Preprints by Clark Robinson
Titles:
 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 threebody 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
EastonMcGehee example with three degrees of freedom is topologically
equivalent
to a product flow, i.e., a GrobmanHartman type theorem in the
degenerate situation.
 Preprint 2012:
Diffusion along Transition Chains of Invariant Tori and
AubryMather 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
twodimensional annulus, (ii) the restriction of the dynamics to the annulus is
an area preserving monotone twist map, (iii) the annulus contains sequences
of invariant onedimensional 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 AubryMather sets. Under these
assumptions, there exist trajectories that follow the transition chains, cross
over the gaps, and follow the AubryMather 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 twodimensional
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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