Direct Hopf Bifurcation in Parametric Resonance of Hybridized Waves.

We study parametric resonance of interacting waves having the same wave vector and frequency. In addition to the well-known period-doubling instability we show that under certain conditions the instability is caused by a Hopf bifurcation leading to quasiperiodic traveling waves. It occurs, for example, if the group velocities of both waves have different signs and the damping is weak. The dynamics above the threshold is briefly discussed. Examples concerning ferromagnetic spin waves and surface waves of ferro fluids are discussed. PACS numbers: 03.40.Kf, 76.50.+g, 47.20.-k Typeset using REVTEX 1 Parametric resonance is an instability phenomenon that is responsible for the excitation of modes of a weakly damped system due to periodic modulation of parameters which determine the frequencies of the modes [1]. The prototypical example is the vertically driven pendulum where the modulated parameter is the acceleration of gravity [2]. In spatially extended systems these modes are waves. Experimentally parametric resonance is well-known for spin waves (Suhl instabilities) [3–5] and surface waves of fluids (Faraday instability) [6–8]. Let us briefly summarize the well-known behavior of the one-wave case where only one wave type is involved. The ground state (e.g., the flat surface of a fluid) becomes unstable if the driving (i.e., modulation) amplitude exceeds some threshold which is proportional to the damping constant. The instability is caused by the excitation of waves fulfilling the first-order parametric resonance condition