Low-dimensional phase-locked states in the zakharov equations

Abstract

In this paper we identify phase-locked states among the solutions of the Zakharov equations. Locked states appear as resonant island chains in the appropriate Poincaré plots, with the relevant surface of section obtained by projecting out the full dynamical set on a subspace defined in terms of a pair of center-manifold variables. This pair allows an accurate canonical description of the system immediately after an inverse pitchfork bifurcation destabilizes an initial homogeneous steady state. If one is very close to the bifurcation point, nonlinear saturation of the initial instability is provided by quasistatic integrable ion-acoustic fluctuations, but as one proceeds away from that point, resonant nonintegrable ion-acoustic fluctuations become gradually more important; we show that the phase-locked states result from those resonant fluctuations. If one is not too far from the pitchfork bifurcation, locking is the stable asymptotic state of the interaction. As one moves farther away, locking exists only over long but finite amounts of time. In addition, the resonance separatrix appears to bring the first chaotic activity into the system.