Geometrical Resonance in Spatiotemporal Systems

– We generalize the concept of geometrical resonance to perturbed sine-Gordon, Nonlinear Schrödinger and Complex Ginzburg-Landau equations. Using this theory we can control different dynamical patterns. For instance, we can stabilize breathers and oscillatory patterns of large amplitudes successfully avoiding chaos. On the other hand, this method can be used to suppress spatiotemporal chaos and turbulence in systems where these phenomena are already present. This method can be generalized to even more general spatiotemporal systems. Spatiotemporal chaos [1, 2, 3, 4, 5] is one of the most important (and most studied) phenomena of recent years. Chaos can be advantageous in some situations, while in many other situations, it should be avoided or controlled [6,7,8,9,10,11,12,13,14]. In certain cases, the desired effect is a high-amplitude periodic oscillation. We should drive a nonlinear system with a large external force to produce such a high-amplitude oscillation. However, this should be done in such a way that chaos is avoided. Different feedback mechanisms have been devised to control chaos [7,15,16,17]. A great deal of research has been dedicated also to the problem of suppresing chaos by harmonic (or just periodic) perturbations [11, 14, 18, 19, 20, 21, 22, 23, 24, 25]. Among those works are the ones that use the concept of Geometrical Resonance (GR) [14, 21, 23, 24, 25, 26, 27]. In Ref. [14] the concept of GR was used as a chaos-eliminating mechanism for the perturbed φ equation. In this letter we generalize the concept of Geometrical Resonance to a very general class of spatiotemporal systems which includes the sine-Gordon, Nonlinear Schrödinger, Boussinesq, Toda lattice and Complex Ginzburg-Landau equations (among others). We will use this concept as a method of chaos control when these equations are nonintegrable because of the presence of perturbations. GR is an extension of the linear notion of resonance to a nonlinear formulation based on a local energy conservation requirement [23].