Preprints by Clark Robinson

Book: "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.

Book: "Dynamical Systems: Continuous and Discrete"

Information about the undergraduate textbook

Preprint 2014: Topological Decoupling near Planar Parabolic Orbits

In two different 3 body problems, oscillatory orbits have been shown to exist for the three-body problem in Celestial Mechanics: Sitnikov, Alekseev, Moser, and McGehee considered a spatial problem which had one degree of freedom; Easton, McGehee, and Xia considered a planar problem which had at least three degrees of freedom. Both situations involve analyzing the motion as one particle with mass m_3 goes to infinity while the other two masses stay bounded in elliptic motion. Motion with m_3 at infinity corresponds to a periodic orbit in the first problem and the Hopf flow on S^3 in the second problem, both of which are normally degenerately hyperbolic. The proof of the existence of oscillatory orbits uses stable and unstable manifolds for these degenerate cases. In order to get the symbolic dynamics which shows the existence of oscillation, the orbits which go near infinity need to be controlled for an unbounded length of time. In this paper, we prove that the flow near infinity for the Easton-McGehee example with three degrees of freedom is topologically equivalent to a product flow, i.e., a Grobman-Hartman type theorem in the degenerate situation.

Preprint 2012: Diffusion along Transition Chains of Invariant Tori and Aubry-Mather Sets

We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus, (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map, (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains, i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence, (iv) the transition chains of tori are interspersed with gaps created by resonances, (v) within each gap there is a designated, finite collection of Aubry-Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry-Mather sets within each gap, in any prescribed order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.

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.
Appeared in Discrete and Continuous Dynamical Systems, Series S 1 (2008), pp 647 - 652,

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.
Appeared in Discrete and Continuous Dynamical Systems, Series S 2 (2009), pp 393 - 416.

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.
Appeared in Qualitative Theory of Dynamical Systems, 7 (2008), pp 227 - 236.

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.
SIAM J. Math. Analysis, volume 32 (2000), pages 119–141.

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.
Appeared in Contemporary Mathematics, volume 198 (1996), pages 45–53.

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.

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.
Appeared in Monografias de Matematica 4, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brazil (1971), Chapter 9.

Reprint: "Differentiable Conjugacy Near Compact Invariant Manifolds"

This is a retyping of the paper that appeared in {i>Bolletim da Sociedade Brasileira de Matematica 2 (1971). I have made slight changes in the wording in a few places as well as reformated the paper. I contains results related to differentiable linearization of a diffeomorphism near a normally hyperbolic invariant manifold. There are also statements about when the set of strong stable manifolds of points is a differentiable folitaion of the stable manifold of the invariant manifold.
Boletim de Sociedade Brasileira de Matematica 2 (1971), 33–44.

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