξ2)>. Hence for given(x, y)> the coordinate ξ defining the point of contact must satisfy the equation
G(ξ, x, y) := 2ξ3 + ξ(1− 2y)− x = 0. (8.31)
Clearly, by 8.5.1 the set M of all solutions of (8.31) is a 2-dimensional submanifold ofR3. If we use a rotation, such that the base plane is spanned by the parameters x, y, thenFigure 8.18 shows that the manifold has a fold.
Since the potential energy of the lamina is proportional to the distance between thecontact-point and the center of gravity, it follows, as in subsection 8.4.1, that the stabilityboundary is the curve on the manifold defined by the system of equations(
G(ξ, x, y)DξG(ξ, x, y)
):=
(ξ3 + ξ(1− 2y)− x
6ξ2 + (1− 2y)
)= 0. (8.32)
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The orthogonal projection of this curve onto the (x, y)–plane satisfies
x = −4t3, y =12
+ 3t2, t ∈ R1, (8.33)
and is shown in Figure 8.19. The curve on M , above the line segment marked (1)→ (4),is depicted in Figure 8.20, and, on it, the points between (3a) and (2b) are unstableequilibria. If, the center of gravity moves from (1) to (4) on the line segment of Figure
x
yξ < 0ξ > 0
(4) (3) (2) (1)
Figure 8.19: Stability boundary
(1)
(2a)
(2b)
(3a)
(3b)
(4)
Figure 8.20: Hysteresis loop
8.19, then on M we reach the stability boundary at the point (3a) (above (3)). There weexpect the point of contact to jump from (3a) to the stable equilibrium (3b) and then toproceed to (4). Conversely, if we move from (4) toward (1), then the stability boundaryis reached at (2b) (above (2)) and the point of contact will jump to the point (2a) on thestable branch and continue along it to (1). The sequence of points
(1)→ (3a)→ (3b)→ (4)→ (2b)→ (2a)→ (1)
represents a so–called hysteresis loop.
8.6 Some Numerical Aspects
As before, consider a dynamical system (8.26) defined by a mapping G from Rn := Rm×Rd
to Rm such that G ∈ C1(Ω) for some open set Ω ⊂ Rn. Here d is the dimension of theparameter space. For ease of notation, we assume from now on, that Ω = Rn.
For the study of local bifurcations, we may concentrate on the properties of the set ofequilibria
M := (x, µ) ∈ Rn : G(x, µ) = 0 . (8.34)
If rankDG(x, µ) = m for all (x, µ) ∈ M , then M is a d-dimensional submanifold of Rn,and hence we want to investigate numerically the shape of this manifold.
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Consider any point (x0, µ0)> ∈ M , where x0 is a hyperbolic equilibrium and hence weknow that rankDxG(x0, µ0) = m. Then, by the implicit function theorem, there existopen neighborhoods Ud ⊂ Rd and Um ⊂ Rm of µ0 and x0, respectively, and a C1 mappingξ : Ud −→ Um, such that
We call (Ud, ϕ) a local parametrization of the manifold M near (x0, µ0)>.
A regular, parametrized C1–curve in Rn is a C1 function γ : J −→ Rn on some openinterval J ⊂ R1, such that Dγ(t) 6= 0, ∀τ ∈ J . The so–called immersion conditionDγ(t) 6= 0 ∀t ∈ J , excludes, for instance, the occurrence of cusps, as in the case ofγ : R1 −→ R2, γ(t) := (t2, t3)>, at t = 0.
A standard approach for analyzing the manifold M is to probe it along suitable curvesegments, but there are also numerical methods for a direct computation of parts of themanifold itself (see, e.g., M. L. Brodzik and W. C. Rheinboldt, Comp. Math. and Appl.,28, 1994, 9-21)
Let (x0, µ0)> ∈M , where x0 is a hyperbolic equilibrium, and construct a local parametriza-tion (Ud, ϕ) of M near (x0, µ0). We vary µ only along a straight line
τ ∈ R1 7−→ µ = µ0 + cτ ∈ Rd, c ∈ Rd, c 6= 0.
(For d = 1 simply set c = 1). Since Ud is an open neighborhood of µ0, there exists a δ > 0,such that µ0 + cτ ∈ Ud for τ in J := τ : |τ | < δ. Since Dγ(τ) = Dϕ(cτ)c 6= 0, itfollows that
γ : J −→M, γ(τ) = ϕ(ct) ∀ τ ∈ J , (8.38)
is a regular, parametrized curve on M .
The task is now to generate computational approximations of these regular, parametrizedC1 curves on the equilibrium manifold M . In general, however, the local parametrization(Ud, ϕ) is not explicitly known; i.e., we have to work with the solution set of the equationG(x, µ0 + cτ) = 0 for x near x0 and τ near zero.
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For ease of notation, it is useful to consider a continuously differentiable mapping
F : Γ ⊂ Rm × R1 = Rm+1 −→ Rm, Γ open, (8.39)
such that
rankDF (x, τ) = m, ∀ (x, τ) ∈ N :=
(x, τ)> ∈ Γ : F (x, τ) = 0, (8.40)
where now the scalar parameter is denoted by τ . Thus the set N is a one-dimensionalsubmanifold of Rm+1. In the following, we will often write y = (x, τ)> for the vectors ofthe domain space of the mapping (8.39). Note, that in the original setting, we had
F (x, τ) = G(x, µ0 + cτ), Γ := Um × J , rankDG(x0, µ0) = m,
which implies that (8.40) holds in some neighborhood of (x0, 0)>.
Algorithms for approximating such a one-dimensional submanifold N of Rm+1 by a se-quence of points yk := (xk, τk)> ∈ Rm+1, k = 0, 1, 2, . . ., have been a topic of long-standingin numerical mathematics and engineering. Usually, they are called numerical continuationmethods, although alternate names are also in use, such as imbedding methods, homotopymethods, parameter variation methods, or incremental methods. For some references tothe large literature, see, e.g., E. L. Allgower and K. Georg, Numerical Continuation Meth-ods, Springer Verlag, 1990. We outline only briefly some of the ideas behind a typicalcontinuation method:
Let yk ∈ Rm+1, k ≥ 0, be a ’current’ point of the process on or near N . Then a suitablevector vk ∈ Rm+1 is chosen, such that the augmented mapping
Fk(y) :=
(F (y)
(vk)>(y − yk)
)(8.41)
has at y = yk an invertible derivative
DFk(yk) =
(DF (yk)(vk)>
). (8.42)
Then, for a sufficiently small ηk ∈ R1 the equation
Fk(y) =
(0ηk
)(8.43)
has a unique solution y = yk+1 ∈ N , which can be computed by some iterative process,e.g., a Newton-type method, started from some appropriately chosen point near N . Con-vergence properties of the iteration are used to control the choice of the stepsize to thenext predicted point.
With this four major decisions arise in the design of this type of continuation process,namely the
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(i) construction of the augmenting vectors vk,
(ii) design of the iterative process and its controls,
(iii) choice of the increments ηk,
(iv) prediction of the starting point of the iteration.
A frequent choice for vk is a normalized tangent vector of N at yk specified by
DF (yk)uk = 0, ‖uk‖2 = 1. (8.44)
An (un-normalized) tangent vector can be obtained as solution of a linear system(DF (yk)v>
)u =
(01
),
where v is any vector, such that the matrix is nonsingular. The choice of a tangent vectorfor vk also suggests the use of a point yk ± hkvk on the tangent line as start-point for theiterative process. Here the sign has to be chosen such that we follow the manifold N ina specified direction. H. B. Keller in 1978 introduced the name pseudo-arclength methodsfor continuation methods based on this choice of the direction vectors.
During the continuation process we are interested in detecting bifurcation points. Lety(s) = (x(s), τ(s)) ∈ Rm × R1 denote a smooth local parameterization of a ”branch”of the manifold N . Then we are looking for points where Re λ(s) = 0 for at least oneeigenvalue λ(s) of DxF (x(s), τ(s)). Typically, such a point is detected by means of a testfunction; that is, a smooth scalar function ψ(s) = ψ(y(s)), which has a simple zero atbifurcation points of a particular type. Hence, such a point is found between consecutivecontinuation points y(sk) and y(sk+1), when
ψ(sk)ψ(sk+1) < 0. (8.45)
Once a bifurcation point has been detected, it is computed as the solution of some aug-mented system of equations. Of special interest are here minimally extended systems(
F (x, τ)g(x, τ)
)= 0, (8.46)
constructed with a suitably chosen real function g. But, in many cases much largerextended systems are required. Theoretically, one might use in (8.46) the test functionitself; but, from the viewpoint of the required iterative process, this is often not a goodchoice.
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Note, that the construction of a test function has to work with a smooth branch of themanifold N specified by a suitable local parametrization. On the other hand, the contin-uation process may well jump between disjoint parts of N and hence produce consecutivepoints, that are not connected by a smooth branch. Figure 8.21 shows sketches of a fewpossible situations, such as a jump from the neighborhood of a limit point to another partof N , a jump across a gap between two neighboring branches, and – near a transcriticalbifurcation – to an intersecting branch.
Figure 8.21: Problems during a Continuation Process
It should also be noted, that at a limit point or Hopf-bifurcation point (x∗, τ∗)> ∈ Γwe have rankDF (x∗, τ∗) = m. Hence (x∗, τ∗)> is a well-defined point of N and a well-designed continuation process should pass through such points. But at a transcritical ora cusp bifurcation the mapping F is no longer a submersion and hence we should expectcomputational difficulties.
These situations need to be taken into account in the design of continuation methods, ingeneral, and of detection algorithms for bifurcation points, in particular. This cannot bediscussed here; instead we consider only a few techniques for detecting and computingsimple bifurcation points.
At a limit point y∗ = (x∗, τ)> on the one-dimensional manifold N the matrix DxF (x∗, τ∗)has a zero eigenvalue. Hence, if, as before, s 7→ y(s) = (x(s), τ(s))> denotes a localparametrization of a branch of N , then we can use the test function
ψ(s) = detDxF (x(s), τ(s)) (8.47)
for detecting limit points by means of (8.45). However, the test function (8.47) vanishesnot only at limit points, but, for example, also at a cusp bifurcation point. Therefore,a second test is needed to identify the particular bifurcation. This is not an untypicalsituation. In fact, in most cases, it is not possible to represent a singularity with onlyone test function, and one has to work with a decision-tree involving various tests forquantities to be zero or nonzero, etc.
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In the case of limit points with respect to the (scalar) parameter τ another characterizationderives from the fact, that at the point y∗ ∈ N the tangent space Ty∗N of N is perpendic-ular to the parameter space. Hence, suppose that the smooth function s 7→ u(s) ∈ Ty(s)N ,represents a normalized tangent vector of N along our local branch. Then we can use thetest function
ψ(s) = em+1u(s), DF (x(s), τ(s))u(s) = 0. (8.48)
If a simple zero of ψ has been found between two points computed by a continuationprocess, then one can employ very effectively some interpolation coupled with the correctoriteration of the process to evaluate the limit point (see, e.g., R. Melhem and W. Rheinboldt,Computing, 29, 1982, 201-226).
The condition (8.45) is designed to ensure that the test function has a simple zero betweenthe two computed points. But, for example, the test function (8.48) is zero also at aninflection point, which would not agree with the normal form (8.6) of limit points. Hence,once again, there is a need for excluding such a situation, which, for limit points, leads tothe use of the second derivative of F .
The detection of a Hopf bifurcation point (x(s∗), τ(s∗))> along our current branch of N ismore demanding. Such a point is characterized by a simple pair of purely imaginary eigen-values ±iγ of DxF (x(s∗), τ(s∗)). Hence this matrix must have two distinct eigenvalues,that sum to zero, which means, that the test function
ψ(s) :=m∏
j,`=1
[λj(s) + λ`(s)] (8.49)
vanishes for s = s∗.
Obviously, it would be costly to evaluate the entire spectrum of DxF (y(s)) at each com-puted point, in order to compute the test function (8.49). Here the Kronecker product ofmatrices can be used. Generally, the Kronecker product of any two matrices A,B ∈ Rn×n
is the n2 × n2 dimensional block-matrix
A⊗B =
a11B a12B · · · a1nB
a21B a22B · · · a2nB
· · · · · · · · · · · ·an1B an2B · · · annB
.
If σ(A) = λA1 , . . . , λAn and σ(B) = λB1 , . . . , λBn denote the spectra of the two matrices,then a fundamental result of C. Stephanos, 1900, states, that the eigenvalues of
(In ⊗A) + (B ⊗ In) =
A+ b11In b12In · · · b1nIn
b21In A+ b22In · · · b2nIn...
.... . .
...bn1In bn2In · · · A+ bnnIn
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are the n2 numbers λAj + λB` , j, ` = 1, . . . , n.
Since the determinant of any square matrix is the product of its eigenvalues, it follows,that the test function
must vanish at a Hopf bifurcation point. While the matrix has a relatively simple form,the computation of this test function is certainly costly for larger dimensions.
Note, that (8.50) vanishes not only at Hopf points, but also at points, where two realeigenvalues of DxF sum to zero, so called neutral saddles. Thus, once again, a secondarytest is needed, to distinguish the cases. In essence, this reduces to a check whether theeigenvalues, that sum to zero, have a nonzero real part or not. Here one often uses theso called Lyapunov coefficients, which represent a generalization of the eigenvalues at anequilibrium point.
For the computation of a Hopf bifurcation point (x∗, τ∗)> ∈ N we require a complexeigenvector corresponding to a purely imaginary eigenvalue pair; i.e.,
In order to normalize u and v, assume that c ∈ Rn is some vector, for which c>u = 0and c>v = 1, and consider the following augmented system of (3n + 2) equations in theunknowns x, u, v, γ and µ:
F (x, τ)DxF (x, τ)u+ γv
DxF (x, τ)v − γuc>u
c>v − 1
= 0. (8.51)
A. Griewank and G. W. Reddien (IMA J. Num. Anal, 3, 1983, 295-303) proved, that if(x∗, τ∗)> ∈ N is a Hopf bifurcation point, then – under certain conditions – the system(8.51) has a solution (x∗, u∗, v∗, γ∗, µ∗) and the Jacobian of the system is nonsingular atthis point. They also showed, that in the implementation of Newton’s method it is notnecessary to work with the full system, but that a Newton step can be computed by meansof three evaluations of the derivative DG(x, τ).
As we noted, the mapping F is a submersion at a limit point or Hopf-bifurcation point(x∗, τ∗)> ∈ Γ, and hence, in principle, the continuation process passes through such points
153
and their detection and computation does not cause numerical difficulties. But the situa-tion is different at a cusp bifurcation, where F is no longer a submersion. In particular,such points are sensitive to small changes of the mapping G of the original multi-parameterproblem (8.26). For example, the unfolded problem
x = ν + x(µ− x2), (x, µ, ν)> ∈ R3,
reduces for ν = 0 to the normal form (8.12) of the cusp bifurcation, where, locally nearthe origin of the (x, µ)-plane, the equilibrium set forms the familiar pitchfork of Figure8.9. But, even for very small, nonzero values of ν the pitchfork disappears as Figure 8.22shows.
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x2 (x − mu) + nu
nu = 0.01
Figure 8.22: Perturbed Cusp Bifurcation
Thus, numerically, we expect difficulties in detecting and computing the cusp point. Inpractice, it is likely, that the continuation path does not go through the point itself, butpasses it only in some neighborhood. For a precise computation we should work with theunfolded problem and construct augmented systems of equations involving the mappingG. Here, as before, minimally extended systems are of interest, which involve only asmany additional equations as the dimension d of the parameter space. In other words, ford = 2 we want to use a system of the formG(x, µ)
g1(x, µ)g2(x, µ)
= 0.
with real-valued functions g1 and g2. The construction of such functions is discussed, e.g.,by E. L. Allgower and H. Schwetlick, (Z. angew. Math. Mech. 77 (1997) 2, 83-97) andR.-X. Dai and W. C. Rheinboldt (SIAM J. Num. Anal. Vol 27, 2, 417–446).
Larger augmented systems are also widely discussed in the literature. For example for a
154
scalar problem g(x, µ, ν) = 0, g : R3 −→ R1, the system g(x, µ, ν)Dxg(x, µ, ν)Dxxg(x, µ, ν)
= 0
can be used. For g(x, µ, ν) := ν + x(µ− x2) this becomesν + µx− x3
µ− 3x2
−6x
= 0,
and it straightforward, that (0, 0, 0)> ∈ R3 is the unique solution, where the Jacobian isnonsingular and, hence, Newton’s method is locally convergent.
Suppose that for the reduced system F (x, τ) = G(x, µ0 + cτ) = 0 with the scalar param-eter τ we have computed a cusp bifurcation point (x∗, τ∗)>, where two solution branchesintersect. Then, a further task is the determination of the tangent directions of these inter-secting branches. Of course, for this we must work again on the d–dimensional equilibriummanifold M ⊂ Rm+d of the unfolded problem defined by G.
In the mechanical example of subsection (8.4.1) the cusp bifurcation occurs at the saddlepoint (0, 1, 0)> ∈ R3 of the two-dimensional equilibrium manifold. This reflects a moregeneral connection of these types of points with geometric properties of the equilibriummanifolds, notably its curvature behavior.
It is not feasible to discuss here the theoretical and computational aspects of the curvatureof manifolds. As observed by P. J. Rabier and W. C. Rheinboldt (Numer. Math. 57, 681-694 (1990)), it is computationally advantageous to work on Riemannian manifolds withthe second fundamental tensor rather than the Gaussian curvature tensor.
Suppose, that the dynamical system (8.26) now involves a mapping G : Ω ⊂ Rn :=Rm×Rd −→ Rm, which is twice continuously differentiable on the open set Ω. Of course,we still assume, that rankDG(x, µ) = m on M :=
(x, µ)> : G(x, µ) = 0
and hence
that M is a d-dimensional submanifold of Rn.
The tangent space TyM of M at y := (x, µ)> ∈M is the d-dimensional space kerDG(x, µ)and hence the orthogonal complement NyM = kerDG(y)⊥ is an m-dimensional linearsubspace of Rn. We may call NyM the normal space of M at y. Since G is a submersionon M , the restriction DG(y)|NyM is a nonsingular linear mapping of NyM onto itself.Therefore, for any y ∈M the bilinear mapping
Vy : TyM × TyM −→ NyM
Vy(u, v) = −[DG(y)|NyM ]−1D2G(y)(u, v) ∀u, v ∈ TyM,(8.52)
155
is well defined. More specifically, V represents a symmetric, vector-valued, two-covarianttensor on M and is a characterization of the mentioned second fundamental tensor of Min terms of the submersion G.
In order to connect this tensor with the curvature properties of M , consider a unit tangentvector u ∈ TyM , ‖u‖2 = 1 at a point y ∈M . Then z = Vy(u, u) ∈ NyM is a normal vectorof M at y and the affine plane y + span(u, z) intersects M in a curve γ. It then follows,that κ(y) := ‖z‖2 is the curvature of γ at y and, for nonzero κ(y), the unit vector z/‖z‖2is the principal normal of γ at y.
The representation (8.52) readily translates into a computational algorithm. Let thecolumns of
Q ∈ Rn×d, Q>Q = Id, rgeQ = TyM,
be an orthonormal basis of TyM . Then it is readily verified, that the matrix of the linearsystem (
DG(y)Q>
)z =
(D2G(y)(u, v)
0
), u, v ∈ TyM, (8.53)
is nonsingular and that the (unique) solution equals z = Vy(u, v). Thus, if functions forthe evaluation of the Jacobian matrix DG(y) and the vector D2G(y)(u, v) are available,then the computation of V rquires only the solution of a linear system of dimension n.
In the cited article by Rabier and Rheinboldt it is also shown how to approximate thecomponent Vy(u, u) of V by means of the indicated geometric characterization in terms ofcertain curves on M , without requiring the second derivative of G.
Without further motivation, we consider now a special class of bifurcations on M . A pointy∗ ∈M is a foldpoint of M with respect to the parameter space Π, if
Π ∩Ny∗M 6= 0. (8.54)
The dimension r := dim(Π ∩ Ny∗M) > 0 of the subspace is called the first singularityindex of y∗. Thus, at y∗ the directions of r parameter components are orthogonal to thetangent space. Generally, it is easily seen, that y∗ ∈ M is a foldpoint if DxG(x, µ) has azero eigenvalue. Thus, e.g., for a scalar-parameter problem limit points with respect tothe parameter are foldpoints.
We fix now the r parameter components corresponding to (8.54) to their value at thefoldpoint y∗. This gives a reduced problem in which only the remaining d− r parametercomponents are variable. Then the solution set of this reduced problem in the vicinity ofthe foldpoint equals
y ∈M : 〈y − y∗, z〉 = 0 ∀ z ∈ Π ∩Ny∗M, (8.55)
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and will be called the cutset of M at the foldpoint y∗.
As an example with a two-dimensional parameter space Π, consider again the real valuedequation
At the cusp–bifurcation y∗ = (0, 0, 0)> ∈ R3, we have Π := span(e2, e3), Ty∗M =span(e1, e2), and Ny∗M = span(e3). Thus (8.54) is satisfied and the cutset is the so-lution set of x(µ− x2) = 0; i.e., the familiar pitchfork.
Let z1, . . . , zr be an orthonormal basis of the subspace Π ∩ Ny∗M and extend it toan orthonormal basis z1, . . . , zm of all of Ny∗M . Then, under some non-degeneracyconditions (see, M. Buchner, J. Marsden and S. Schechter, J. Diff. Equ. 48, 1983,404-433,and J. P. Fink and W. C. Rheinboldt, SIAM J. Num. Anal., 24, 1987, 618-633) the formof the cutset is determined – locally near y∗ – by the nontrivial zeroes of the system ofthe r quadratic equations
〈Vy∗(u, u), zk〉 = 0, k = 1, . . . , r. (8.57)
The solutions are the desired bifurcation directions for M at the foldpoint y∗. We usuallyassume that these solutions are normed to Euclidean length one.
In the case of (8.56) a straightforward calculation shows that at y∗ = 0 ∈ R3 the equations(8.57) become
Hence the bifurcation directions are e1, e2 ∈ R3; that is, exactly the direction of theintersecting branches of the pitchfork at the origin.
As a second example, consider the two–point boundary value problem
−u′′ − µu+ au2 = 0, u(0) = u(π) = 0, (8.59)
where a is a constant and µ a parameter. After a straightforward discretization and inunfolded form, this leads to the finite dimensional system
Tx+ h2(aq(x)− µx) + νw = 0, x ∈ Rm, h =π
m+ l, (8.60)
where
q(x) =
x2
1...x2m
, T =
2 −1 0 · · · 0−1 2 −1 · · · 0...
. . . . . . . . ....
0 0 0 · · · 2
.
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Let λ1 < . . . < λm be the eigenvalues of T and w a normalized eigenvector of T corre-sponding to the eigenvalue λk for a given k. Then y∗ = (0, λk/h2, 0) turns out to be afoldpoint of (8.60) with respect to the space of the parameters µ and ν. Moreover, a closeranalysis shows that for odd values of k the bifurcation is transcritical, while for even k thebranches intersect at a right angle.
The system of quadratic equations (8.57) is readily solvable. If u1, . . . , ud is a basisof Ty∗M for which the components Vij = Vy∗(ui, uj), 1 ≤ i ≤ j ≤ d, of the secondfundamental tensor have been computed, then (8.57) reduces to a system of homogeneousquadratic equations
ξ>Akξ = 0, ξ ∈ Rd, ξ 6= 0, u = ξ1u1 + . . .+ ξdu
d, (8.61)
where the symmetric d× d matrices Ak have the elements
akij = 〈Vij , zk〉, 1 ≤ i ≤ j ≤ d, k = 1, . . . r.
This system of quadratic equations can have finitely many isolated solutions only in thecase d = r + 1. The case d = 2, (r = 1) is trivial, and also for d = 3, (r = 2) simplemethods are available. For larger dimensions there exists software for the numerical ofsystems of polynomial equations, which can be applied here.
