Lyapunov-exponent

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. 

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... 

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... 

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. 

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 ... 

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 ... 

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. 

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). 

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... 

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-... 

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). 

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]). 

The largest Lyapunov exponents for the relativistic hydrogenlike atom in a uniform magnetic field... 

Alternate mass-core hard potential channel In the two billiard gas models just discussed there is no local thermal equilibrium. Even though the internal temperature can be clearly defined at any position(Alonso et al, 2005), the above property may be considered unsatisfactory(Dhars, 1999). In order to overcome this problem, we have recently introduced a similar model which however exhibits local thermal equilibrium, normal diffusion, and zero Lyapunov exponent(Li et al, 2004). 

The results of the previous section have already established that classical chaos and quantum mechanics are not incompatible in the macroscopic limit. The question then naturally arises whether observed quantum mechanical systems can be chaotic far from the classical limit This question is particularly significant as closed quantum mechanical systems are not chaotic, at least in the conventional sense of dynamical systems theory (R. Kosloff et.al., 1981 1989). In the case of observed systems it has recently been shown, by defining and computing a maximal Lyapunov exponent applicable to quantum trajectories, that the answer is in the affirmative (S. Habib et.al., 1998). Thus, realistic quantum dynamical systems are chaotic in the conventional sense and there is no fundamental conflict between quantum mechanics and the existence of dynamical chaos. 

Figure 2. f inite-time Lyapunov exponents A( t J tor a driven I lulling oscillator with measurement strengths k = 5 x 10-4, 0.01, 10, averaged over 32 trajectories (linear scale in time, top, and logarithmic scale, bottom bands indicate the standard deviation over the 32 trajectories) (S. Habib et.al., 1998).The (analytic) 1/f fall-off at small k values (dashed red line), prior to the asymptotic regime, is evident in the bottom panel. 

We stress that the chaos identified here is not merely a formal result - even deep in the quantum regime, the Lyapunov exponent can be obtained from measurements on a real system. Quantum predictions of this type can be tested in the near future, e.g., in cavity QED and nanomechanics experiments (H. Mabuch et.al., 2002 2004). Experimentally, one would use the known measurement record to integrate the SME this provides the time evolution of the mean value of the position. From this fiducial trajectory, given the knowledge of the system Hamiltonian, the Lyapunov exponent can be obtained by following the procedure described above. It is important to keep in mind that these results form only a starting point for the further study of nonlinear quantum dynamics and its theoretical and experimental ramifications. 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

Lyapunov exponents then equal the logarithm of the eigenvalues of A (Eckmann and Ruelle, 1985). 

We note that the integral over the energetically allowed phase space— that is, the classical level density (97)—was found in Fig. 20 to be in excellent agreement with the quantum-mechanical level density. This finding indicates that there is a valid correspondence between the quantum-mechanical two-state system and its classical mapping representation. A similar conclusion was drawn in a recent smdy of a mapped two-state problem, which focused on the Lyapunov exponents and the energy level statistics of the system [124, 235]. 

On the other hand, it is well known that there is a relationship between Lyapunov exponents and the divergence of the vector field deduced from the differential equations describing a dynamical system. This relation provides a test on the numerical values obtained from the simulation algorithm. This relationship is, according to the definition of Lyapunov exponents ... 

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