Matematika i Matematičeskoe Modelirovanie (Jan 2015)
The Research of the Dynamical System of Globally Coupled R ossler Oscillators
Abstract
The paper studies a dynamical system of globally coupled Rossler oscillators previously con- sidered in terms of oscillation quenching. There are two types of quenching, namely homogeneous steady state and inhomogeneous steady state. A transition from the former to the latter can cause a disease in biological structure, a defect in synchronized power grid, as well as it can be used for epidemic delimitation.The investigation of the system of Rossler oscillators is conducted in two directions: finding the equilibrium points and localization of invariant compact sets.The equilibrium points are found for the systems of two oscillators, at most. In particular, a system consisting of one oscillator has, depending on the values of parameters, one, two, or infinitely many equilibrium points, and in the latter case the equilibrium points form a line in 3D space. The set of equilibrium points of the system of two oscillators is explicitly described as well; its structure to an even greater degree depends on the values of parameters.Localization of invariant compact sets involves the method concerned with constructing a localizing function. One knows that its values on each invariant compact set are bounded above and below, respectively, by the supremum and the infimum of its values on the set of zeros of its gradient. This method revealed, as a localizing set, the intersection of a family of closed subsets each of them being bounded by some parabolic surface and located on its outer side. The paper presents illustration of this localizing set for one certain set of values of parameters.
