October  1999, 5(4): 765-782. doi: 10.3934/dcds.1999.5.765

Homoclinic and multibump solutions for perturbed second order systems using topological degree

1. 

Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

Received  September 1998 Revised  February 1999 Published  July 1999

We present results on homoclinic and multibump solutions for perturbed second order systems. Using topological degree, we generalize results recently obtained by variational methods. We give Melnikov type conditions for the existence of one homoclinic solution and for the existence of infinitely many multibump solutions. We give also an example for which the set of zeros of the Poincaré-Melnikov function contains an interval and results requiring a simple zero of this function can not be applied. In the case of multibump solutions, when the perturbation is periodic, we prove the existence of approximate Bernoulli shift structures leading to some form of chaos.
Citation: Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765
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