Second Lyapunov value

The second Lyapunov method implies that one uses V values which have maxima at the rest point under study whose derivative [eqn. (98)] is not positive (V < 0) in the vicinity of this point and zero values are admitted only at this point. 

The system (9.3.5) or (9.3.4) is the normal form for the second critical case. The coefficients Lq are called the Lyapunov values. Observe from the above procedure that in order to calculate Lq one needs to know the Taylor expansion of the Eq. (9.3.1) up to order p + g == 2Q + 1. 

The next bifurcation that we will now focus on occurs when the first Lyapunov value vanishes. Here, after getting rid of terms of second and fourth order (the smoothness r of the map is assumed to be not less than five) the map may be reduced to the form... 

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. 

We note that the above results are not limited to the case of linear decay, but also apply to any kind of decay-type or stable reaction dynamics in a flow with chaotic advection (Chertkov, 1999 Hernandez-Garcfa et ah, 2002). In such systems where the reaction dynamics is nonlinear, the decay rate b should be replaced by the absolute value of the negative Lyapunov exponent of the Lagrangian chemical dynamics given by the second equation in (6.25), that represents the average decay rate of small perturbations in the chemical concentration along the trajectory of a fluid element. 

A function L satisfying the equation above is called a Lyapunov function. The second variation of entropy L = —d S may be used as a Lyapunov functional if the stationary state satisfies dKidf > 0. A functional is a set of functions that are mapped to a real or complex value. Hence, a nonequilibrium stationary state is stable if... 

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