Self-oscillatory modes

A. Existence of Spatially Nonuniform Self-Oscillatory Modes... 

Numerical simulations were performed on the planar problem in [83], which showed that the transition to the self-oscillatory combustion occurs when a parameter related to the Zeldovich number is increased. Numerical studies [5, 6, 79] of the one-dimensional problem found transitions to relaxation oscillations and period doublings, and these studies demonstrated two routes to chaotic dynamics as the bifurcation parameter related to the Zeldovich number was increased. Numerical studies [36 1,1,7,4] of the two- and three-dimensional model found spinning modes of propagation as well as standing modes, which describe multiple point propagation, and quasi-periodic modes of propagation. 

Before we can start to develop a model we also have to decide how to interpret the behavior observed in Fig. 2.1. The variations in insulin and glucose concentrations could be generated by a damped oscillatory system that was continuously excited by external perturbations (e.g. through interaction with the pulsatile release of other hormones). However, the variations could also represent a disturbed self-sustained oscillation, or they could be an example of deterministic chaos. Here, it is important to realize that, with a sampling period of 10 min over the considered periods of 20-24 h, the number of data points are insufficient for any statistical analysis to distinguish between the possible modes. We need to make a choice and, in the present case, our choice is to consider the insulin-glucose regulation to operate... 

Besides the phase of the fundamental mode, strictly speaking, the preferred phase, many other characteristics have been studied in [226]. Because a large mismatch was chosen, they have lacked any trend, but an interesting oscillatory behavior has been discovered for the initial two-mode coherent state. Within each period, the phase-matched second-harmonic and second-subharmonic generation processes can be prepared. The model of an ideal Kerr-like medium [223] have been considered for a comparison with cascaded quadratic non-linearities. It follows that these nonlinearities exhibit not only self-phase modulation in the fundamental mode but also a cross-phase modulation of the modes that can be considered for a nondemolition measurement. 

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