Damping effects evolution

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. 

At a nonlinear stage, the evolution of waves can be damped by heat transport to the chamber walls, which has been ignored in the analysis above. We assumed that at the initial stage the effect of heat transport to the chamber walls is small this assumption led to instability. However, as the stack temperature increases, heat flux to the chamber walls also increases and this may damp the further instability growth. This scenario will cause oscillations of cell and stack voltage. The oscillations could be identified by their period, which is in the order from ten minutes to several hours. 

As mentioned in the introduction, the (linear) Bloch equations constitute the simplest possible description of spin relaxation phenomena. Erker and Augustine considered the case of non-linear Bloch equations, corrected for the effect of radiation damping and inhomogeneous broadening. Stockmann and Dubbers generalized the Bloch equations to include the evolution of polarization vectors of various ranks in arbitrary multipole fields. The paper makes extensive use of irreducible tensors and provides a concise derivation of the relaxation effects. 

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