Self-similar organization of Gavrilov-Silnikov-Newhouse sinks

Abstract

The numerical analyses of the Hénon map suggest the following features. As we increase the value of the control parameter around each stable period of the period-1 branch, an infinitely large series of period n-tupled saddle nodes appears in the following sequence (nϭ . . . ,5,4,3). The limit of each series is the infinitely large set of homoclinic points, created at the homoclinic tangency for the respective flip saddle ͑boundary saddle in the case of period 1͒. These observations are in good agreement with the predictions of Gavrilov, Silnikov, and Robinson. Each newly created sink, referred to as Gavrilov-Silnikov ͑GS͒ sink, later constitutes a first-order secondary cascade. The flip ͑boundary͒ saddles of these cascades also exhibit homoclinic tangency. Past such tangency, around the respective GS sink, an infinitely large series of period n-tupled saddle nodes (n ϭ . . . ,5,4,3) seems to appear in a similar manner. The newly created GS sinks later constitute second-order secondary cascades. These phenomena, comprised of the homoclinic tangency of a flip ͑boundary͒ saddle, followed by the sequential appearance of an infinitely large sequence of period n-tupled saddle nodes around the respective GS sink, appear to recur in a self-similar manner, creating higher-order and further higher-order GS sinks and the associated secondary cascades. Each secondary cascade survives within a small subinterval of the control parameter window where the respective GS sink from the immediate lower-order secondary cascade exists. These processes appear to continue ad infinitum. Therefore, in the limiting conditions, an infinitely large sequence of sinks may simultaneously coexist in the phase space for an infinitely large number of control parameter values. These observations are in good agreement with the predictions of Newhouse. Thus, the GS sinks may be identified as Gavrilov-Silnikov-Newhouse ͑GSN͒ sinks that are organized in a self-similar manner in the phase and parameter space. These features are very similar to those we recently observed in a periodically forced, damped Toda oscillator ͓B. K. Goswami, Phys. Rev. E 62, 2068 ͑2000͔͒. Since, the Hénon map and Toda oscillator are standard models ͑one from the maps and the other from the oscillators͒, our observations may provide some strong evidences towards universality in the self-similar organization of GSN sinks in the low-dissipative limit. The two most fascinating fundamental features in nonlinear science are ͑i͒ universality and ͑ii͒ order ͑self-similarity͒ within complexity ͓1-3͔. For instance, let us consider the classic example of chaos that could be an exceedingly complex irregular state of a nonlinear system. Notably, chaos reveals fractal nature in the projected phase space ͑maps͒ with inherent self-similarity ͓4͔.1 Such self-similarity has been observed in a large number of nonlinear systems ͑maps͒ in the interdisciplinary area of nonlinear science. Thus, the self-similarity is a universal feature of chaos.From chaos, we draw attention to another complex yet c...

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