Table of contents

We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon.

Cellular automata are dynamical systems on the compact metric space of subshifts. They leave many classes of subshifts invariant. Here we show that cellular automata leave 'circle subshifts' invariant. These are the strictly ergodic subshifts of (0,1)(Z^d) obtained by a circle sequence x_n=1_J(n. alpha ) where J is a finite union of half-open intervals. For such initial conditions, the evolution of the whole infinite configuration can be computed by evolving the finitely many parameters defining the set J. Moreover, many macroscopic quantities can be computed exactly for the infinite system. We illustrate that in one dimension by rule 18 and in two dimensions by the Game of Life. The ideas also apply to cellular automata acting on (0, ..., N-1)(Z^d). This we illustrate by the HPP model, a lattice gas automaton with N=16.

We consider the equations of motion of a dynamically symmetric sphere, rolling on a convex surface of revolution. We prove that the eight dimensional constraint manifold is filled up with tori of dimension at most three, on which we have quasi-periodic motion. The vector field on these tori depends on three coordinates only.

We modify the recently proposed model of Speight and Ward to make it possess time dependent solutions. We find that for each lattice spacing and for each velocity of the sine Gordon kink we can find a modification of the model for which this kink is a solution. We find that this model has really 3 'kink-like' solutions; the original kink, the static kink and a further kink moving with velocity v approximately 0.97. We discuss various properties of the model, from the point of view of its usefulness for numerical simulations.

In this paper we consider the family of the cubic systems of Kukles (1944) with the condition that one of the parameters a₇ is zero. Under this restriction the centre conditions were given by Kukles. The study of this family exhibits properties and issues which are important in the problem of the full classification of cubic systems with a centre. The family is formed of four strata, one of which is made up of quadratic systems and was studied before by Schlomiuk (1993). If we consider the three strata formed by truly cubic system we have a first (second) stratum consisting of systems symmetric with respect to the x-axis (y-axis) and a third stratum consisting of systems with two invariant straight lines and having an elementary first integral obtained by the Darboux method. Systems in either one of the symmetric strata do not possess elementary first integrals generically. The first stratum is formed by integrable systems having a Liouvillian first integral. We show that systems in the second stratum have no Liouvillian first integral. We give the full bifurcation diagram of each stratum of truly cubic systems.

We characterize the bifurcations of polynomial maps algebraically, by means of the discriminants of the polynomials whose roots are the periodic orbits. These discriminants are computed explicitly, in terms of multiplier polynomials. This approach affords a generalization of the notion of bifurcation to the broader context of iteration of polynomials over an integral domain.

A statistical equilibrium theory is developed to characterize the large-scale coherent structures in a turbulent two-dimensional magnetofluid. Macrostates are defined as local joint probability distributions, or Young measures, on the values of the fluctuating magnetic field and velocity field at each point in the spatial domain. The most probable macrostate is found by maximizing a Kullback entropy functional subject to constraints dictated by the conserved integrals of the ideal dynamics. This maximum entropy macrostate is, for each point in the spatial domain, a Gaussian probability distribution, whose local mean is an exact stationary solution of the equations of ideal magnetohydrodynamics. The predictions of the model are in excellent qualitative and quantitative agreement with recent high resolution numerical simulations of turbulence in slightly dissipative two-dimensional magneto fluids.

The uniqueness of the branch of two-tori in the D₄-equivariant Hopf bifurcation problem is proved in a neighbourhood of a particular limiting case where, after reduction, the Euler equations for the rotation of a free rigid body apply.