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DYNAMICS REPORTED reports on recent developments in dynamical systems. Dynamical systems of course originated from ordinary differential equations. Today, dynamical systems cover a much larger area, including dynamical processes described by functional and integral equations, by partial and stochastic differential equations, etc. Dynamical systems have involved remarkably in recent years. A wealth of new phenomena, new ideas and new techniques are proving to be of considerable interest to scientists in rather different fields. It is not surprising that thousands of publications on the theory itself and on its various applications are appearing DYNAMICS REPORTED presents carefully written articles on major subjects in dynam­ ical systems and their applications, addressed not only to specialists but also to a broader range of readers including graduate students. Topics are advanced, while detailed expo­ sition of ideas, restriction to typical results - rather than the most general ones - and, last but not least, lucid proofs help to gain the utmost degree of clarity. It is hoped, that DYNAMICS REPORTED will be useful for those entering the field and will stimulate an exchange of ideas among those working in dynamical systems Summer 1991 Christopher K. R. T Jones Drs Kirchgraber Hans-Otto Walther Managing Editors Table of Contents Hyperbolicity and Exponential Dichotomy for Dynamical Systems Neil Fenichel 1. Introduction . . . . . . . . . . . . . . . . . . I 2. The Main Lemma . . . . . . . . . . . . . . . . 2 3. The Linearization Theorem of Hartman and Grobman 5 4. Hyperbolic Invariant Sets: €-orbits and Stable Manifolds 6 5.



Hyperbolicity and Exponential Dichotomy for Dynamical Systems

The aim of this paper is to show that many of the interesting topological consequences of hyperbolicity follow from just one lemma about exponential dichotomy. Our main lemma asserts that there are local stable manifolds and local unstable manifolds associated with a sequence of maps which are close to hyperbolic linear maps, and that certain local stable manifolds and local unstable manifolds have unique points of intersection. Our main lemma also includes detailed estimates for the positions of the local stable and unstable manifolds, and for the behavior of orbits in the local stable and unstable manifolds. This is an exponential dichotomy result because the hypotheses guarantee that orbits diverge either in the forward direction or in the backward direction. In applications the maps represent a given dynamical system, or dynamical systems in a neighbor hood of a given dynamical system, in local coordinates.
Neil Fenichel

Feedback Stabilizability of Time-Periodic Parabolic Equations

Differential equations (partial and ordinary) have traditionally occupied a prominent place within mathematics. One of the main reasons for this is the fact that they have served as models for the evolution of systems arising in physics, chemistry, biology and various other disciplines. However, the traditional topics in the theory of differential equations do not encompass many important problems which fall into the realm of what is today known as control theory. In this paper we describe the basis for a geometric theory of time-periodic abstract linear control systems of ‘parabolic’ type, concentrating on stabilization by feedback, and discuss some applications to second order time-periodic parabolic initial-boundary value problems on bounded domains. A theory of this kind has already been developed in the finite dimensional case (cf. [17], [16]) and in infinite dimensions when the system is autonomous (cf. [7], [40]). For the time-periodic infinite dimensional case some first steps have been made by A. Lunardi (cf. [33]). But before we embark on a description of our results we give an example as motivation for the kind of problems in control theory we shall be concerned with.
Pablo Koch Medina

Homoclinic Bifurcations with Weakly Expanding Center Manifolds

Interaction of homoclinic bifurcation and bifurcation on the center manifold is studied. We show that the occurrence of different types of solutions near the homoclinic orbit is determined asymptotically by a reduced system on the center manifold. The method is applied to cases where the center manifold is one- or two-dimensional. When the center manifold is one-dimensional, we can obtain all the solutions near the homoclinic orbit. When a Hopf bifurcation occurs on a two-dimensional center manifold, the system can have infinitely many periodic and aperiodic solutions. These solutions disappear in a manner predicted by the reduced system when the perturbation term is increased. We prove that certain periodic and aperiodic solutions disappear through inverse period doubling or saddle-node bifurcation.
Xiao-Biao Lin

Homoclinic Orbits in a Four Dimensional Model of a Perturbed NLS Equation: A Geometric Singular Perturbation Study

This paper contains a detailed and explicit study of an important mechanism leading to chaotic dynamics in a perturbation of the integrable nonlinear Schrödinger equation, a perturbation which contains damping and driving terms. Specifically, we study, both analytically and numerically, homoclinic and chaotic behavior in a two mode ode truncation. First, we summarize recent results of numerical experiments which establish the presence of irregular (chaotic) temporal behavior in this two mode system. We then establish the existence of an orbit which is homoclinic to a fixed point q - a saddle in a “resonance band”. This analytical argument begins from a representation of certain invariant manifolds by fibers, a representation which we explicitly illustrate with concrete examples. The existence of the homoclinic orbit then follows from a Melnikov argument combined with methods from geometric singular perturbation theory. Next these homoclinic orbits are constructed, and studied, numerically with a bifurcation algorithm. These numerical studies find some members of the family of homoclinic orbits which were predicted by the theory. Finally, the existence of a chaotic symbol dynamics is established through a “Smale horseshoe” which is shown to exist near the homoclinic orbit.
David W. McLaughlin, E. A. Overman, Stephen Wiggins, C. Xiong


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