Simulation of Multidimensional Nonlinear Dynamics by One-Dimensional Maps with Many Parameters

Abstract

We propose a concrete class of discrete dynamical systems as nonlinear matrix models to describe the multidimensional multiparameter nonlinear dynamics. In this article we simulate the system asymptotic behavior. A two-step algorithm for the computation of ω-limit sets of the dynamical systems is presented. In accordance with the qualitative theory which we develop for this class of systems, we allocate invariant subspaces of the system matrix containing cycles of rays on which ω-limit sets of the dynamical systems are situated and introduce the dynamical parameters by which the system behavior is described in the invariant subspaces. As the first step of the algorithm, a cycle of rays which contains the ω-limit set of the system trajectory, is allocated using system matrix. As the second step, the ω-limit set of the system trajectory is computed using the analytical form of one-dimensional nonlinear Poincare map dependent on the dynamical parameters. The proposed algorithm simplifies calculations of ω-limit sets and therefore reduces computing time. A graphic visualization of ω-limit sets of n-dimensional dynamical systems, n > 3 is shown.