arXiv subjects

G. J. Lord

Publications and source records attributed to G. J. Lord.

4 recordsLinked to original sources

arXiv2026

Mean-square Stability and Bifurcations for Dissipative SDEs

We investigate the dynamics of dissipative systems with stochastic forcing and focus in particular on mean-square stability. First we show, under a natural condition on the drift and diffusion, that the stochastic system is mean-square dissipative. Next we examine the linearised system and state conditions ensuring that perturbations of a linear system with affine noise are bounded. We then relate the mean-square dynamics of the nonlinear and linearised systems. The approach gives a straightforward deterministic method to examine the effects of stochastic forcing on the stability of equilibria of deterministic systems and to obtain bifurcation diagrams that can be included into standard numerical continuation packages. The technique is illustrated numerically on some standard and non-standard examples.

math.PR↗

arXiv2011

Freezing Stochastic Travelling Waves

We consider in this paper travelling wave solutions to stochastic partial differential equations and corresponding wave speed. As a particular example we consider the Nagumo equation with multiplicative noise which we mainly consider in the Stratonovich sense. A standard approach to determine the position and hence speed of a wave is to compute the evolution of a level set. We compare this approach against an alternative where the wave position is found by minimizing the $L^2$ norm against a fixed profile. This approach can also be used to stop (or freeze) the wave and obtain a stochastic partial differential algebraic equation that we then discretize and solve. Although attractive as it leads to a smaller domain size it can be numerically unstable due to large convection terms. We compare numerically the different approaches for estimating the wave speed. Minimization against a fixed profile works well provided the support of the reference function is not too narrow. We then use these techniques to investigate the effect of both \Ito and Stratonovich noise on the Nagumo equation as correlation length and noise intensity increases.

math.NA↗

arXiv2010

Finite to infinite steady state solutions, bifurcations of an integro-differential equation

We consider a bistable integral equation which governs the stationary solutions of a convolution model of solid--solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is varied to examine the transition from an infinite number of steady states to three for the continuum limit of the semi--discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.

math.DS↗

arXiv2008

Bifurcations in the regularized Ericksen bar model

We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of Müller's conjecture \cite{Muller} concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of Müller's conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a ``loop'' structure that characterizes $(k,3k)$ bifurcations.

math.DS↗