Critical behavior of crisis-induced transition to spatiotemporal chaos in parameter space

In previous works we reported a transition mechanism from a temporal chaos (TC) to spatiotemporal chaos (STC) through a crisis due to a collision to a saddle steady wave (SSW). However, the transition also displays as a critical phenomenon in parameter space. In the present work the time variations of mode interaction energy, deltaE(I)(k)(t), of the perturbation wave (PW) with its carrier SSW are calculated. In the TC state in all the dimensions the motion is dominated by negative deltaE(I)(k)(t). With variation of the parameter in one dimension deltaE(I)(k=1)(t) becomes smaller and smaller while statistically more balanced in its negative and positive values. The critical parameter point for the crisis is right at the place where the time-averaged negative and positive deltaE(I)(k=1)(t) are equal. A power-law behavior is observed when approaching to the point. After the crisis in the STC state the motion with positive deltaE(I)(k=1)(t) suddenly becomes much stronger than that with negative ones. In addition, it is shown that stable orbit of the SSW is a boundary of the PW motion, it behaves like a potential well that constrains the PW motion.