Glen R. Hall

Resonance Zones in Two-Parameter Families of Circle Homeomorphisms

We consider a two-parameter family of diffeomorphisms of the circle where one of the parameters controls the amount of rigid rotation while the second controls the nonlinearity. In particular, we show that the regions in the parameter plane for which the map has a periodic orbit of a particular rotation number (resonance zones) increase in size linearly as the second parameter is increased from zero. This is a discretization of the phenomenon known as “phase locking” for ordinary differential equations. Using this, we obtain some results on the smoothness of the curves between the resonance zones.

A Topological Version of a Theorem of Mather on Twist Maps.

Abstract : In this report shows that a twist map of an annulus with a periodic point of rotation number p/q must have a Birkhoff periodic point of rotation number p/q. Topological techniques are used so no assumption of area-preservation or circle intersection property is needed. If the map is area preserving then this theorem and the fixed point theorem of Birkhoff imply a recent theorem of Mather. It is also shown that periodic orbits of (significantly) smallest period for a twist map must be Birkhoff. (Author)

Coorbital periodic orbits in the three body problem

We consider the dynamics of coorbital motion of two small moons about a large planet which have nearly circular orbits with almost equal radii. These moons avoid collision because they switch orbits during each close encounter. We approach the problem as a perturbation of decoupled Kepler problems as in Poincares periodic orbits of the first kind. The perturbation is large but only in a small region in the phase space. We discuss the relationship required among the small quantities (radial separation, mass, and minimum angular separation). Persistence of the orbits is discussed.

Bifurcations of Periodic Orbits

This chapter and Chapter 13 use the theory of normal forms developed in Chapter 9. They contain an introduction to generic bifurcation theory and its applications. Bifurcation theory has grown into a vast subject with a large literature; so, this chapter can only present the basics of the theory. The primary focus of this chapter is the study of periodic solutions, their existence and evolution. Periodic solutions abound in Hamiltonian systems. In fact, a famous Poincaré conjecture is that periodic solutions are dense in a generic Hamiltonian system, a point that was established in the C case by Pugh and Robinson (1983).

Robustness of periodic point free maps of the annulus

Abstract We construct a diffeomorphism of the annulus into itself which has no periodic points such that when it is composed with any sufficiently small rigid rotation the resulting map still has no periodic points.

An Example of a Universally Observable Flow on the Torus

In this paper we examine the question of existence of a two-dimensional universally observable system, i.e., dynamics which are observable by every continuous nonconstant real-valued function on the state space. We are motivated by the work of D. McMahon, who proved that a class of three-dimensional manifolds with horocycle flow have this property. We examine this example and are able to give sufficient conditions for a flow to be universally observable. We then use these conditions to show the existence of a continuous universally observable flow on the torus. The proofs involve techniques and concepts from topological dynamics and dynamical systems on the torus.

A Limit Case of the “Ring Problem”: The Planar Circular Restricted

We study the dynamics of an extremely idealized model of a planetary ring. In particular, we study the motion of an infinitesimal particle moving under the gravitational influence of a large central body and a regular n-gon of smaller bodies as n tends to infinity. Our goal is to gain insight into the structure of thin, isolated rings.

1+n

We investigate the dynamics of maps and flows which arise from a class of models of closed queueing networks in computer science theory. The network consists of n+/ servers, one of which is a central server with a queue of size n-1. A protocol or scheduling discipline must be specified in this server to define the queueing network. The standard model gives rise to a flow on an n-torus. We consider the service protocols first in-first out (FIFO) and last in-first out (LIFO) in dimension three, for which the state spaces are modifications of a 3-torus. We present a sufficient condition on the time it takes each call to complete one cycle for the FIFO protocol which guarantees that the set of periodic orbits which involve no waiting in the queue is a global attractorfor the associated semi-flow. We also investigate the dynamics for the LIFO service protocol via a return map derived from the associated area preserving flow.

Body Problem

This chapter gives an introduction to the geometric theory of autonomous Hamiltonian systems by studying some local questions about the nature of the solutions in a neighborhood of a point or a periodic solution. The dependences of periodic solutions on parameters is also presented in the case when no drastic changes occur, i.e., when there are no bifurcations. Bifurcations are addressed in Chapter VIII. Several applications to the 3-body problem are given. The chapter ends with a brief introduction to hyperbolic objects and homoclinic phenomena.

Computer protocol and torus maps

In the last chapter, some local results about periodic solutions of Hamiltonian systems were presented. The systems contain a parameter, and the conditions under which a periodic solution can be continued in the parameter were discussed. Since Poincare used these ideas extensively, it has become known as Poincare’s continuation method. By Lemma V.E.2, a solution o(t, ξ′) of an autonomous differential equation is T-periodic if and only if o(T’) = o is the general solution. This is a finite-dimensional problem since is a function defined in a domain of ℝm+1 into ℝm. Thus, periodic solutions can be found by the finite-dimensional methods, i.e., the finite-dimensional implicit function theorem, the finite-dimensional fiixed point theorems, the finite-dimensional degree theory, etc. This chapter will present results which depend only on the finite-dimensional implicit function theorem. Chapter X will present a treatment of fixed point methods as they apply to Hamiltonian systems. In this chapter the periodic solutions vary continuously with the parameter (“can be continued”), but Chapter VII will discuss the bifurcations of periodic solutions.