Nonlinearly perturbed heat equation

Abstract

We consider the linear heat equation with appropriate boundary conditions describing the temperature on a wire with adiabatic endpoints. We also consider a perturbation, which provokes a global change in the temperature of the wire. This perturbation occurs periodically and is modeled by an iterated nonlinear map of the interval belonging to a one-parameter family of quadratic maps, f_μ. We observe a long term stabilization, under time evolution, of the number of new critical points of the temperature function. However, for certain values of the parameter μ, even with the stabilization effect of the number of critical points, the evolution of the temperature function is chaotic. We study the parameters of the system, that is, difusion coeficient and μ, in order to characterize the observed behaviour and its dependence on the topological invariants of f_μ, in particular the dependence on the chaotic behaviour of f_μ.