Preprints by Clark Robinson

Book: Contents, Additions, and Errata for "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos"

This book is a mathematical graduate text in Dynmamical Systems. The table of contents, a list of know errata, and a few additional homework problems are given. Finally, publication data and publisher address is given.

Reprint June 2008: "Uniform Subharmonic Orbits for Sitnikov Problem"

We highlight the argument in Moser's monograph that the subharmonic periodic orbits for the Sitnikov problem exist uniformly for the eccentricity sufficiently small. We indicate how this relates to the uniformity of subharmonic periodic orbits for a forced Hamiltonian system of one degree of freedom with a symmetry.

Reprint May 2008: "Obstruction argument for transition chains of tori interspersed with gaps"

We consider a dynamical system that exhibits a two-dimensional normally hyperbolic invariant manifold diffeomorphic to an annulus. We assume that in the annulus there exist transition chains of invariant tori interspersed with Birkhoff zones of instability. We prove the existence of orbits that follow the transition chains and cross the Birkhoff zones of instability.

Reprint September 2006: "Shadowing orbits for transition chains of invariant tori alternating with gaps"

We describe a topological method for proving the existence of diffusing orbits that shadow transition chains of invariant tori and cross Birkhoff zones of instability. This approach is motivated by the large gap problem in Hamiltonian systems.

Preprint 2005: "What is a Chaotic Attractor?"

We discuss various definitions of a chaotic attractor and give several types of examples of attractors that should not be called chaotic attractors.

Reprint April 2001: "Symbolic Dynamics for transition tori"

This paper considers transition tori of Arnold which have transverse heteroclinic intersections. Using a sequence of correctly aligned windows, we prove the existence of an orbit which comes near an arbitrary infinite sequence of these transition tori. By our proof, we aim to clarify the reason that it is possible to take an infinite or bi-infinite chain of such tori. This fact is implicit in the treatment of obstructing sets used by Arnold, but does not seem transparent. Our treatment of the transition tori is based on an earlier article of Easton about this topic, but removes some of the undesirable assumptions which he made.

The explanation is given in terms of correctly aligned windows introduced by Easton. Recently, these ideas have been developed using the Conley index by Carbinatto, Kwapisz, and Mischaikow and several others. We use the statement given by M.\ Gidea.

Besides seeking to clarify the situation for transition tori, we hope to make people in Celestial Mechanics aware of the results on topological horseshoes without the necessity of proving the differentiably transverse homoclinic intersection.

Paper: "Melnikov Method for Autonomous Hamiltonians"

This paper presents the method of applying the Melnikov method to autonomous Hamiltonian systems in dimension four. Besides giving an application to Celestial Mechanics, it discusses the problem of convergence of the Melnikov function and the derivative of the Melnikov function.

Preprint: "The Subharmonic Melnikov Method"

This preprint considers the subharmonic Melnikov method as applied to an autonomous Hamiltonian systems with two degrees of freedom which completely decouples for epsilon equal to zero. The question is which periodic orbits persist for epsilon not equal to zero.

Preprint: "Nonsymmetric Lorenz Attractors from a Homoclinic Bifurcation"

This preprint considers a bifurcation of a flow in three dimensions from a double homoclinic connection to a fixed point satisfying a resonance condition between the eigenvalues. For correctly chosen parameters in the unfolding, we prove that there is a transitive attractor of Lorenz type. We do not assume any symmetry condition, so we need to discuss nonsymmetric one dimensional Poincare maps with one discontinuity and absolute value of the derivative always greater than one.

Reprint: "Stability of Anosov diffeomorphisms" with A. Verjovsky

This is a retyping of the paper written by myself and A. Verjovsky which appeared in "Seminario de Sistemas Dinamicos" edited by J. Palis, Monografias de Matematica 4 (1971), IMPA Rio de Janeiro Brazil, Chapter 9. It contains a proof of the stability of Anosov diffeomorphisms following the idea of Mather. Mather's original proof had a gap because the composition map is not continuously differentiable. This paper supplies the necessary uniformities to make this proof work.

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