The Numerical Computation of Heteroclinic Connections in Systems of Gradient Partial Differential Equations

Abstract

The numerical computation of heteroclinic connections in partial differential equations (PDEs) with a gradient structure, such as those arising in the modeling of phase transitions, is considered. Initially, a scalar reaction diffusion equation is studied; structural assumptions are made on the problem to ensure the existence of absorbing sets and, consequently, a global attractor. As a result of the gradient structure, it is known that, if all equilibria are hyperbolic, the global attractor comprises the set of equilibria and heteroclinic orbits connecting equilibria to one another. Thus it is natural to consider direct approximation of the set of equilibria and the connecting orbits.Results are proved about the Fourier spanning basis for branches of equilibria and also for certain heteroclinic connections; these results exploit the oddness of the nonlinearity. The reaction-diffusion equation is then approximated by a Galerkin spectral discretization to produce a system of ordinary differential equations...