Numerical verifications for homo/heteroclinic orbits and their applications to partial differential equations

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The purpose of this research is to present fast algorithms to prove existence of homoclinic or heteroclinic orbits appearing dynamical systems by means of computers. The main result obtained in the research period is that we succeeded in showing reasonably fast algorithms to prove existence of homoclinic orbits in reversible dynamical systems. The key idea of this algorithm is to rigorously estimate Melnikov type integrals by computers. On the other hand, when we remove the assumption of reversibility, it does not yet succeeded in obtaining a good algorithm for this purpose.