Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers

Kazuyuki Yagasaki

doi:10.3934/dcds.2011.29.387

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2011, Volume 29, Issue 1: 387-402. Doi: 10.3934/dcds.2011.29.387

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Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers

Kazuyuki Yagasaki^1,

1.
Department of Mechanical and Systems Engineering, Gifu University, Gifu 501-1193

Received: August 31, 2007

Revised: August 31, 2009

Published: December 2010

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Abstract

We consider two-degree-of-freedom Hamiltonian systems with saddle-centers, and develop a Melnikov-type technique for detecting creation of transverse homoclinic orbits by higher-order terms. We apply the technique to the generalized Hénon-Heiles system and give a positive answer to a remaining question of whether chaotic dynamics occurs for some parameter values although it is known to be nonintegrable in a complex analytical meaning.

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Mathematics Subject Classification: Primary: 37J45, 70H07, 70H09; Secondary: 34C37, 34E10, 37J30.

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