Thomas Erneux

The slow passage through a Hopf bifurcation: delay, memory effects, and resonance

This paper explores analytically and numerically, in the context of the FitzHugh–Nagumo model of nerve membrane excitability, an interesting phenomenon that has been described as a delay or memory effect. It can occur when a parameter passes slowly through a Hopf bifurcation point and the systems response changes from a slowly varying steady state to slowly varying oscillations. On quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects; the dynamic aspect of the parameter variation. This delay and its dependence on the speed of the parameter variation are described.The model involves several parameters and particular singular limits are investigated. One in particular is the slow passage through a low frequency Hopf bifurcation where the systems response changes from a slowly varying steady state to slowly varying relaxation oscillations. We find in this case the onset o...

Propagating waves in discrete bistable reaction-diffusion systems

Abstract We consider a discrete bistable reaction-diffusion system modeled by N coupled Nagumo equations. We develop an asymptotic method to describe the phenomenon of propagation failure. The Nagumo model depends on two parameters: the coupling constant d and the bistability parameter a. We investigate the limit a→0 and d(a)→0 and construct traveling front solutions. We obtain the critical coupling constant d = d ∗ (a) above which propagation is possible and determine the propagation speed c = c(d) if d>d ∗ . We investigate two different cases for the initiation of a propagating front solution. Case 1 considers a uniform steady state distribution. A propagating front appears as the result of a fixed boundary condition. Case 2 also considers a uniform steady state distribution but a propagating front appears as the result of a localized perturbation.

Nonlinear rupture of free films

A free viscous film is subject to van der Waals attractions that lead to film rupture. Long‐wave asymptotics is used to derive approximate equations that govern the unstable evolution of the film. The solution of the nonlinear evolution equation is then considered using bifurcation techniques leading to an estimate for the nonlinear rupture time.

Slow passage through a Hopf bifurcation: from oscillatory to steady state solutions

This paper investigates the slow passage through a supercritical Hopf bifurcation from a branch of slowly varying periodic solutions to a branch of slowly varying steady states. This analysis is motivated by a recent numerical study of bursting oscillations in an enzymatic system. It was found that the transition from oscillations to steady states is delayed even if the rate of change of the control parameter is extremely small.The delay due to the slow passage is characterized by determining the amplitude of the oscillations at the bifurcation point. Defining

Synchronization properties of network motifs : Influence of coupling delay and symmetry

\varepsilon

Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers

as the rate of change of the bifurcation parameter, it is shown that the amplitude is an

The slow passage through a steady bifurcation: Delay and memory effects

O( \varepsilon^{1/4} )

Analytical stability boundaries for a semiconductor laser subject to optical injection

quantity as

Free boundary problems in controlled release pharmaceuticals. I: diffusion in glassy polymers

\varepsilon \to 0

Optically injected quantum-dot lasers

.In addition, a particular class of equations leading to relaxation oscillations is considered. It is assumed that frequency