Charles C. Conley

The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold

Abstract : The following conjecture of V. I. Arnold is proved: every measure preserving diffeomorphism of the torus T2, which is homologeous to the identity, and which leaves the center of mass invariant, possesses at least 3 fixed points. The proof of this global fixed point theorem does not make use of the generating function technique. The theorem is a consequence of the statement that a Hamiltonian vector-field on a torus T2n, which depends periodically on time, possesses at least (2n+1) forced oscillations. These periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows. A second conjecture of V. I. Arnold proved concerns a Birkhoff-Lewis type fixed point theorem for symplectic maps. Additional keywords; Periodic functions; Variational principles; Equations; Reprints. (Author)

On the ultimate behavior of orbits with respect to an unstable critical point I. Oscillating, asymptotic, and capture orbits

This work concerns an analytic autonomous Hamiltonian system of differential equations with two degrees of freedom which admits and unstable equilibrium point. More specifically, the eigenvalues corresponding to the equilibrium point include one real pair and one imaginary pair and so we know (by a theorem of A. Liapunov) that passing through the equilibrium point there is an invariant two dimensional manifold on which all solutions are periodic. Furthermore, the level surfaces of the Hamiltonian function are invariant three dimensional manifolds which, for appropriate values of that function, contain exactly one of these periodic solutions. Our aim is to study the orbits on such a level surface. The nature of the work is most easily described in terms of the following example: consider two bowls connected by a trough so that, when inverted they look like two mountains with a pass between. The differential equations are taken to be those describing the motion of a point mass sliding without friction on this “double bowl”. The Hamiltonian function is the sum of the potential energy, i.e. the height in the bowl, and the kinetic energy, also obtained as usual. Since the kinetic energy is positive, fixing the value of the Hamiltonian function corresponds to limiting the height to which the mass can go. Our problem concerns the case where the mass can go high enough to get from one bowl to the other with just a little room to spare. Having fixed on an appropriate level surface (of the Hamiltonian function) we first study the behavior of orbits near the equilibrium point, which in the example above, corresponds to the saddle point in the trough connecting

On traveling wave solutions of nonlinear diffusion equations

An existence proof for periodic traveling wave solutions of an equation of Nagumo is outlined. The proof begins with the analysis of a limiting case in which one set of dependent variables moves infinitely fast compared to the remainder. Singular “orbits” are defined for this limiting system and a perturbation argument using isolating blocks allows one to find actual solutions.

Some Aspects of the Qualitative Theory of Differential Equations

This chapter discusses a few aspects of the qualitative theory of differential equations. It presents some problems of a qualitative nature. A salient feature of these problems is that they do not require any difficult computations. All the examples presented in the chapter concern the existence of orbits that are situated in some special way; in a general sense they are boundary-value problems. The chapter illustrates a picture of a tube with square cross section. There is a differential equation whose solutions behave in the following way: on the top, bottom, and left end they cross into the tube; on the sides and the right end they cross out of the tube. In the case discussed in the chapter, it is assumed that there is no invariant set in the tube; thus, every point is mapped both to the exit set and the entrance set under the mentioned mappings. An orbit segment is said to cross the tube if it runs from the left end to the right end without leaving the tube.

Remarks on the Stability of Steady-State Solutions of Reaction-Diffusion Equations

In this note we shall describe a new approach to the stability problem for steady-state solutions of reaction- diffusion equations. We shall explain our techniques and results by discussing a few examples which model quite well the general theory.

Hyperbolic sets and shift automorhpisms

An elementary discussion of hyperbolic invariant sets is presented and a proof of the theorem that “near any nonperiodic chain recurrent point there lies an embedded shift automorphism” is indicated.

Algebraic and Topological Invariants for Reaction-Diffusion Equations

In the last few years, a good deal of progress has been made in the understanding of the qualitative properties of solutions of reaction-diffusion equations. This has been due to the introduction of new topological techniques into the field; in particular, we mention the concept of an isolated invariant set, and its index, as developed in [2].

Bifurcation and Stability of Stationary Solutions of the Fitz-Hugh-Nagumo Equations

The Fitz-Hugh-Nagumo equations have been of some interest to both mathematicians and theoretical biologists for several years. The reason for this stems from the fact that they can be considered as a simpler model for the celebrated Hodgkin-Huxley equations, in that they exhibit many of the features of this latter system. Indeed, mathematicians have studied them because their structure is different from the usually encountered equations in physics, and they therefore admit solutions with less familiar properties: homoclinic travelling waves, threshold effects, etc. The equations can be written as