Nonchaotic systems

The trajectory q pl (t) is determined by minimizing S in (20) on the set of all classical deterministic trajectories determined by the Hamiltonian H (37), that start on a stable limit cycle as t — —oo and terminate on a saddle cycle as t > oo. That is, qopt(t) is a heteroclinic trajectory of the system (37) with minimum action, where the minimum is understood in the sense indicated, and the escape probability assumes the form P exp( S/D). We note that the existence of optimal escape trajectories and the validity of the Hamiltonian formalism have been confirmed experimentally for a number of nonchaotic systems (see Refs. 62, 95, 112, 132, and 172 and references cited therein). 

In short, under the effect of small fluctuations the system will visit the phase space with probabilities close to the ones corresponding to the deterministic attractor, provided that the above-mentioned conditions are satisfied. Therefore deterministic chaos should fully retain its relevance and its implications in the presence of noise, in that it determines (up to a small correction) the probabilistic structure of the system in the limit of long times. Interestingly, in this perspective it is on nonchaotic systems that fluctuations seem to have the most drastic effect, since the perturbation of the (now) singular distribution of the deterministic system by the noise, however small, leads to a smooth distribution. 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

Chaotic behavior of the system predominates over nonchaotic behavior— for most values of the parameter , one Lyapunov exponent is positive. The most spectacular behavior of the coupled oscillators is observed in the region... 

These simple and mostly intuitive arguments apply mainly to chemical and biological reaction engineering systems. For other systems such as fluid flow systems, the sources and causes of bifurcation and chaos can be quite different. It is well established that the transition from laminar flow to turbulent flow is a transition from nonchaotic to chaotic behavior. The synergetic interaction between hydrodynamically induced bifurcation and chaos and that resulting from chemical and biological reaction/diffusion is not well studied and calls for an extensive multidisciplinary research effort. 

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