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 (ODEs). Analogous results to those holding for the PDE are proved for the ODEs—in particular, the existence and structure of the global attractor and appropriate spanning bases for the equilibria and certain heteroclinic connections, are studied. Heteroclinic connections in the system of ODEs are then computed using a generalization of known methods to cope with the gradient structure. Suitable parameterizations of the attractor are introduced and numerical continuation used to find families of connections on the attractor. Special connections, which are stable in certain Fourier spanning bases, are used as starting points for the computations. The methods used allow the calculation of connecting orbits that are unstable as solutions of the initial value problem, and thus provide a computational tool for understanding the dynamics of dissipative problems in a manner that could not be achieved by use of standard initial value methods. Numerical results are given for the Chafee–Infante problem and for the Cahn–Hilliard equation. A one-parameter family of PDEs connecting these two problems is introduced, and it is demonstrated numerically that the global attractor for the Chafee–Infante problem can be continuously deformed into that for the Cahn–Hilliard equation.

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