Effect of a cutoff on pushed and bistable fronts of the reaction-diffusion equation.

We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction-diffusion equation when a small cutoff is applied to the reaction term at the unstable or metastable equilibrium point. The results are valid for arbitrary reaction terms and include the case of density-dependent diffusion.

Speed of pulled fronts with a cutoff.

We study the effect of a small cutoff epsilon on the velocity of a pulled front in one dimension by means of a variational principle. We obtain a lower bound on the speed dependent on the cutoff, for which the two leading order terms correspond to the Brunet-Derrida expression. To do so we cast a known variational principle for the speed of propagation of fronts in different variables which makes it more suitable for applications.

Minimal speed of fronts of reaction-convection-diffusion equations.

We study the minimal speed of propagating fronts of convection-reaction-diffusion equations of the form u(t)+microphi(u)u(x)=u(xx)+f(u) for positive reaction terms with f(')(0)>0. The function phi(u) is continuous and vanishes at u=0. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained.

Linear and nonlinear marginal stability for fronts of hyperbolic reaction diffusion equations.

We study traveling fronts of equations of the form u(tt)+phi(u)u(x)=u(xx)+f(u). A criterion for the transition from linear to nonlinear marginal stability is established for positive functions phi(u) and for any reaction term f(u) for which the usual parabolic reaction diffusion equation u(t)=u(xx)+f(u) admits a front. As an application, we treat reaction diffusion systems with transport memory.

Variational principle for limit cycles of the Rayleigh-van der Pol equation.

We show that the amplitude of the limit cycle of Rayleigh's equation can be obtained from a variational principle. We use this principle to reobtain the asymptotic values for the period and amplitude of the Rayleigh and van der Pol equations. Limit cycles of general Liénard systems can also be derived from a variational principle.

Variational approach to a class of nonlinear oscillators with several limit cycles.

We study limit cycles of nonlinear oscillators described by the equation x + nuF(x) + x = 0 with F an odd function. Depending on the nonlinearity, this equation may exhibit one or more limit cycles. We show that limit cycles correspond to relative extrema of a certain functional.