Basin boundaries and bifurcations near convective instabilities: a case study
The paper aims to consider the transition between convective and absolute nonlinearities in a nonlinear parabolic (reaction-diffusion) equation. As an example, a model equation is taken: \[ u_t=u_{xx}+u_x+\mu u+\kappa u^3. \] The analysis is focused on the case \(0<\mu<1/4\), when the equation, if considered on a finite interval \(0<x<l\), features convective instability of the trivial solution, \(x=0\). As a measure of proximity to the absolute instability, the distance of the trivial solution to a nearest border of its attraction basin is estimated. It is shown that this distance becomes exponentially small with the increase of the domain's size \(l\) if the nonlinearity in the equation is also unstable, \(\kappa >0\). In the opposite case of the stable nonlinearity, \(\kappa <0\), the distance to the border of the attraction basin is bounded away from zero. Bifurcations accounting for the transition to the absolute instability from its convective counterpart are investigated too.
scientific article; zbMATH DE number 2134395