Chaos in a long rectangular wave channel Public Deposited

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/sn00b192r

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  • The Melnikov method is applied to a model of parametrically generated cross-waves in a long rectangular channel in order to determine if these cross-waves are chaotic. A great deal of preparation is involved in order to obtain a suitable form for the application of the Melnikov method. The Lagrangian for water waves, which consists of the volume integrals of the kinetic energy density, potential energy density, and a dynamic pressure component, is transformed to surface integrals in order to avoid constant conjugate momenta. The Lagrangian is simplified by subtracting the zero variation integrals and written in terms of generalized coordinates, the time dependent components of the crosswave and progressive wave velocity potentials. The conjugate momenta are calculated after expanding the Lagrangian in a Taylor series. The Hamiltonian is then determined by a Legendre transformation of the Lagrangian. Ordinarily, the first order evolution equations obtained from derivatives of the Hamiltonian are suitable for applications of the Melnikov method. However, the crosswave model results in extremely complicated evolution equations which must be simplified before a Melnikov analysis is possible. A sequence of seven canonical transformations are applied and yield a final set of evolution equations in fairly simple form. 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 for the perturbed system which must also satisfy KAM conditions. The Melnikov results indicate the system is chaotic near resonance. Furthermore, the heteroclinic orbits, about which chaotic motions occur, are transformed back to the original set of variables and found to be extremely complicated; this orbit would be impossible to determine analytically without the canonical transformations. The theoretical results were verified by experiments. Poincare maps obtained from measurements of the free surface displacement indicate both quasi-periodic and chaotic motions of the water surface. Power spectra and time series of the water surface displacement are also analyzed for chaotic behavior, with less conclusive results. Stability diagrams of cross-wave generation confirm behavior consistent with parametric excitation.
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