Phase-locking and resonant islands in the Zakharov equations

We identify phase-locked states among the solutions of the Zakharov equations. The phase-locked states appear as resonant island chains in the appropriate Poincaré plots. The relevant surface of section is 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 de-stabilizing an initial homogeneous steady-state. If one is very close to the bifurcation point, nonlinear saturation of the initial instability is provided by quasi-static integrable ion-acoustic fluctuations but as one proceeds away from the bifurcation point, resonant non-integrable ion-acoustic fluctuations become gradually more important; we show that the phase-locked states result from those resonant fluctuations. The resonance separatrix appears to bring the first chaotic activity into the system.