Numerical approximation and spectral analysis of infinite-dimensional dynamical systems
03-008 Yong-Kui Zou, Wolf-Jürgen Beyn.
On the existence of transversal heteroclinic orbits in
discretized dynamical systems
In this paper we prove the existence of
transversal heteroclinic orbits for maps that are obtained from
one-step methods applied to a continuous dynamical system. It
is assumed that the continuous system exhibits a heteroclinic
orbit at a specific value of a parameter. While it is known
that analytic vector fields lead to exponentially small
splittings of separatrices in the discrete system, we analyze
here the case of a continuous system that is smooth of finite
order only. Assuming that a certain derivative has a jump
discontinuity at a specific hyperplane we show that the
discretized systems have transversal heteroclinic orbits. The
essential step in deriving such a result is a refinement of a
previously developed error analysis which applies exponential
dichotomy and Fredholm techniques to the discretized
system.
