On the effect of noise on quasiperiodicity of different dimensions, including the quasiperiodic Hopf bifurcation

Abstract

Background and Objectives: The basic model of study is the simplest three - dimensional map with two-frequency and three-frequency quasiperiodicity at adding of noise. The main objective is to examine the effect of noise on the quasiperiodic Hopf bifurcation of the 3-torus birth. Materials and Methods: To study the torus map in the presence of noise we use such numerical methods as computing of Lyapunov exponents, calculation of Fourier spectra, drawing of attractor portraits. Results: Quasi-periodic bifurcations under the influence of noise occupy certain intervals in the parameter, but their main classification features (equality or not of the corresponding Lyapunov exponents) are preserved at the qualitative level. Conclusion: We considered the effect of noise on the simplest system with two- and three-frequency quasiperiodicity. The three-frequency quasiperiodicity is preserved at certain noise amplitudes, but then turns into a two-frequency one. In the Fourier spectra, this process develops according to the scenario of "blurring" the noise components of the corresponding spectral components.

Keywords

• noise
• torus map
• quasi-periodic dynamics
• lyapunov exponent
• fourier spectrum