Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation

In this article existence and stability of N-front travelling wave solutions of partial differential equations on the real line is investigated. The N-fronts considered here arise as heteroclinic orbits bifurcating from a twisted heteroclinic loop in the underlying ordinary differential equation describing travelling wave solutions. It is proved that the N-front solutions are linearly stable provided the fronts building the twisted heteroclinic loop are linearly stable. The result is applied to travelling waves arising in the FitzHugh-Nagumo equation

Instability of localised buckling modes in a one-dimensional strut model

Stability of localised equilibria arising in a fourth-order partial differential equation modelling struts is investigated. It was shown in Buffoni, Champneys & Toland (1996) that the model exhibits many multi-modal buckling states bifurcating from a primary buckling mode. In this article, using analytical and numerical techniques, the primary mode is shown to be unstable under dead loading in a large range of parameter values, while is likely to be stable under rigid loading for small axial loads. Furthermore, for general reversible or Hamiltonian systems, stability of the multi-modal solutions is established assuming stability of the primary state. As this hypothesis is not satisfied for the buckling mode arising in the strut model, any multi-modal buckling state will be unstable for both loading devices

Toward nonlinear stability of sources via a modified Burgers equation

Coherent structures are solutions to reaction-diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at $x=\pm\infty$ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a modified Burgers equation as a model problem that captures some of the essential features of coherent structures, we show how this phenomenon can be analysed and nonlinear stability be established in this simpler context.Comment: revised version with some typos fixe

Stability of multiple-pulse solutions

In this article, stability of multiple-pulse solutions in semilinear parabolic equations on the real line is studied. A system of equations is derived which determines stability of N-pulses bifurcating from a stable primary pulse. The system depends only on the particular bifurcation leading to the existence of the N-pulses. As an example, existence and stability of multiple pulses are investigated if the primary pulse converges to a saddle-focus. It turns out that under suitable assumptions infinitely many N-pulses bifurcate for any fixed N > 1. Among them are infinitely many stable ones. In fact, any number of eigenvalues between 0 and N-1 in the right half plane can be prescribed

Convergence estimates for the numerical approximation of homoclinic solutions

This article is concerned with the numerical computation of homoclinic solutions converging to a hyperbolic or semi-hyperbolic equilibrium of a system u̇ = ƒ (u, μ). The approximation is done by replacing the original problem by a boundary value problem on a finite interval and introducing an additional phase condition to make the solution unique. Numerical experiments have indicated that the parameter μ is much better approximated than the homoclinic solution. This was proved in Schecter (1995) for phase conditions fulfilling an additional "niceness" assumption, which is unfortunately not satisfied for the phase condition most commonly used in numerical experiments and which actually suggested the super-convergence result. Here, this result is proved for arbitrary phase conditions. Moreover, it is shown that it is sufficient to approximate the original boundary value problem to first order when considering semi-hyperbolic equilibria extending a result of Schecter (1993)

Constructing dynamical systems possessing homoclinic bifurcation points of codimension two

A procedure is derived which allows for a systematic construction of three-dimensional ordinary differential equations possessing homoclinic solutions. These are proved to admit homoclinic bifurcation points of codimension two. The examples include the non-orientable resonant bifurcation, the inclination-flip and the orbit-flip. In addition, an equation is constructed which admits a homoclinic orbit converging to a saddle-focus satisfying Shilnikov's condition. The vector fields are polynomial and non-stiff in that the eigenvalues are of moderate size

Frequency Regions for Forced Locking of Self-Pulsating Multi-Section DFB Lasers

A method is developed which allows for the calculation of locking regions of self-pulsating multi-section lasers which are exposed to external optical data sequences. In particular, resonant locking is investigated where both wavelength detuning and detuning of the power frequencies are important