Abstract
Hysteresis is a common phenomenon in a type of nonlinear waves. Corresponding to the negative tangential branch of a hysteretic curve the steady wave solutions are unstable due to saddle instability; such a saddle steady wave (SSW) is a moving saddle point in the system. In a model system based on a nonintegrable equation derived in fluid and plasma physics, a clear connection is found between turbulent solutions and the appearance of hystereses. A physical cause underlying this phenomenon is the embedded saddle point. In this review paper we focus on a senario of strong turbulent motion that is related to the saddle point. It is shown that in the reference frame moving with the SSW the nonlinear wave can be transformed into a set of coupled oscillators for which the SSW provides a potential; when the orbit of the oscillators collides with the saddle point a crisis onsets; subsequently occurs another critical event during which the master oscillator gets free from the trapping of the potential, inducing a transition to spatiotemporal chaos. The turbulent state after the transition is actually a special kind of self-organized motion characteristic by on-off imperfect phase synchronization among these oscillators.
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