Izvestiya of Saratov University.

Physics

ISSN 1817-3020 (Print)
ISSN 2542-193X (Online)


For citation:

Kuznetsov A. P., Sedova Y. V. On the effect of noise on quasiperiodicity of different dimensions, including the quasiperiodic Hopf bifurcation. Izvestiya of Sarat. Univ. Physics. , 2021, vol. 21, iss. 1, pp. 29-35. DOI: 10.18500/1817-3020-2021-21-1-29-35

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.03.2021
Full text:
(downloads: 35)
Language: 
Russian
Article type: 
Article
UDC: 
517.9

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

Autors: 
Kuznetsov Alexander Petrovich, Saratov Branch of Kotel’nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences
Sedova Yuliya Viktorovna, Saratov Branch of Kotel’nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences
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.

Reference: 
  1. Schuster H. G., Just W. Deterministic chaos: An introduction. Weinheim, Wiley-VCH, 2006. 283 p.
  2. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A universal concept in nonlinear sciences. Cambridge, Cambridge University Press, 2001. 432 p.
  3. Kuznetsov S. P. Dinamicheskij haos [Dynamical Chaos]. Moscow, Fizmatlit Publ., 2001. 296 p. (in Russian).
  4. Anishchenko V. S., Astakhov V. V., Neiman A. B., Vadivasova T. E., Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems. Tutorial and Modern Development. Berlin, Springer Science & Business Media, 2007. 446 p.
  5. Kuznetsov A. P., Kuznetsov S. P., Stankevich N. V. A simple autonomous quasiperiodic self-oscillator. Communications in Nonlinear Science and Numerical Simulation, 2010, vol. 15, pp. 1676–1681.
  6. Anishchenko V. S., Nikolaev S. M., Kurths J. Peculiarities of synchronization of a resonant limit cycle on a two-dimensional torus. Phys. Rev. E, 2007, vol. 76, pp. 046216.
  7. Broer H., Simó C., Vitolo R. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol’d resonance web. Bulletin of the Belgian Mathematical Society-Simon Stevin, 2008, vol. 15, no. 5, pp. 769‒787.
  8. Vitolo R., Broer H., Simó C. Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3Ddiffeomorphisms. Nonlinearity, 2010, vol. 23, pp. 1919‒ 1947.
  9. Broer H., Simó C., Vitolo R. Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regular and Chaotic Dynamics, 2011, vol. 16, no. 1‒2, pp. 154‒184.
  10. Anishchenko V., Nikolaev S., Kurths J. Winding number locking on a two-dimensional torus: Synchronization of quasiperiodic motions. Phys. Rev. E, 2006, vol. 73, pp. 056202.
  11. Kuznetsov A. P., Kuznetsov S. P., Sedova J. V. Effect of noise on the critical golden-mean quasiperiodic dynamics in the circle map. Physica A, 2006, vol. 359, pp. 48‒64.
  12. Kuznetsov A. P., Sedova Yu. V. The simplest map with three-frequency quasi-periodicity and quasi-periodic bifurcations. International Journal of Bifurcation and Chaos, 2016, vol. 26, iss. 8, pp. 1630019.
Received: 
21.10.2020
Accepted: 
31.01.2021
Published: 
31.03.2021