Maximal Lyapunov exponent

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. 

Fig. 15. Maximal Lyapunov exponent versus maximum scale. If Sm > 5, then Am < 0 for all cases.

Fig. 16. Maximal Lyapunov exponent versus time evolution, (a) =0.87301,...

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

This system in its linear version (i.e., when e = 0) is a dynamical filter. Suppose that the oscillators interact with each other with the interaction parameter a = 0.9. The frequency 00 of the external driving field varies in the range 0 < < 4.2. The other parameters of the system are A 200, coq 1, c 0.1, and = 0.05. The autonomized spectrum of Lyapunov exponents A-4, >,5 versus the frequency to is presented in Fig. 23. In the range 0 < < 0.2 the system does not exhibit chaotic oscillation. Here, the maximal Lyapunov exponent Xi = 0 and the spectrum is of the type 0, —, —, —, (limit cycles). 

It is readily seen that the set of equations (76) consists of three equations of motion in the real variables ReIm c, w. If, (x) = constant, chaos in the system does not appear since the set (76) becomes a two-dimensional autonomous system. The maximal Lyapunov exponents for the systems (75) and (72)-(74) plotted versus the pulse duration T are presented in Fig. 36. We note that within the classical system (75) by fluently varying the length of the pulse T, we turn order into chaos and chaos into order. For 0 < T < 0.84 and 1.08 < 7) < 7.5, the maximal Lyapunov exponents Li are negative or equal to zero and, consequently, lead to limit cycles and quasiperiodic orbits. In the points where L] = 0, the system switches its periodicity. The situation changes dramatically if,... 

Figure 36. Maximal Lyapunov exponents for the system before (solid line) and after quantum...

Figure 38. The calssical (dashed) and quantum (solid line) maximal Lyapunov exponents (a) and the appropriate bifurcation maps (b,c) versus the modulated parameter Q. The parameters are...

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

A two-dimensional surface of section of phase space with the same parameters as Fig. 4a is shown in Fig. 4b. We measure maximal Lyapunov exponents averaged over a finite time interval and plot each of them to the initial point set on the section of phase space (q = qi = 0). The brighter region represents the more unstable region, and two vertical lines in the center of Figure 4b... 

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