Christian Elphick

A simple global characterization for normal forms of singular vector fields

We derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems. Using an appropriate scalar product in the space of homogeneous vector polynomials, we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented.

Patterns of propagating pulses

The complex dynamics that arise in certain nonlinear partial differential equations in time and in one space dimension are studied. In the general case considered, the equation admits a solitary wave in the form of a pulse tailing off exponentially, fore and aft, with possibly oscillatory character. Complicated solutions are described by a superposition of many such solitary structures in interaction. The description is asymptotic in terms of a parameter that becomes exponentially small as the ratio of typical pulse separation to pulse width becomes large. The outcome is a set of dynamical equations for the motion of the individual pulses with nearest neighbor interactions. This system of ordinary differential equations (ODEs) admits a wide range of patterns, both regular and chaotic. The stability theory of such patterns is sketched and the continuum limit of the lattice-dynamical equations of the pulses is given.

Normal form reduction for time-periodically driven differential equations

Abstract We present a global characterisation for the normal form of a time-periodically driven differential equation describing the behaviour of a physical system in the neighborrhood of a multiple instability.

Dynamics of fronts in thermally bistable fluids

Fronts between different thermal phases of a fluid, when the cooling function allows two thermally stable phases around an unstable one (bistability), are investigated. Fluid motion is included, in addition to thermal conduction. For a one-dimensional case the front dynamics are investigated by introducing an appropriate Lyapunov functional. It is assumed that the coefficient of thermal conductivity is small, so that the front thickness is very small compared with front separations. Pairs of adjacent fronts define a cloud (or an intercloud region), and their motion gives rise to the growth of the cloud (condensation) or of the intercloud region (evaporation). The properties of various types of fronts separating the different thermal phases of the fluid are discussed. Interaction between fronts and its effect on front motion is found using an approximation method

Topological defects dynamics and Melnikov's theory

Abstract We use Melnikovs analysis to construct recurrence time maps near homoclinic and heteroclinic bifurcations. We show that a method developed by Kawasaki and Ohta in order to study defects dynamics is indeed equivalent to this version of Melnikovs theory.

Global aspects of hamiltonian normal forms

Abstract We derive simple global characterizations for the normal forms of hamiltonian systems with constant and periodic coefficients.

Classes of exactly solvable nonlinear evolution equations for Grassmann variables: The normal form method

A systematic procedure is presented to solve analytically differential equations for Grassmann variables with the most general nonlinearity. The method consists in the reduction of the original equation to its simplest form (normal form). The classes of solvable normal forms are determined only by the structure of the linear part of the original equation and are parametrized in terms of the number of critical eigenvalues.

Spatial Disorder in Extended Systems

During the last few years much effort has been devoted to the study of temporal chaotic behavior of simple dynamical systems modeling turbulent-like phenomena in a great variety of macroscopic physical systems [1]. On the other hand the study of pattern formation occurring in extended systems has only recently become a very popular subject [2].

Localized and Extended Patterns in Reactive Media

We study the dynamics of interacting localized structures in homogeneous reactive media. Equations of motion for solitary waves in excitable media and for vortices in oscillatory media are derived under the assumption of weak interactions. We show that excitable media with oscillatory recovery can support a multitude of stable, nonuniform spatial patterns and that phase field effects in oscillatory media may lead to the formation of bound vortex pairs. The implications of the latter result on the transition to turbulence in oscillatory media are discussed.

Bound States of Interacting Localized Structures

Localized or solitary structures are frequently formed in extended systems under the combined effects of instability and dissipation.1 The effective particle approach widely used for integrable systems2 and in quantum field theory3 can also be used for such systems. Many of the nonlinear PDEs encountered in macroscopic physics can be thus reduced to ODEs that give insight into the full problem, particularly when the original system is invariant under a continuous group. Each solitary structure is assigned a set of the group parameters and these become collective coordinates characterizing the state of the system.4,5