Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

Abstract

The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to non-autonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Luke Lagrangian density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates (or, equivalently, three degrees of freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The generalized momenta are computed from the Lagrangian, and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains both autonomous and non-autonomous components that must be suspended by applying an extension of the Herglotz algorithm for non-autonomous transformations in order to apply the Kolmogorov-Arnold-Moser (KAM) averaging operation and the GMM. Three canonical transformations are applied to (i) eliminate cross product terms by a rotation of axes; (ii) to transform to action-angle canonical variables and to eliminate two degrees of freedom; and (iii) to suspend the non-autonomous terms and to apply the Hamilton-Jacobi transformation. The system of nonlinear non-autonomous evolution equations determined from Hamilton’s equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic separatrices are computed from the unperturbed autonomous system. The non-dissipative perturbed Hamiltonian system with surface tension satisfies the KAM non-degeneracy requirements, and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic separatrix; consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents.