Chaos Near Resonance

verfasst von: G. Haller

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

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Resonances are ubiquitous in dynamical systems with many degrees of freedom. They have the basic effect of introducing slow-fast behavior in an evolutionary system which, coupled with instabilities, can result in highly irregular behavior. This book gives a unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, a general finite dimensional theory of homoclinic jumping is developed and illustrated with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context. Previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds are described. The results are applied to a variety of different problems, which include applications from beam oscillations, surface wave dynamics, nonlinear optics, atmospheric science and fluid mechanics. The theory is further used to study resonances in Hamiltonian systems with applications to molecular dynamics and rigid body motion. The final chapter contains an infinite dimensional extension of the finite dimensional theory, with application to the perturbed nonlinear Schrödinger equation and coupled NLS equations.

Inhaltsverzeichnis

Frontmatter

1. Concepts From Dynamical Systems

Abstract

This chapter is a survey of facts and results from dynamical systems theory that are used throughout the book. Our primary goal is to fix notation and terminology for later chapters, and recall some aspects of the theory that are usually not discussed in introductory books. For the sake of easier reading, we keep the discussion informal, avoiding technical notation and proofs as much as possible. While most omitted details can be found in graduate level introductory books to dynamical systems, some of the more advanced topics are covered only in research articles. In either case, we shall supply references that treat the subject in greater depth. In addition, we encourage the reader to use Appendices A and B as references on the basics of differential geometry and functional analysis.

G. Haller

2. Chaotic Jumping Near Resonances: Finite-Dimensional Systems

Abstract

The evolution of physical problems can be decomposed into slow and fast components near resonances. The slow variables are typically slowly varying amplitudes and nearly resonant phase combinations, while the fast variables describe the remaining degrees of freedom. In many cases there are sets of solutions on which the fast oscillatory components vanish. These solutions then form a slow manifold, whose geometry and stability determines the nature of the dynamics near the resonance. The slow variation on this manifold is due to some small detuning or perturbation from the exact resonant states, which form a resonant,or critical, manifold. Viewing slow manifolds as small perturbations of critical manifolds is the main idea of geometric singular perturbation theory (cf. Section 1.18), and this is the approach we shall take in this book.

G. Haller

3. Chaos Due to Resonances in Physical Systems

Abstract

Here we survey several applications of the theory developed in Chapter 2. The problems are picked from rigid body dynamics, fluid mechanics, atmospheric science, and nonlinear optics. Almost all physical models we study are dissipative and have been noted to display complex dynamics, yet the usual Melnikov method (see Section 1.27) does not reveal any chaotic behavior when applied to perturbations of their integrable limits. Rather, most of these models tend to develop their chaotic attractors, as the perturbation increases, around orbits homoclinic to slow or partially slow manifolds. While the techniques are still missing for the analytic study of such attractors, the attracting nature of heteroclinic cycles near resonances can be shown explicitly in some examples (cf. Section 3.7).

G. Haller

4. Resonances in Hamiltonian Systems

Abstract

In this chapter we consider resonances in multi-degree-of-freedom Hamiltonian systems. The resonances are assumed to occur either among the eigenvalues of an elliptic equilibrium (Section 4.1) or among the frequencies of an invariant torus (Sections 4.3 and 4.5). In all cases, slow manifolds or partially slow manifolds exist in local normal forms computed near the resonant object. For resonant equilibria, fast transients among slow manifolds turn out to be responsible for resonance energy transfer, while for invariant tori, irregular motion across resonances can be shown to exist. We examine these phenomena in more detail in a model of the classical water molecule (Section 4.2) and in two concrete examples of rigid body systems (Sections 4.4 and 4.5). The first of the latter examples involves tori of elliptic stability type, while the second one deals with elliptic—hyperbolic tori.

G. Haller

5. Chaotic Jumping Near Resonances: Infinite-Dimensional Systems

Abstract

In this chapter we study the global effect of resonances in near-integrable evolution equations. As in Chapter 2, the unperturbed equation is assumed to have a set of resonant states, which appear as fixed points in a frame moving with the forcing. The resonant states are connected through a heteroclinic network of orbits before perturbation and admit infinitely many neutrally stable directions. These directions correspond to infinitely many Fourier modes exhibiting fast oscillations around the finitely many slowly varying modes. As earlier, we seek fast multipulse transitions between the resonant states after perturbation. These recurrent jumping solutions are again expected to cause irregular or chaotic dynamics, but their exact implications are mostly unknown as of yet. While we discuss some partial results in this direction, future developments in infinite-dimensional chaotic dynamics will still have a lot to offer to this theory.

G. Haller

Backmatter

Titel: Chaos Near Resonance
verfasst von: G. Haller
Verlag: Springer New York
Electronic ISBN: 978-1-4612-1508-0
Print ISBN: 978-1-4612-7172-7
DOI: https://doi.org/10.1007/978-1-4612-1508-0

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Concepts From Dynamical Systems

2. Chaotic Jumping Near Resonances: Finite-Dimensional Systems

3. Chaos Due to Resonances in Physical Systems

4. Resonances in Hamiltonian Systems

5. Chaotic Jumping Near Resonances: Infinite-Dimensional Systems

Backmatter