Chaotic attractors nonhyperbolic attractor

In this section the application of the optimal path approach to the problem of escape from a nonhyperbolic and from a quasihyperbolic attractor is examined. We discuss these two different types of chaotic attractor because it is known [160] that noise does not change very much the structure and properties of quasi-hyperbolic attractors, but that the structure of non-hyperbolic attractors is abruptly changed in the presence of noise, with a strong dependence on noise intensity. Note that for optical systems both types of chaotic attractor [161-163] (nonhyperbolic and quasihyperbolic) are observed, but a nonhyperbolic attractor is much more typical. 

Here u(t) is the control function. It is a system where chaos can be observed at relatively small values h m 0.1 of the driving force amplitude and the chaotic attractor is a nonhyperbolic attractor or a quasiattractor [167]. 

A simplified parameter space diagram obtained numerically [168] is shown in Fig. 13. The dashed lines bound the region in which both the linear and nonlinear responses of period 1 coexist. The upper line marks the boundary of the linear response, and the lower line marks that for the nonlinear responses. The boundaries of hysteresis for the period 1 resonance are shown by solid lines. The region in which linear response coexists with one or two nonlinear responses of period 2 is bounded by dotted lines. This region is similar to the one bounded by dashed lines. The region of coexistence of the two resonances of period 2 is bounded by the dashed-dotted line. Chaotic states are indicated by small dots. The chaotic state appears as the result of period-doubling bifurcations, and thus corresponds to a nonhyperbolic attractor [167]. Its boundary of attraction Sfl is nonfractal and is formed by the unstable manifold of the saddle cycle of period 1 (SI). 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

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See also in sourсe #XX -- [ , , , , , , , , , , , ]

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