Robert L. Devaney

Reversible diffeomorphisms and flows

We generalize the classical notion of reversibility of a mecfcaisietl system. The generic qualitative properties of symmetric orbits of such sx-ttco* are studied using transversality theory. In particular, we prove analogues alette closed orbit, Liapounov, and homoclinic orbit theorems for R-reversible

Homoclinic orbits in Hamiltonian systems

The object of this paper is to study the orbit structure of a Hamiltonian system in a neighborhood of a trajectory which is doubly asymptotic to an equilibrium solution, i.e., an orbit which lies in the intersection of the stable and unstable manifolds of a critical point. Such an orbit is called a homoclinic orbit. For diffeomorphisms, the analogous situation is fairly well understood. By a theorem of Smale [7], in every neighborhood of a transversal homoclinic point of a periodic point, there is a compact invariant set on which some iterate of the diffeomorphism is topologically conjugate to the Bernoulli shift on N symbols. Via Poincare maps on local transversal sections, there is thus an analogous result for hyperbolic closed orbits of vector fields. For critical points of vector fields, however, the situation is somewhat different. In the first place, by the Kupka-Smale theorem [4], the existence of orbits doubly asymptotic to an equilibrium point is not generic. In fact, the set of vector fields which admit homoclinic orbits at critical points is of the first Baire category in the set of all smooth vector fields. This follows since the sum of the dimensions of the stable and unstable manifolds at the critical point is at most equal to the dimension of the manifold itself. Hence, the stable and unstable manifolds cannot intersect transversely in a one-dimensional (homoclinic) orbit. For Hamiltonian systems, however, this is no longer true. The stable and unstable manifolds of hyperbolic critical points must both lie in a fixed energy surface, and by [lo], th e y are generically transverse within that surface. Since the codimension of energy surfaces is one, it follows that the stable and unstable mnaifolds may thus intersect transversely within the energy surface along a homoclinic orbit. Hence these orbits cannot be removed by small perturbations of the Hamiltonian. We remark that, with certain restrictions on the characteristic exponents at the critical point, Silnikov [5] has found horseshoe mappings similar to those of Smale near homoclinic orbits of vector fields. However, his assumptions

Shift automorphisms in the Hénon mapping

We investigate the global behavior of the quadratic diffeomorphism of the plane given byH(x,y)=(1+y−Ax2,Bx). Numerical work by Hénon, Curry, and Feit indicate that, for certain values of the parameters, this mapping admits a “strange attractor”. Here we show that, forA small enough, all points in the plane eventually move to infinity under iteration ofH. On the other hand, whenA is large enough, the nonwandering set ofH is topologically conjugate to the shift automorphism on two symbols.

Dynamics of exp ( z )

We describe the dynamical behaviour of the entire transcendental function exp( z ). We use symbolic dynamics to describe the complicated orbit structure of this map whose Julia Set is the entire complex plane. Bifurcations occurring in the family c exp( z ) are discussed in the final section.

Chaos and Fractals: The Mathematics behind the Computer Graphics

Overview: Dynamics of simple maps by R. L. Devaney Nonlinear oscillations and the Smale horseshoe map by P. J. Holmes Fractal basin boundaries and chaotic attractors by K. T. Alligood and J. A. Yorke Julia sets by L. Keen The Mandelbrot set by B. Branner Introduction to fractals by J. Harrison Iterated function systems by M. F. Barnsley.

Singularities in Classical Mechanical Systems

Singularities in the equations of motion of a classical mechanical system usually play a dominant role in the global phase portrait of the system. By a singularity we mean a point or set of points where the system is undefined, as in the case of a collision between two or more of the particles in the n-body problem. Such singularities often lead to a complicated global orbit structure. Not only do certain solutions tend to run off the phase space, but also nearby solutions tend to behave in an erratic or unpredictable manner. Numerical studies of such systems are often inconclusive because of this erratic behavior. And power series or other analytic techniques often yield only a very local description of solutions near the singularity, one which gives no hint of the global complexity of the system.

Triple collision in the planar isosceles three body problem

We employ a device due to McGehee to discuss the qualitative behavior of orbits which reach or come close to triple collision in a special case of the planar three body problem. We show that there exist infinitely many orbits which both begin and end in triple collision. Nearby orbits behave in different ways depending on whether they pass close to the collinear or equilateral triangle central configuration. Finally, we discuss a new type of orbit in the three body problem which we call “billiard shots”.

Collision Orbits in the Anisotropic Kepler Problem

The anisotropic Kepler problem is a one-parameter family of classical mechanical systems with two degrees of freedom. When the parameter # = 1, we have the well known Kepler or central force problem. As # increases beyond 1, we introduce more and more anisotropy into the Kepler problem. As we show below, this changes the orbit structure of the system dramatically. When p = 1, the system is completely integrable and the orbit structure is well understood. With the exception of certain collision orbits, all orbits are closed and lie on two dimensional tori in the case of negative total energy. For p > 1, we keep the same potential energy, but make the kinetic energy anisotropic, i.e. the kinetic energy becomes

A piecewise linear model for the zones of instability of an area-preserving map

Abstract In this note we study the global behavior of the piecewise linear area-preserving transformation x 1 = 1 − y 0 + | x 0 |, y 1 = x 0 , of the plane. We show that there are infinitely many invariant polygons surrounding an elliptic fixed point. The regions between these invariant polygons serve as models for the “zones of instability” in the corresponding smooth case. For our model we show that some of these annular zones contain only finitely many elliptic islands. The map is hyperbolic on the complement of these islands and hence exhibits stochastic behavior in this region. Unstable periodic points are dense in this region.

Sierpinski-curve Julia sets and singular perturbations of complex polynomials

In this paper we consider the family of rational maps of the complex plane given by z 2 + λ z 2 where λ is a complex parameter. We regard this family as a singular perturbation of the simple function z 2 . We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the correspondingmaps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However,we also showthatparameterscorrespondingto differentopensets havedynamics that are not conjugate.