Table of contents

The time-dependent Ginzburg - Landau equations (TDGL) model a thin-film superconductor of finite size placed under a magnetic field. For numerical computation, we use a staggered grid discretization, a technique well known in numerical fluid mechanics. Some properties of the solutions are established. An efficient explicit - implicit method based on the forward Euler method is developed. In our simulations, we impose natural boundary conditions at the edge of the superconductor. With suitable choices of parameters (corresponding to physical superconductors of type II) and the strength of the external magnetic field, the steady-state solutions exhibit vortices. When a variable strength magnetic field, simulating a transient current, is introduced, we observe motion of the vortices in a periodic pattern.

In the context of equivariant dynamical systems with a compact Lie group, , of symmetries, Field and Krupa have given sharp upper bounds on the drifts associated with relative equilibria and relative periodic orbits. For relative equilibria consisting of points of trivial isotropy, the drifts correspond to tori in . Generically, these are maximal tori. Analogous results hold when there is a nontrivial isotropy subgroup , with replaced by .

In this paper, we generalize the results of Field and Krupa to noncompact Lie groups. The drifts now correspond to tori or lines (unbounded copies of ) in and generically these are maximal tori or lines. Which of these drifts is preferred, compact or unbounded, depends on : there are examples where compact drift is preferred (Euclidean group in the plane), where unbounded drift is preferred (Euclidean group in three-dimensional space) and where neither is preferred (Lorentz group).

Our results partially explain the quasiperiodic (Winfree) and linear (Barkley) meandering of spirals in the plane, as well as the drifting behaviour of spiral bound pairs (Ermakova et al). In addition, we obtain predictions for the drifting of the scroll solutions (scroll waves and scroll rings, twisted and linked) considered by Winfree and Strogatz.

We study the modulational instability in discrete lattices and we show how the discreteness drastically modifies the stability condition. Analytical and numerical results are in very good agreement. We predict also the evolution of a linear wave in the presence of noise and we show that modulational instability is the first step towards energy localization.

Every positive -quadratic differential form defined on an oriented surface has two transversal -one-dimensional foliations with common singularities associated with it. In this article we begin the description of the simplest patterns of topological change - bifurcation - in one-parameter families of positive -quadratic differential forms , depending smoothly on a real parameter t, which occur at values where has a non-locally stable singular point.

We consider the large time asymptotics for the evolution of a planar curve subject to mean curvature flow and constant forcing. Depending on the sign of the forcing we prove convergence for large times to either a travelling wave or a self-similar profile. The context of our work is the study of the motion of a superconducting vortex.

We address the issue of spatially localized periodic oscillations in coupled networks - so-called discrete breathers - in a general context. This context is concerned with general conditions which allow continuation of periodic solutions of vector fields. One advantage of our approach is to encompass in the same mathematical framework the cases of conservative and dissipative systems. An essential feature is that we allow the period to vary. In particular, we deduce existence of discrete breathers in networks where each site has an equilibrium and some sites have a limit cycle, and in Hamiltonian networks without requiring local anharmonicity. The latter case is dealt with by considering the persistence of families of periodic solutions in the more general context of systems with an integral, not just Hamiltonian ones.

We consider a stochastic perturbation of weakly coupled expanding circle maps. We construct the dynamics and its natural invariant measure via a polymer expansion and show the stochastic stability of the system.

PACS Numbers: 0520, 0570L

We construct a kneading theory à la Milnor - Thurston for Lozi mappings (piecewise affine homeomorphisms of the plane). As a two-dimensional analogue of the kneading sequence, the pruning front and the primary pruned region are introduced, and the admissibility criterion for symbol sequences known as the pruning front conjecture is proven under a mild condition on the parameters. Using this result, we show that topological properties of the dynamics of the Lozi mapping are determined by its pruning front and primary pruned region only. This gives us a solution to the first tangency problem for the Lozi family, moreover the boundary of the set of all horseshoes in the parameter space is shown to be algebraic.

We use the cross-correlation sum introduced recently by Kantz to study symmetry properties of chaotic attractors or, more precisely, of invariant measures supported by such attractors. In particular, we apply it to a system of six coupled nonlinear oscillators which was shown by Kroon et al to have attractors with several different symmetries, and compare our results with those obtained by `detectives' in the sense of Golubitsky et al.

The Lagrangian

has been studied by several authors. Here we consider the case and build an analytic perturbation f such that has, for and suitably small, an orbit (Q,q) satisfying for . We will show that this `diffusion speed' is the best possible that can be obtained in this case.