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Structural Stability and Bifurcations

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Structural Stability and Bifurcations 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 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 x0. By ODE-theory, the solution of (8.1) depends continuously on the initial point, as is obvious for the solution x(t) = exp(tA)x0 of the linear problemx _ = Ax, x(0) = x0. 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 k+1 x = g(xk); 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 1 z± := (µ + 1) ± p(µ + 1)(µ − 3); 2µ 130 that is, the attractor consisting of the single point fx∗∗g has become an attractor-set fz+; z−g of cardinality two. Such sudden changes in the dynamics caused by smooth alterations of a system occur in many problems. Bridges bend continuously under an increasing load, but then suddenly break. 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 structure suddenly ruptures in an earthquake. A living cell suddenly changes its reproductive rythm and doubles and redoubles cancerously. In each example, continuous changes suddenly lead to an abrupt change in the structure of the solution field. Accordingly, these situations are also called structural instabilities. 8.2 Structural Stability In this section we outline { without proofs { the basic ideas behind the concept of structural stability. In essence, a dynamical systems (8.1) is structurally stable if its phase portrait remains topologically invariant under small perturbations of the mapping G. This is illustrated by the two pairs of pictures in Figures 8.1 and 8.2. The two phase portraits 8.1 can be continuously deformed into each other, while, in the second pair, the connection between 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 equivalence n of mappings. If x 7! Xx is a coordinate transformation on R defined by some nonsingular n×n n×n −1 X 2 R , then any matrix A 2 R transforms into X AX. Analogously, suppose n n n that R is mapped onto itself by a homeomorphism h : R −! R ; i.e., a bijective, n n continuous mapping with a continuous inverse, then a nonlinear mapping f : R −! R is transformed into the mapping g = h−1fh. Accordingly, the mappings f and g are 131 n called topologically equivalent, if for some homeomorphism h on R the following diagram is commutative: f n n R / R h h g n n R / R For the discussion of the topological equivalence of phase portraits of dynamical systems it 1 n n is customary to use vector fields. Recall, that for a C -mapping G : R −! R , the unique n solution of (8.1) passing through x 2 R has at x the tangent vector G(x). Accordingly, 1 n n we will view G as defining a C vector field on R to be denoted by (R ;G). The solutions of (8.1) are the orbits of this vector field. n n 1 2 n Two vector fields (R ;G1), (R ;G2) are topologically equivalent near x ; x 2 R , if there n 1 2 exist neighborhoods U1; U2 ⊂ R of x ; x and a homeomorphism h : U1 −! U2 with 1 2 h(x ) = x , which maps each orbit of the first vector field in U1 onto an orbit of the second vector field in U2, preserving the direction of time. n n×n A linear vector field (R ;A) is hyperbolic, if A 2 R has no eigenvalues with zero real part. Here, the next result shows, that a characteristic quantity is the number of eigenvalues with negative real part, called the index of A: n n 8.2.1. The hyperbolic (linear) vector fields (R ;A) and (R ;B) are topologically equivalent if 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 ! ! −1 −3 2 0 A = ;B = : −3 −1 0 −4 Then we have ! 1 1 −1 B = XAX−1;X = p 2 1 1 2 and hence X exp(tA) = exp(tB)X. Since x 7−! Xx is a homeomorphism on R , the 2 2 vector fields (R ;A) and (R ;B) are indeed topologically equivalent. n As indicated, structural stability of a vector field (R ;G) requires, that all sufficiently small perturbation of G produce topologically equivalent vector fields. For this we have to specify what perturbations are allowed. In the following definition we work with the normed linear 1 n n space (C (Ω); k · k1) of all continuously differentiable mappings G :Ω ⊂ R −! R on a given open set Ω, with the norm 1 kGk1 := sup kG(x)k2 + sup kDG(x)k2; 8G 2 C (Ω): x2Ω x2Ω 132 0 1 8.2.2 Definition. For G 2 C (Ω) the vector field (Ω;G0) is structurally stable in 1 1 0 (C (Ω); k · k1), if there is an > 0, such that for all G 2 C (Ω) with kG − G k1 < , 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 a considerable mathematical depth. n For a linear vector field (R ;A) defined by some matrix A it is natural to restrict the n×n perturbations to the normed linear space (R ; k · k2). Recall that the eigenvalues of n×n A 2 R depend continuously on the elements of the matrix. Hence, if the vector n n×n field (R ;A) is hyperbolic, then there exists > 0, such that for each B 2 R with n kB − Ak2 < , the vector field (R ;B) is also hyperbolic. Moreover, it can be shown, that, for sufficiently small , the index of any such B is the same as that of A. Hence 8.2.1 implies, that the vector fields of A and B are topologically equivalent. This leads to the following result: n n×n 8.2.3. A linear vector field (R ;A) is structurally stable in (R ; k · k2) if and only if A is hyperbolic. For nonlinear mappings the Hartman-Grobman theorem provides a connection with this linear result: n 1 8.2.4. Let Ω ⊂ R be an open set and for given G 2 C (Ω) denote by Φ the flow of the dynamical system (8.1). Suppose, that x∗ 2 Ω is a hyperbolic equilibrium of (8.1). Then ∗ there exists a homeomorphism h of an open neighborhood U1 ⊂ Ω of x onto an open n ∗ neighborhood U2 ⊂ R of the origin, such that h(x ) = 0 and ∗ h(Φ(t; x)) = exp(tDG(x )) h(x) 8t 2 J ; x 2 U1; (8.3) where the open interval J , 0 2 J , is locally determined. Under the conditions of the theorem, (8.3) implies that the homeomorphism h maps each ∗ solution 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, locally ∗ n near the hyperbolic equilbrium x , the vector field (R ;G) is topologically equivalent with n ∗ the hyperbolic linear vector field (R ; DG(x )) near the origin. 133 8.3 Introduction to Bifurcations Bifurcation theory is the mathematical study of changes in the topological properties of the phase portraits of a family of dynamical systems. We sketch only some elementary ideas underlying this large topic. Many physical, chemical, and biological problems lead to parameter-dependent dynamical systems 0 n m d m x_ = G(x; µ); x(0) = x ;G :Ω ⊂ R := R × R −! R ; n = m + d; (8.4) where G 2 C1(Ω) on an open set Ω. Then, in essence, a bifurcation occurs at a parameter value 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, bifurcations identify the failure of the structural stability of the vector field (Ω;G(·; µ)) for specific parameter values. For instance, the planar system ! µ −1 x_ = A(µ)x; A(µ) := ; (8.5) 1 µ has the origin as an equilibrium for each value of µ.
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