Instabilities and bifurcations in geomechanics

Bifurcations and Instabilities in Dissipative Systems

In addition to describing the nonlinear evolution of the Benjamin–Feir instability, the KS equation has been studied as a model for interfacial instabilities, for example in flame fronts and solidification problems. Coefficients in front of any of the terms may be removed by appropriate rescalings of u, t and x, leaving the canonical form above. The one remaining parameter is the domain size L:...

Instabilities and bifurcations of nonlinear impurity modes.

We study the structure and stability of nonlinear impurity modes in the discrete nonlinear Schrödinger equation with a single on-site nonlinear impurity emphasizing the effects of interplay between discreteness, nonlinearity, and disorder. We show how the interaction of a nonlinear localized mode (a discrete soliton or discrete breather) with a repulsive impurity generates a family of stationar...

Basin boundaries and bifurcations near convective instabilities: A case study

Using a scalar reaction-diffusion equation with drift and a cubic nonlinearity as a simple model problem, we investigate the effect of domain size on stability and bifurcations of steady states. We focus on two parameter regimes, namely the regions where the steady state is convectively or absolutely unstable. In the convective-instability regime, the trivial stationary solution is asymptotical...

Computational Geomechanics

Pulse bifurcations and instabilities in an excitable medium: computations in finite ring domains.

We investigate the instabilities and bifurcations of traveling pulses in a model excitable medium; in particular, we discuss three different scenarios involving either the loss of stability or disappearance of stable pulses. In numerical simulations beyond the instabilities we observe replication of pulses ("backfiring") resulting in complex periodic or spatiotemporally chaotic dynamics as well...