An Overview of Bifurcation Theory

Appendix A An Overview of Bifurcation Theory A.I Introduction In this appendix we want to provide a brief introduction and discussion of the con cepts of dynamical systems and bifurcation theory which has been used in the pre ceding sections . We refer the reader interested in a more thorough discussion of the mathematical results of dynamical systems and bifurcation theory to the books of Wiggins (1990) and Kuznetsov (1995). A discussion intended to more econom ically motivated problems of dynamical systems and chaos can be found in Medio (1992) or Day (1994). It is noteworthy here that chaos is only possible in the non autonomous case; that is, the dynamical system depends explicity on time; but not for the autonomous case where time does not enter directly in the dynamical sys tem (cf. Wiggins (1990». Since both models we have considered in the preceding sections are autonomous they cannot reveal any chaos . However, as the reader has seen, the dynamical systems we have derived reveal some bifurcations. Roughly spoken, bifurcation theory describes the way in which dynamical sys tem changes due to a small perturbation of the system-parameters. A qualitative change in the phase space of the dynamical system occurs at a bifurcation point, that means that the system is structural unstable against a small perturbation in the parameter space and the dynamic structure of the system has changed due to this slight variation in the parameter space. We can distinguish bifurcations into local, global and local/global types, where only the pure types are interesting here. Local bifurcations such as Andronov-Hopfor Fold bifurcations can be detected in a small 182 An Overview of Bifurcation Theory neighborhood ofa single fixed point or stationary point. In a similar way we can find bifurcations in limit cycles by the construction of a Poincare Map. Bifurcations that cannot be detected by an analysis of the properties of the fixed points as parameter values vary are called global (cf. Wiggins (1990) ; Kuznetsov (1995); Glendinning (1996)) . A.2 Some Definitions and Results Let us now introduce some important definitions and results for dynamical systems with continuous time T = IR with t E T. In this section we use the same notation and definitions as Kuznetsov (1995) . Since we are just interested in dynamical sys tems with a finite-dimensional state space, we define the state space ofthe dynamical system as X = R' of some finite dimension l. The evolution law that determines I the state of the variable Xt E IR at time t under the condition that the initial state value X o is known is defined by (A.l) which transforms the initial state X o E Rl into some state Xt E ]Rl at time t > 0, i.e. Xt = cr:/xo. We call the map cpt evolution operator. The formal definition of a dynamical system is given by I I Definition A.t. A dynamical system is a pair {IR , cpt}, where IR is the state space I I and cpt : IR ---+ IR is a family ofevolution operators satisfying the properties and (A.2) I where the first property in (A.2) means the identity map on IR , that is idx = x for all x E ]Rl and the second property means that cpt+s = cpt (cp sx) for all x E ]Rl. The second property states that we can evolve the dynamical system in the course of t + s units of time starting at x and get the same final state value as if we had evolved the system first over s units of time and then over the next t units of time starting on the same initial value x. Definition A.2. An orbit starting at Xo is an ordered subset ofthe state space ]Rl, (A .3) By an orbit through a point Xo we mean a curve in the state space ]Rl passing through the point Xo E ]Rl that is parameterized by the time t. We use here the term A.2 Some Definitions and Results 183 orbit and trajectory as synonyms. Note that in the literature of bifurcation theory sometimes both terms are distinguished (cf. (Wiggins, 1990, p. 2)). The simplest solutions we might expect for a dynamical system are solutions that remain constant over time. Such solutions are called equilibria, fixed points or stationary points. A formal definition is given below l O Definition A.3. A point X O E IR is called an equilibrium (fixed point) ifcptxo = X for all t > 0. A second solution type is a periodic orbit. Definition A.4. A cycle is a periodic orbit for each point X o E L o which satisfies cpt+Toxo = cpt xo with some To > °for all t > O. We understand under a periodic orbit the smallest possible amount of time To > 0 under which a system placed at a point X o will return exactly to this point. In this case the solution is periodic of period To and the system exhibits periodic oscillations. Now let us introduce a formal definition of a phase space plot which contains a lot of information about the dynamical behavior of a system under consideration. Definition A.5. The phase portrait ofa dynamical system is a partitioning ofthe state space into orbits. l Definition A.6. A (positively) invariant set ofa dynamical system {IR , cpt} is a l subset S C IR such that Xo E S implies cpt xo E S for all t > O. In the Definition A.6 we restrict ourselves to positive times. The definition means that a system which starts on the set S with initial state X o remains on the set S as time moves forward. Obviously, the orbit defined in Definition A.2 is an invariant set. Other examples of invariant sets are equilibria and cycles which are l l closed in IR , since the space IR is endowed with the Euclidean metric. Notice that the equilibria and the cycles are both orbits, the former are equilibria orbits whereas the latter are periodic orbits due to the Definition A.2. Definition A.7. An invariant set S C IRl is said to be a C", (r 2: 1) invariant man ifold iff S has the structure ofa C" differentiable manifold. Similarly, a positively invariant set S C IRl is said to be a C"; (r 2: 1) positively invariant manifold ifS has the structure ofa C" differentiable manifold. The term C" means continuously differentiable at every xES for every integer r 2: 1. Roughly spoken, Definition A.7 states that an invariant set S where the sur face is r-times continuously differentiable at every xES is called an O" invariant manifold. 184 An Overview of Bifurcation Theory To avoid the technical details concerning our understanding of the term "mani fold" we give here only a intuitive description of a manifold. A manifold is a set that has locally the structure of Euclidean space. The most familiar examples of mani l folds are the sphere or the torus which are embedded in IR • Furthermore, we obtain manifolds by the solution of a system of nonlinear equations, the so-called solution manifold. In contrary the cone is not a manifold. Although each point but one on the cone has a small Euclidean neighborhood but no neighborhood of the vertex has Eu clidean structure. For a more thorough treatment of a manifold we refer the reader to Guillemin and Pollack (1974) or Brocker and Janich (1990). In the case of dy namical systems we get two types ofmanifolds. Firstly, a linear vector subspace and secondly, a smooth surface embedded in IRl which can be represented locally with the implicit function theorem as a graph. Since the surface has no singular points if the surface is smooth and the derivative of the function representing the surface has maximal rank, the surface can be locally represented by a graph using the implicit function theorem (for more details see (Wiggins, 1990, p.14-15)). Once we have found a solution such as an equilibrium orbit or periodic orbit we want to find out if the solution is stable. An invariant set So must attract nearby solutions to classify as a stable orbit. Definition A.S. An closed invariant set So is called stable if (i) for any sufficient small neighborhood U ::J So there exists a neighborhood V ::J So such that <.pt x E U for all x E V and all t > 0; (ii) there exists a neighborhood U ::J So such that <.pt x ....... Sofor all x E U as t ....... +00. Property (i) in Definition A.8 is the so-called Lyapunov stability. This property states that an invariant set Soin order to be Lyapunov stable must assure that nearby orbits cannot leave its neighborhood. Property (ii) is the so-called asymptotic stabil ity which states that an invariant set So attracts nearby orbits as time goes by. Definition A.8 gives us a clear designation of stability. Now, we provide a suffi cient condition for an equilibrium X o to be stable by the following theorem . l Theorem A.I. Consider a dynamical system {IR , <.pt} defined by x = j(x), (A.4) where j is smooth. Suppose that it has an equilibrium xO,i.e.f(xO) = 0, and denote by A the Jacobian matrix of j(x) evaluated at the equilibrium, A = j ",(XO). Then XOis stable ifall eigenvalues J.Ll, J.L2, ••• , J.L l ofA satisfy Re J.L < O. A.2 Some Definitions and Results 185 Proof Hint : The theorem can be proved by construction of a Lyapunov function.

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