This example is studied using Floquet theory. A classic example is k(t) = 1 + ε cosωt, where one finds regions of instability in the ε, ω plane for which there are solutions which grow exponentially in time. The instability regions are open and with closures touching ε = 0 at critical frequencies (see [A], [MW]). For any t, the constant coefficient problems with k frozen at k(t) is conservative...

High-order ARX models can be used to approximate a quite general class of linear systems in a parametric model structure, and wellestablished methods can then be used to retrieve the true plant and noise models from the ARX polynomials. However, this commonly used approach is only valid when the plant is stable or if the unstable poles are shared with the true noise model. In this contribution,...

We discuss the structure of parametric resonance which occurs in the process of reheating after inflation with two interacting scalar fields. It is found that, for the case of a not too large coupling constant, a quasi-homogeneous part of the second, initially subdominant scalar field may not be neglected due to its stochastic growth during inflation. This fact has strong consequences for the r...

We address the problem of studying the decay of the inflaton field φ to another scalar field χ through parametric resonance in the case of a coupling that involves several decay modes. This amounts to the presence of extra harmonic terms in the perturbation of the χ field dynamics. For the case of two frequencies we compute the geometry of the resonance regions, which is significantly altered d...

We analyze pattern formation in doubly resonant second-harmonic generation in the presence of a competing parametric process, also named the internally pumped optical parametric oscillator. Different scenarios are established where either the upor down-conversion processes dominate the spatiotemporal behavior. The possibility of obtaining exact solutions above threshold for the parametric oscil...

We report results on a model of two coupled oscillators that undergo periodic parametric modulations with a phase difference θ. Being to a large extent analytically solvable, the model reveals a rich θ dependence of the regions of parametric resonance. In particular, the intuitive notion that anti-phase modulations are less prone to parametric resonance is confirmed for sufficiently large coupl...

We introduce a complete analytical and numerical study of the modulational instability process in a system governed by a canonical nonlinear Schrödinger equation involving local, arbitrary nonlinear responses to the applied field. In particular, our theory accounts for the recently proposed higher-order Kerr nonlinearities, providing very simple analytical criteria for the identification of mul...