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Additional info for The Transition to Chaos: In Conservative Classical Systems: Quantum Manifestations
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Let us next introduce the partition aVTa which consists of elements, A~O)nAy) (iJ=1,2), where n denotes the intersection of the elements A~O) and AJI). The partition, aVTo:, is shown in Fig. a). Similarly the elements of the partition, aVTaVT2a , are shown in Fig. b). VTn-I a ) , (2612) .. 13) and the sum is taken over all elements of partition 0.. The maximum entropy occurs when the elements of a partition all have equal area. If we assume that our partitions do have equal area then it is easy to see that h(avTaV...
Al. 1976]: 1. The n~CX) lim kn{T,Xo " 0, Yo 0) = k{T,Xo• 0, Yo 0) exists; • 2. k{T, Xo,o, Yo,o) is independent of T; 3. k{T, Xo,o, Yo,o) is independent of do; 4. k{T, Xo,o, Y0,0) = 0 if Xo,o is chosen to lie in a regular region of the energy surface; 5. k{T, Xo,o, Y0,0) is independent of Xo,o and is positive if Xo,o is chosen to lie in a chaotic region of the energy surface. Therefore, in a chaotic region of the energy surface we can write k{E) = k{T,Xo,o, Yo,o). The quantity, k{E) obtained in this manner is the largest Lyapounov exponent, AN-I.
1. Lyapounov Characteristic Exponent Oseledec [Oseledec 1974] was first to show that a procedure analogous to that used to study exponential divergence of flow in the neighborhood of hyperbolic fixed points could be used to study the nature of the flow in the neighborhood of a moving point in phase space. To see how this works, consider a system with N degrees of freedom (2N dimensional phase space). We shall denote the 2N dimensional vector describing the state of the system at time, t, by X{" = X N (Pl(t), ...