High-dimensional discrete map based on coupled quasi-periodic generators

Abstract

Background and Objectives: Quasi-periodic oscillations are widespread in nature and technology. In the phase space, quasi-periodic oscillations with a different number of incommensurable frequencies correspond to invariant tori of different dimensions. Multiparametric analysis of high-dimensional systems is quite difficult. The way out of this situation can be the transition from systems with continuous time to discrete maps. Materials and Methods: In the paper we use a discretization method for the transition from a continuous-time system (two coupled quasi-periodic generators) to a new high-dimensional map. The time derivatives are replaced by finite differences. There is a new additional parameter corresponding to the discretization step, with the variation of which the system can demonstrate new interesting properties. Results: A new high-dimensional map has been obtained by the discretization method of differential equation system of coupled quasi-periodic generators. For this map, charts of Lyapunov exponents have been constructed in the plane of the frequency detuning of generators and the coupling magnitude. The existence of invariant tori of different dimensions has been demonstrated. Graphs of Lyapunov exponents and Fourier spectra have been presented. The evolution of maps with an increase of the discretization parameter has been investigated, the destruction of high-dimensional tori has been demonstrated. The influence of noise of different intensity has been studied. Conclusion: The two-parameter Lyapunov analysis of the new high-dimensional map made it possible to identify regions of invariant tori of different dimensions, up to fivefrequency ones. The map demonstrates Fourier spectra characteristic of quasi-periodicity of increasing dimension. Quasi-periodic bifurcations of invariant tori and an Arnold resonance web based on invariant tori of different dimensions have been observed. With the growth of the discretization parameter, the destruction of high-dimensional tori occurs. An increase in the discretization parameter leads to the destruction of high-dimensional tori with an increase in the noise intensity.

Keywords

• quasiperiodicity
• invariant tori
• lyapunov exponents