Are Cross-Waves Chaotic?

Abstract

The Wiggins-Holmes extension of the generalized Melnikov method (GMM) is applied to parametrically forced cross-waves in a long rectangular channel in order to determine if these cross-waves are chaotic. A great deal of effort is required in order to obtain a suspended system for the application of the GMM. The Lagrangian for surface gravity water waves is transformed to surface integrals in order to eliminate constant conjugate momenta. The Lagrangian is simplified by subtracting the zero variation integrals and is then expressed in terms of generalized coordinates given by the time dependent components of the cross-wave and progressive-wave velocity potentials. The conjugate momenta and the Hamiltonian are computed by the Legendre transform of the Lagrangian. The first-order differential equations obtained from derivatives of the Hamiltonian are usually suitable for applications of the GMM. However, the cross-wave Hamiltonian contains a variation in the third direction (3.3c) that is 100% non-autonomous and will not survive the averaging theorem required for the Wiggins-Holmes extension of the GMM. A sequence of seven canonical transformations are required in order to suspend the Hamiltonian in the third direction. The unperturbed system is analyzed to determine hyperbolic fixed points and the equations describing the heteroclinic orbits for near resonance cases. The Melnikov function is calculated from the perturbed system that must also satisfy KAM conditions. The Melinkov results indicate that the system is chaotic near resonance. When the heteroclinic orbit about which the chaotic motions occur is transformed back to the original set of variables, the orbit is shown to be extremely complicated and would be impossible to determine analytically without the canonical transformations and the suspended Hamiltonian.