Abstract
It is well-known [l] that the existence of a transversal homoclinic orbit of a diffeomorphism implies chaos, i.e. the diffeomorphism possesses a compact domain which is invariant for some of its iterates and such that the dynamics of such an iterate on it is homeomorphic to the Bernoulli shift on a finite number of symbols. However, the location of transversal homoclinic points is generally difficult. One of the most common approaches is a perturbation method based on the derivation of the so-called Melnikov functions [2], i.e. we look for a transversal homoclinic point of a perturbed diffeomorphism when the unperturbed diffeomorphism has a known dynamics. Analytical methods, based on the Lyapunov-Schmidt procedure, are presented in [3-51 for periodically, regularly perturbed autonomous ordinary differential equations (o.d.e.), and in [6-81 for diffeomorphisms. Regularly perturbed impulsive o.d.e. are studied in [8]. Singular perturbation problems are studied analytically in [9, lo] for o.d.e., and in [l 11 for impulsive o.d.e. In all these papers it is assumed that the so-called reduced o.d.e. has a homoclinic orbit. In particular, the following impulsive o.d.e. is studied in [ll]
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