Numerical Computations of Connecting Orbits in Discrete and Continuous Dynamical Systems

The aim of this paper is to present a numerical technique for the computation of connections between periodic orbits in non{autonomous and autonomous systems of ordinary diierential equations. First the existence and computation of connecting orbits between xed points in discrete dy-namical systems is discussed; then it is shown that the problem of nding connections between equilibria and periodic solutions in continuous systems may be reduced to nding connections between xed points in a discrete system. Implementation of the method is considered: the choice of a linear solver discussed and phase conditions are suggested for the discrete system. The paper concludes with some numerical examples: connections for equilibria and periodic orbits are computed for discrete systems and for non{autonomous and autonomous systems, including systems arising from the discretization of a partial diierential equation.