Kerr oscillators

Coherent External Field Modulated External Field Pulsed External Field Final Remarks Chaos in Kerr Oscillators A. Introduction Basic Equations... 

After an overview of the main papers devoted to chaos in lasers (Section I.A) and in nonlinear optical processes (Section I.B), we present a more detailed analysis of dynamics in a process of second-harmonic generation of light (Section II) as well as in Kerr oscillators (Section III). The last case we consider particularly in the context of coupled nonlinear systems. Finally, we present a cumulant approach to the problem of quantum corrections to the classical dynamics in second-harmonic generation and Kerr processes (Section IV). 

One of the best known and most intensively studied optical models is an oscillator with Kerr nonlinearity. Mutually coupled Kerr oscillators can be successfully used for a study of couplers the systems consist of a pair of coupled Kerr fibers. The first two-mode Kerr coupler was proposed by Jensen [136] and investigated in depth [136,137]. Kerr couplers affected by quantization can... 

In this section we consider a model of interactions between the Kerr oscillators applied by J. Fiurasek et al. [139] and Perinova and Karska [140]. Each Kerr oscillator is externally pumped and damped. If the Kerr nonlinearity is turned off, the system is linear. This enables us to perform a simple comparison of the linear and nonlinear dynamics of the system, and we have found a specific nonlinear version of linear filtering. We study numerically the possibility of synchronization of chaotic signals generated by the Kerr oscillators by employing different feedback methods. 

The Hamilton function for a single Kerr oscillator is defined by... 

For e = 0, the quantities (10) and (17) become first integrals for the harmonic oscillator [141]. It is obvious from (15)—(16) that a trajectory in phase space (p, q) for the Kerr oscillator is analytically the same ellipse as for the harmonic oscillator... 

The only difference is that for the harmonic oscillator the phase point draws the ellipse with the frequency too, whereas for the Kerr oscillator with the frequency, = o>o[l + e(pq + 0) 2)]. The frequency depends on the initial conditions, which is a feature typical of nonlinear conservative systems [143]. 

The single Kerr anharmonic oscillator has one more interesting feature. It is obvious that for Cj = 0 and y- = 0, the Kerr oscillator becomes a simple linear oscillator that in the case of a resonance 00, = (Do manifests a primitive instability in the phase space the phase point draws an expanding spiral. On adding the Kerr nonlinearity, the linear unstable system becomes highly chaotic. For example, putting A t = 200, (D (Dq 1, i = 0.1 and yj = 0, the spectrum of Lyapunov exponents for the first oscillator is 0.20,0, —0.20 1. However, the system does not remain chaotic if we add a small damping. For example, if yj = 0.05, then the spectrum of Lyapunov exponents has the form 0.00, 0.03, 0.12 1, which indicates a limit cycle. 

These Kerr oscillators, with j = 2 = 0, are linear subsystems that in the case of resonance (oa = (Oi = (02) exhibit a common instability—the solutions of Eqs. (43) and (44) for t > 00 grow linearly without bound. This resonance instability of our linear subsystems vanishes for C / 0 and 2 / 0. The subsystems become stable but only for small values of ei and 2. For example, beats generated by the first oscillator for C = 10 9, A = 200, and fi>o = i = 1 are illustrated in Fig. 29a, and the appearing beats originate from the Kerr nonlinearity. 

Let us now consider a system of two nonlinearly coupled Kerr oscillators. Now, we write the Hamiltonian (25) in the form... 

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