Discretization of homoclinic orbits, rapid forcing and ''invisible'' chaos

One-step discretizations of order p and step size epsilon of autonomous ordinary differential equations can be viewed as time-epsilon maps of x(t) = f(lambda, x(t)) + epsilon(p) g(epsilon, lambda, t/epsilon, x(t)), x is an element of IR(N), lambda is an element of IR, where g has period epsilon in t. This is a rapidly forced nonautonomous system. We study the behavior of a homoclinic orbit Gamma for epsilon = 0, lambda = 0, under discretization. Under generic assumptions we show that Gamma becomes transverse for positive epsilon. The transversality effects are estimated from above to be exponentially small in epsilon. For example, the length l(epsilon) of the parameter interval of lambda for which Gamma persists can be estimated by l(epsilon) less than or equal to C exp(-2 pi eta/epsilon) where C, eta are positive constants. The coefficient eta is related to the minimal distance from the real axis of the poles of Gamma(t) in the complex time domain. Likewise, the region where complicated, ''chaotic'' dynamics prevail is estimated to be exponentially small, provided x is an element of IR(2) and the saddle quantity of the associated equilibrium is nonzero. Our results are visualized by high precision numerical experiments. The experiments show that, due to exponential smallness, homoclinic transversality becomes practically invisible under normal circumstances, already for only moderately small discretization steps.