Discrete and Continuous Dynamical Systems
April 2006 , Volume 14 , Issue 2
Topological and Analytical Shadowing Techniques
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This volume includes contributions of the invited participants to the special session on "Topological and Analytical Shadowing Techniques'', part of the AIMS' Fifth International Conference on Dynamical Systems and Differential Equations, Pomona, California, June 16 - 19, 2004. The volume also includes contributions of invited scientists who were unable to participate to the conference.
Shadowing techniques refer to methods for detection of true orbits closely following pseudo-orbits in dynamical systems. The first shadowing results were proved by Anosov (1967) and Bowen (1975), in the case of hyperbolic systems. Many systems of interest, however, are not hyperbolic. In such cases, one may employ topological or analytical methods to prove --- either rigorously or numerically --- the existence of orbits with prescribed itineraries.
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We consider a Hamiltonian system modeling the plane restricted elliptic 3 body problem with one of the masses small and prove the existence of periodic and chaotic orbits shadowing chains of collision orbits. Periodic orbits of this type were first studied by Poincaré for the non-restricted 3 body problem. The present paper contains general results which hold for time periodic Hamiltonian systems with a small Newtonian singularity. Applications to celestial mechanics will be given in a subsequent paper.
In this paper a method for finding homoclinic and heteroclinic connections between Lyapunov orbits using invariant manifolds in a given energy surface of the planar restricted circular three body problem is developed. Moreover, the systematic application of this method to a range of Jacobi constants provides a classification of the connections in bifurcation families. The models used correspond to the Sun-Earth+Moon and the Earth-Moon cases.
Covering relations are a topological tool for detecting periodic orbits, symbolic dynamics and chaotic behavior for autonomous ODE. We extend the method of the covering relations onto systems with a time dependent perturbation. As an example we apply the method to non-autonomous perturbations of the Rössler equations to show that for small perturbation they possess symbolic dynamics.
We use topological methods to investigate some recently proposed mechanisms of instability (Arnol'd diffusion) in Hamiltonian systems.
In these mechanisms, chains of heteroclinic connections between whiskered tori are constructed, based on the existence of a normally hyperbolic manifold $\Lambda$, so that: (a) the manifold $\Lambda$ is covered rather densely by transitive tori (possibly of different topology), (b) the manifolds $W^\s_\Lambda$, $W^\u_\Lambda$ intersect transversally, (c) the systems satisfies some explicit non-degeneracy assumptions, which hold generically.
In this paper we use the method of correctly aligned windows to show that, under the assumptions (a), (b), (c), there are orbits that move a significant amount.
As a matter of fact, the method presented here does not require that the tori are exactly invariant, only that they are approximately invariant. Hence, compared with the previous papers, we do not need to use KAM theory. This lowers the assumptions on differentiability.
Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.
A shadow is an exact solution to an iterated map that remains close to an approximate solution for a long time. An elegant geometric method for proving the existence of shadows is called containment, and it has been proven previously in two and three dimensions, and in some special cases in higher dimensions. This paper presents the general proof using tools from differential and algebraic topology and singular homology.
We show how the shadowing property can be used in connection to rotation sets. We review the concept of periodic chains and the shadowing rotation property, and study a class of diffeomorphisms with invariant sets that have such property. In particular, we consider invariant sets that arise from homoclinic and heteroclinic connections for twist maps in higher dimensions. As a consequence, we can show the existence of a family of twist maps, each one with an open set of rotation vectors that are realized by points that are close to a fixed point.
Let $f:X\to X$ be the restriction to a hyperbolic basic set of a smooth diffeomorphism. If $G$ is the special Euclidean group $SE(2)$ we show that in the set of $C^2$ $G$-extensions of $f$ there exists an open and dense subset of stably transitive transformations. If $G=K\times \mathbb R^n$, where $K$ is a compact connected Lie group, we show that an open and dense set of $C^2$ $G$-extensions satisfying a certain separation condition are transitive. The separation condition is necessary.
We extend some previous results concerning the relationship between weak stability properties of the geodesic flow of manifolds without conjugate points and the global geometry of the manifold. We focus on the study of geodesic flows of compact manifolds without conjugate points satisfying either the shadowing property or topological stability, and we prove for three dimensional manifolds that under these assumptions the fundamental groups of certain quasi-convex manifolds have the Preissmann's property. This result generalizes a similar one obtained for manifolds with bounded asymptote.
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