Lyapunov exponents

Chapter 4 covers much of the same ground as chapter 3 but from a more formal dynamical systems theory approach. The discrete CA world is examined in the context of what is known about the behavior of continuous dynamical systems, and a number of important methodological tools developed by dynamical systems theory (i.e. Lyapunov exponents, invariant measures, and various measures of entropy and  [c.18]

Since the speed of information propagation is, as we shall see in chapter 4, related to the Lyapunov exponent for the CA evolution, and is a direct measure of the sensitivity to initial conditions, it should not be surprising to learn that various rules can also be distinguished by the degree of predictability for the outcome of  [c.65]

For regular motion, T> t) grows only linearly with time, so that the exponents are all zero. On the other hand, because chaotic flows are characterized by exponential divergences of initial nearby trajectories, a characteristic signature of such flows is the existence of at least one positive Lyapunov exponent.  [c.202]

Lyapunov exponents can also be used to predict the mean expansion rate of a volume, AF(f) = Sxi(t) Sxn(t) in the phase space, F  [c.202]

It has been suggested that the signature, S = ( sign (Ai), sign (A2),. .., sign (A )), of the Lyapunov exponents may be used to provide a qualitative characterization of of attractors for dissipative flows. Among the possibilities for n = 4, for example, we have the following  [c.202]

Except for simple cases, it is generally a nontrivial task to compute the Lyapunov exponents of a flow. In trying to estimate A(x(0)) in equation 4.59, for example, the exponentially increasing norm, V t), may lead to computer overflow problems.  [c.202]

Fig. 4.13 Lyapunov exponent versus a for 2.9 < a < 4 for the logistic equation see text. Fig. 4.13 Lyapunov exponent versus a for 2.9 < a < 4 for the logistic equation see text.
As defined above, the Lyapunov exponents effectively determine the degree of chaos that exists in a dynamical system by measuring the rate of the exponential divergence of initially closely neighboring trajectories. An alternative, and, from the point of view of CA theory, perhaps more fundamental interpretation of their numeric content, is an information-theoretic one. It is, in fact, not hard to see that Lyapunov exponents are very closely related to the rate of information loss in a dynamical system (this point will be made more precise during our discussion of entropy in the next section).  [c.205]

The CA analogs of Lyapunov exponents are the left (=Al) and right (=Xr) slopes of the difference pattern between two configurations differing at one site. i and r thus measure the average rate of information transmission to the left and right of the lattice. If the sole differing site is located at i = 0, and Xr are defined by  [c.206]

For example, the time average definition of the Lyapunov exponent for one-dimensional maps, A = lim v->oo (which is often difficult to calculate in prac-  [c.208]

Lyapunov Characteristic Exponents  [c.201]

Lyapunov Characteristic Exponents  [c.201]

As we have seen, a fundamental property of chaotic motion is sensitivity to small changes to initial conditions. Initially closely separated starting conditions evolving along regular dynamical trajectories diverge only linearly in time a chaotic evolution, on the other hand, leads to an exponential divergence in time. Lyapunov characteristic exponents quantify this divergence - for both conservative and dissipative systems - by measuring the mean rate of exponential divergence of initially neighboring trajectories.  [c.201]

It can be shown that A both exists and is finite. Moreover, we can always find a set of n tangent-space basis vectors, c (i = 1,... n), such that Ax = Sxi,..., Sx ) — "The divergence (or contraction) along a given basis direction, e, is then measured by the j Lyapunov characteristic exponent, A. These n (possibly  [c.202]

Lyapunov Characteristic Exponents  [c.203]

Lyapunov Characteristic Exponents  [c.205]

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension.  [c.3059]

Mankin [18] who also used reconstmction from tire electrode potentials of Pt and Br , and d[Pt]/dt as independent variables. These demonstrations of tme chemical chaos were achieved by a number of tlien new methods power-spectral analysis [16,19], trajectory reconstmction in phase space [16], and next-amplitude-map analysis [1 20, 21]. The existence of tme chemical chaos was signalled by a positive Lyapunov exponent calculated from tire experimental return map. Since tliese early investigations, chaos has been documented in a variety of chemical systems. One aspect of tliese CSTR experiments was the observation tliat tire stirring rate moved tire bifurcation point(s) even if tliis rate was very large [22]. This effect depends on turbulent mixing and can be controlled but not eliminated by keeping tire stirring rate constant. We now give examples of two related dynamical systems teclmiques used by experimentalists phase-space reconstmction of chaotic attractors and tire analysis of tire associated next-amplitude maps. First, we discuss a study where an attractor was reconstmcted from experimental data and tlien used to obtain a next-amplitude map [17].  [c.3060]

If there is no defect present, the temporal change of this zigzag region is period-2. The teinporsl change of the domain boundary is chaotic (i.e. it has a positive Lyapunov exponent).  [c.392]

Many of the conventional measures used in studying dynamical systems - power spectra, entropy, Lyapunov exponents, etc. - can in fact be used to quantify the difference among Regimes I-IV ([kaneko89a], [kaneko89c]).  [c.394]

Lyapunov Dimension An interesting attempt to link a purely static property of an attractor, - as embodied by its fractal dimension, Dy - to a dynamic property, as expressed by its set of Lyapunov characteristic exponents, Xi, was, first made by Kaplan and Yorke in 1979 [kaplan79]. Defining the Lyapunov dimension, Dp, to be  [c.213]

See pages that mention the term Lyapunov exponents : [c.203]    [c.206]    [c.215]    [c.215]    [c.732]    [c.3060]    [c.203]    [c.206]    [c.210]    [c.735]   
Cellular automata (2001) -- [ c.201 ]