Dynamics of weakly coupled parametrically forced oscillators

Abstract

The dynamics of two weakly coupled parametric oscillators are studied in the neighborhood of the primary subharmonic instability. The nature of both primary and secondary instabilities depends in a critical way on the permutation symmetries, if any, that remain after coupling is considered, and this depends on the relative phases of the parametric forcing terms. Detailed bifurcation sets, revealing a complex series of transitions organized in part by Bogdanov-Takens points, are calculated for representative sets of parameters. In the particular case of out-of-phase forcing the predictions of the coupled oscillator model are compared with direct numerical simulations and with recent experiments on modulated cross waves. Both the initial Hopf bifurcation and the subsequent saddle-node heteroclinic bifurcation are confirmed.

The CFVib Experiment: Control of Fluids in Microgravity with Vibrations

“…Lowviscosity fluids support more prominent surfaces waves, which can destabilize the underlying vibroequilibria state through the excitation of sloshing modes. This instability is correlated with the presence of modulated (quasiperiodic) subharmonic surfaces waves, which are typical at onset (Varas and Vega 2007;Porter et al 2013;Perez-Gracia et al 2014;Tinao et al 2014;Salgado Sanchez et al 2016). Even without the complications of surface wave dynamics, an interesting dependence on depth was found for the symmetric connected vibroequilibria state, which develops a large central trough (crater) as forcing is increased and the fluid is pushed toward the vibrating endwalls.…”

Finite-size effects on pattern selection in immiscible fluids subjected to horizontal vibrations in weightlessness

“…The oscillatory pressure gradient can also excite subharmonic waves, known as cross-waves [8], through a parametric forcing mechanism analogous to that of Faraday waves, but lacking the spatial homogeneity of the vertical case [9,10]. Depending on the extent to which these localized cross-waves penetrate inwards from the lateral walls, their interaction can induce temporal modulations [9,11] and a range of secondary bifurcations [12].…”

A review of fluid instabilities and control strategies with applications in microgravity

“…By selecting frequency and detuning (in the crosswise direction), one can control the nature and extent of these modulated solutions and produce patterns that cycle back and forth from one side to the other with variable period. The primary Hopf bifurcation as well as the saddle-node heteroclinic bifurcation that destroys the modulated states is well-captured by a coupled oscillator model [135], which suggests that a rich variety of additional dynamics can be selected by varying the forcing and container size. Figure 6 shows a cross-wave solution obtained experimentally by controlling, via the forcing frequency, the strength of interaction in the interior.…”

“…Since the forcing at the two endwalls is out of phase by π, the subharmonic waves they produce are out of phase by π/2. This phase difference leads to a primary Hopf bifurcation and correspondingly modulated solutions [120,124,135,165]. The modulation frequency is controlled by the strength of the interaction and, thus, by the damping, forcing frequency and container length.…”