Chaos signature

In addition to tire period-doubling route to chaos tliere are otlier routes tliat are chemically important mixed-mode oscillations (MMOs), intennittency and quasi-periodicity. Their signature is easily recognized in chemical experiments, so tliat tliey were seen early in the history of chemical chaos. 

Signatures of quantum chaos in open chaotic billiards... 

The complex scattering wave function can be specified by nodal points at which u = 0,v = 0. They have great physical significance since they are responsible for current vortices. We have calculated distribution functions for nearest distances between nodal points and found that there is a universal form for open chaotic billiards. The form coincides with the distribution for the Berry function and hence, it may be used as a signature of quantum chaos in open systems. All distributions agree well with numerically computed results for transmission through quantum chaotic billiards. 

F. Haake, Quantum Signatures of Chaos, Springer, Berlin, 2000. 

S. Mukamel While there are some signatures of chaos in the linear response, my point is that the nonlinear response carries much more direct and sensitive information. The reason is that the stability matrix enters the nonlinear response directly, reflecting interference of initially close trajectories. Such interference is absent in the linear response. 

S.D. Chao, R.T. Skodje, Signature of reactive resonance Three case studies, Theor. Chem. Acc. 108 (2002) 273. 

With Fig. 6.5 we estabhshed that the phase space of the classical version of the SSE system contains chaotic regions. But since the SSE system is a manifestly quantum mechanical system, the central question is whether the classical chaos in the SSE system is at all relevant for the quantum dynamics, and, if yes, what are the signatures ... 

Kudrolli, A., Sridhar, S., Pandey, A. and Ramaswamy, R. (1994). Signatures of chaos in quantum billiards Microwave experiments, Phys. Rev. E49, R11-R14. 

Rutzky CP, Kaye CJ, Siciliano, MJ, Chao M Kahan BD (1980) Longitudinal karyotype and genetic signature analysis of cultured human colon adenocarcinoma cell Unes LS180 and LS174T. Cancer Research 40 1443-1448. 

Apart from the reactivity descriptors discussed so far, an important diagnostic of the quantum signature of classical chaos [63], viz., the uncertainty product is also calculated [28] which is defined as below,... 

Here s the intuition. Given some initial condition x, consider a nearby point Xq -t- Sg, where the initial separation Sg is extremely small. Let S be the separation after n iterates. If 5 then A is called the Liapunov exponent. A positive Liapunov exponent is a signature of chaos. 

Subsection A contains a summary of the formal definitions of chaotic behavior, derived from ergodic theory detailed discussions of this topic may be found elsewhere.11 We comment, in this section, on the gap that must be bridged in order to apply these concepts to chemical dynamics. Subsection B discusses some recent developments in computational signatures of chaos. In Subsection C we review a number of studies that have provided some of these links and that, in some instances, have resulted in new useful computational methods for treating the dynamics of reactions displaying chaotic dynamics. In addition, it includes a subsection on connection between formal ergodic theory and statistical behavior in unimolecular decay. 

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