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2013
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# Vassilios Rothos

GR 265 00, Patras, Greece
Center of Research and Applications of Nonlinear Systems Department of Mathematics, University of Patras

## Publications:

 Rothos V. M., Bountis A. The Second Order Mel'nikov Vector 1997, vol. 2, no. 1, pp.  26-35 Abstract Mel'nikov's perturbation method for showing the existence of transversal intersections between invariant manifolds of saddle fixed points of dynamical systems is extended here to second order in a small parameter $\epsilon$. More specifically, we follow an approach due to Wiggins and derive a formula for the second order Mel'nikov vector of a class of periodically perturbed $n$-degree of freedom Hamiltonian systems. Based on the simple zero of this vector, we prove an $O(\epsilon^2)$ sufficient condition for the existence of isolated homoclinic (or heteroclinic) orbits, in the case that the first order Mel'nikov vector vanishes identically. Our result is applied to a damped, periodically driven $1$-degree-of-freedom Hamiltonian and good agreement is obtained between theory and experiment, concerning the threshold of heteroclinic tangency. Citation: Rothos V. M., Bountis A.,  The Second Order Mel'nikov Vector, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 26-35 DOI:10.1070/RD1997v002n01ABEH000023