Bifurcation backward

The transition was found to be mediated by nucleation and traveling of sharp fronts (Fig. 11a) that indicates a backward bifurcation, although hysteresis has not been identified directly. Rather, a sharp jump in the contrast (pattern amplitude) with increasing voltage has been detected, with some indications that a low contrast pattern already arises at voltages before the jump occurs in Fig. 1 lb. A preliminary, weakly non-linear analysis has exhibited a bifurcation, which is in fact weakly supercritical at low frequencies. If small changes of the parameters and/or additional effects are included (e.g. flexoelectricity and weak-electrolyte effects) the bifurcation could become a more expressed subcritical one [32, 33]. 

The changes of lateral force F in forward and backward motions follow the curve 1 and 2, respectively. It can be observed that there is one saddle-node bifurcation for the repulsive pinning center, but two bifurcations for the attractive piiming center. This suggests that the interfacial instability results from different mechanisms. On one hand, the asperity suddenly looses contact as it slides over a repulsive pinning center, but in the attractive case, on the other hand, the... 

Linxiang, W, Yueun, F., Ying, C., Animation of chaotic mixing by a backward Poincare cell-map method, Int. J. Bifurcation Chaos 11, 7 (2001) 1953-1960. 

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