Graduate Thesis Or Dissertation
 

Application of the generalized Melnikov method to weakly damped parametrically excited cross waves with surface tension

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  • The Wiggins-Holmes extension of the generalized Melnikov method (GMM) is applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to determine if these cross waves are chaotic. The Lagrangian density function for surface waves with surface tension is simplified by transforming the volume integrals to surface integrals and by subtracting the zero variation integrals. The Lagrangian is written in terms of the three generalized coordinates (or, equivalently the three degrees of freedom) that are the time-dependent components of the velocity potential. A generalized dissipation function is assumed to be proportional to the Stokes material derivative of the free surface. The generalized momenta are calculated from the Lagrangian and the Hamiltonian is determined from a Legendre transformation of the Lagrangian. The first order ordinary differential equations derived from the Hamiltonian are usually suitable for the application of the GMM. However, the cross wave equations of motion must be transformed in order to obtain a suspended system for the application of the GMM. Only three canonical transformations that preserve the dynamics of the cross wave equations of motion are made because of an extension of the Herglotz algorithm to nonautonomous systems. This extension includes two distinct types of the generalized Herglotz algorithm (GHA). The system of nonlinear nonautonomous evolution equations determined from Hamilton's equations of motion of the second kind are averaged in order to obtain an autonomous system. The unperturbed system is analyzed to determine hyperbolic saddle points that are connected by heteroclinic orbits The perturbed Hamiltonian system that includes surface tension satisfies the KAM nondegeneracy requirements; and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the Melnikov integral is identically zero implying that a higher dimensional GMM is necessary in order to demonstrate by the GMM that the motion is chaotic. However, numerical calculations of the largest Liapunov characteristic exponent demonstrate that the perturbed dissipative system with surface tension is also chaotic. A chaos diagram is computed in order to search for possible regions of the damping parameter and the Floquet parametric forcing parameter where chaotic motions may exist.
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