A Center-Unstable Manifold Theorem for Parametrically Excited Surface Waves

A Center-Unstable Manifold Theorem for Parametrically Excited Surface Waves

When fluid in a rectangular tank sits upon a platform which is oscillating with sufficient amplitude, surface waves appear in the "Faraday resonance." Scientists and engineers have done bifurcation analyses which assume that there is a center manifold theory using a finite number of excited spatial modes. We establish such a center manifold theorem for Xiao-Biao Lin’s model in which potential f...

Control of spatiotemporal disorder in parametrically excited surface waves.

Interacting surface waves, parametrically excited by two commensurate frequencies, yield a number of nonlinear states. Near the system's bicritical point, a state, highly disordered in space and time, results from competition between nonlinear states. Experimentally, this disordered state can be rapidly stabilized to a variety of nonlinear states via open-loop control with a small-amplitude thi...

Grid states and nonlinear selection in parametrically excited surface waves.

Interacting surface waves, parametrically excited by two commensurate frequencies (Faraday waves), yield a rich family of nonlinear states, which result from a variety of three-wave resonant interactions. By perturbing the system with a third frequency, we selectively favor different nonlinear wave interactions. Where quadratic nonlinearities are dominant, the only observed patterns correspond ...

Parametrically Forced Surface Waves

Necessary conditions for mode interactions in parametrically excited waves.

We study the spatial and temporal structure of nonlinear states formed by parametrically excited waves on a fluid surface (Faraday instability), in a highly dissipative regime. Short-time dynamics reveal that 3-wave interactions between different spatial modes are only observed when the modes' peak values occur simultaneously. The temporal structure of each mode is functionally described by the...