Transition of One Mode-Phase at the Crisis and Onset of Spatiotemporal Chaos

In our previous work Phys. Rev. Lett. 80 (1998) 696 we investigated a crisis-induced transition from temporal chaos to spatiotemporal chaos (STC) due to a collision between the unstable orbit of the carrier saddle steady wave (SSW) and the attractor of its perturbation wave (PW). In this letter, we find that at the crisis one PW mode-phase experiences a state transition. The variations of PW mode-phases are governed by the linear dispersion as well as two competitive effects arising from the system nonlinearity: (1) interaction between the SSW and the PW (SP); (2) self-nonlinearity of the PW (PP). With the increasing importance of system nonlinearity we find that before the crisis for all PW modes the SP effects dominate and the mode-phases vibrate chaotically; while after the crisis for one PW mode the PP effect surpasses that of SP, in which case the phase transits to a state of combination of chaotic whirling and vibrating, which is different from the case of weak nonlinearity. In the meantime, the PW partial wave is free from the trapping of its carrier, and can be driven or damped chaotically depending on the phase difference relative to the carrier SSW, causing the STC motion after the crisis.