Weakly Nonlinear Oscillators

The elementary example above reveals a more general truth There are going to be (at least) two time scales in weakly nonlinear oscillators. We ve already met this phenomenon in Figure 7.6.1, where the amplitude of the spiral grew very slowly compared to the cycle time. An analytical method called two-timing builds in the fact of two time scales from the start, and produces better approximations... 

The same steps occur again and again in problems about weakly nonlinear oscillators. We can save time by deriving some general formulas. 

Consider the equation for a general weakly nonlinear oscillator ... 

The effects of coupling are represented by an interaction function F which, in general, is a 27r-periodic function of the phase difference with F(0) = 0. For weakly coupled, weakly nonlinear oscillators F has the universal form [37, 42]... 

Analytical approaches applicable for small and large amplitudes (for weak and strong nonlinearity) of the oscillations in a nonlinear dynamic system subjected to the influence of a wave has been developed (Damgov, 2004 Damgov, Trenchev and Sheiretsky, 2003). 

The nonlinear directional coupler is potentially a useful device because it has four ports, two input and two output, and because the outputs can be manipulated with either one or two inputs. Optimally the two channels are identical, and the coupling occurs through field overlap between the two channels. As a result, when only one of the channels is excited with low powers at the input, the power oscillates between the two channels with a beat length Lb, just like what occurs in a pair of weakly coupled identical pendulii. As the input power is increased, a mismatch is induced in the wavevectors of the two channels, which decreases the rate of the power transfer with propagation distance. This leads to an increase in the effective beat length. There is a critical power associated with this device, for which an infinitely long, lossless NLDC acts as a 50 50 splitter, that is, Lb oo. 

Interaction with External Fields. The models considered exhibit cooperative behaviour through nonlinear internal oscillations (models 1, 2, 4) or through nonlinear resonances (model 3). This makes plausible the existence of effects, when the system is driven by weak external fields of appropriate frequency. 

Exercises 8.4.5-8.4.11 deal with the forced Duffing oscillator in the limit where the forcing, detuning, damping, and nonlinearity are all weak ... 

It is obvious that in the real physical situations we are not able to avoid dissipation processes. For dissipative systems, we cannot take an external excitation too weak (the parameter e cannot be too small) since the field interacting with the nonlinear oscillator could be completely damped and hence, our model could become completely unrealistic. Moreover, the dissipation in the system leads to a mixture of the quantum states instead of their coherent superpositions. Therefore, we should determine the influence of the damping processes on the systems discussed here. To investigate such processes we can utilize various methods. For instance, the quantum jumps simulations [38] and quantum state diffusion method [39] can be used. Description of these two methods can be found in Ref. 40, where they were discussed and compared. Another way to investigate the damping processes is to apply the approach based on the density matrix formalism. Here, we shall concentrate on this method [12,41,42]. 

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