A small change of one of the system parameters may not in general convert a bistable system to a monostable system. However, an external control in the form of a slow periodic parameter modulation can annihilate one of the coexisting states, and thus results in controlled monostability. The annihilation takes place because the state becomes chaotic via the period doubling route and the chaotic state undergoes boundary crisis within a small range of the control amplitude. These features are observed theoretically in two standard models, namely, Henon map and laser rate equations, and confirmed experimentally in a cavity loss modulated CO2 laser.

In multistable regimes, noise can create "multistate hopping intermittency," i.e., intermittent transitions among coexisting stable attractors. We demonstrate that a small periodic perturbation can significantly control such hopping intermittency. By "control" we imply a qualitative change in the probability distribution of occupation in the phase space around the stable attractors. In other words, if the uncontrolled system exhibits a preference to stay around a given attractor (say " A ") in comparison to another attractor (say " B "), the control perturbation creates a contrasting scenario so that attractor B is most frequently visited and consequently, the occupation probability becomes maximum around B instead of A . The control perturbation works in the following way: It destroys attractor A by boundary crisis while attractor B remains stable. As a result, even if the system is pushed by noise into the erstwhile basin of attractor A , the system does not remain there for long and therefore stays longer around attractor B . Significantly, such a change in the intermittent scenario can be obtained by a small-amplitude and slow-periodic perturbation. The control is theoretically demonstrated with two standard models, namely, Lorenz equations (for autonomous systems), and the periodically driven, damped Toda oscillator (for nonautonomous systems). Recent experiments with a cavity-loss modulated CO2 laser and an analog circuit of Lorenz equations have validated our theoretical demonstrations excellently.

A small perturbation of any system parameters may not in general create any significant qualitative change in dynamics of a multistable system. However, a slow-periodic modulation with properly adjusted amplitude and frequency can do so. In particular, it can control the number of coexisting attractors. The basic idea in this controlling mechanism is to introduce a collision between an attractor with its basin boundary. As a consequence, the attractor is destroyed via boundary crisis, and the chaotic transients settle down to an adjacent attractor. These features have been observed first theoretically with the Hénon map and laser rate equations, and then confirmed experimentally with a cavity-loss modulated CO 2 laser and a pump-modulated fiber laser. The number of coexisting attractors increases as the dissipativity of the system reduces. In the low-dissipative limit, the creation of attractors obeys the predictions of Gavrilov, Shilnikov and Newhouse, when the attractors, referred to as Gavrilov–Shilnikov–Newhouse (GSN) sinks, are created in various period n-tupling processes and remain organized in phase and parameter spaces in a self-similar order. We demonstrate that slow small-amplitude periodic modulation of a system parameter can even destroy these GSN sinks and the system is suitably converted again to a controllable monostable system. Such a control is robust against small noise as well. We also show the applicability of the method to control multistability in coupled oscillators and multistability induced by delayed feedback. In the latter case, it is possible to annihilate coexisting states by modulating either the feedback variable or a system parameter or the feedback strength.

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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