ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Kuznetsov A. P., Sedova Y. V. Maps with quasi-periodicity of different dimension and quasi-periodic bifurcations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, iss. 4, pp. 33-50. DOI: 10.18500/0869-6632-2017-25-4-33-50

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Language: 
Russian
Article type: 
Article
UDC: 
517.9

Maps with quasi-periodicity of different dimension and quasi-periodic bifurcations

Autors: 
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Sedova Yuliya Viktorovna, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

The paper discusses the construction of convenient and informative three-dimensional mappings demonstrating the existence of 2-tori and 3-tori. The first mapping is obtained by discretizing the continuous time system – a generator of quasi-periodic oscillations. The second is obtained via discretization of the Lorentz-84 climate model. The third mapping was proposed in the theory of quasi-periodic bifurcations by Simo, Broer, Vitolo. The necessity of discussing such mappings is connected with the possibility for them of a quasi-periodicity of different dimensions, as well as quasi-periodic bifurcations, i.e. bifurcations of invariant tori. This issue has not yet been adequately covered both in scientific and educational literature. The main method of investigation is the construction of Lyapunov exponents charts. Charts are obtained by numerical methods. On such charts regions of periodic modes, two-frequency quasi-periodicity, three-frequency quasi-periodicity, and chaos are marked by different colors. Illustrations of dynamics in the form of phase portraits are also presented. Specific features and classification features of quasi-periodic bifurcations – bifurcations of invariant tori – are discussed. Quasi-periodic bifurcations are analyzed using graphs of Lyapunov exponents and bifurcation trees. The difference between the quasi-periodic Hopf bifurcation and the saddlenode bifurcation of invariant tori is discussed. The dependence of the picture on the parameter – the discretization step – is discussed. At small values of this parameter, the picture is close to the traditional system of Arnold’s tongues, which, however, are now observed on the basis of two-frequency regimes and are immersed in a three-frequency region. The new moment is the appearance of regions of periodic high-order resonances built into these languages. As the sampling parameter increases, the picture changes. Tongues with characteristic cuspoidbases are replaced by bands of two-frequency modes with built-in transverse bands of periodic resonances, from which, in turn, a new system of fan-like tongues of two-frequency modes departs. The phase portraits inside languages change from multi-turn curves to a system of isolated ovals. Thus, it is shown that the picture associated with the quasi-periodic Hopf bifurcation is quite complex and requires three parameters for its analysis. The cases of different mappings are compared. It is shown that the «torus-mapping» most fully describes the range of essential phenomena in systems with quasi-periodicity of different dimensions.

Reference: 
  1. Guckenheimer J., Holmes P. J. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences. Vol. 42. 3rd ed. New York: Springer, 1990.
  2. Kuznetsov S.P. Dynamic Chaos. 2nd ed. Fizmatlit: Moscow, 2006. 356 p. (in Russian).
  3. Anishchenko V.S. Dynamical Chaos: Models and Experiments: Appearance Routes and Structure of Chaos in Simple Dynamical Systems. World Scientific: Singapore. 1995.
  4. Schuster H.G., Just W. Deterministic Chaos: An Introduction. John Wiley & Sons, 2006.
  5. Thompson J.M.T., Stewart H.B. Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists. Chichester: Wiley, 1986. 376 p.
  6. Postnov D.E. Introduction to the Dynamics of Iterable Maps. Saratov: Saratov University Press, 2007. 160 p. (in Russian).
  7. Kuznetsov Yu.A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1998. 591 p.
  8. Meijer H.G.E. Codimension 2 Bifurcations of Iterated Maps // Doctoral thesis. Utrecht University, 2006.
  9. Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, 2003. 843 p.
  10. Elhadj Z., Sprott J.C. A minimal 2-D quadratic map with quasi-periodic route to chaos. Int. J. of Bifurcation and Chaos. 2008. Vol. 18, No. 5. P. 1567.
  11. Kuznetsov A.P., Kuznetsov S.P., Pozdnyakov M.V., Sedova Y.V. Universal two-dimensional map and its radiophysical realization. Nelineinaya dinamika. 2012. Vol. 8, No3. P. 461 (in Russian).
  12. Arrowsmith D.K., Cartwright J.H.E., Lansbury A.N., Place C.M. The Bogdanov map: bifurcations, mode locking, and chaos in a dissipative system. Int. J. of Bifurcation and Chaos. 1993. Vol.3, No 4. P. 803.
  13. Kuznetsov A.P., Savin A.V., Sedova Y.V. Bogdanov-Takens bifurcation: From flows to discrete systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009. Vol. 17, No 6. P. 139 (in Russian).
  14. Gonchenko S.V., Ovsyannikov I.I., Simo C., Turaev D.V. Three-dimensional Henonlike maps and wild Lorenz-like attractors. Int. J. of Bifurcation and Chaos. 2005. Vol. 15, No 11. P. 3493.
  15. Gonchenko S. V., Meiss J. D., Ovsyannikov I. I. Chaotic dynamics of three-dimensional Henon maps that originate from a homoclinic bifurcation. Regul. Chaotic Dyn. 2006. Vol. 11, No. 2. P. 191.
  16. Adilova A.B., Kuznetsov A.P., Savin A.V. Complex dynamics in the system of two coupled discrete Rossler oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics 2013. Vol. 21, No 5. P. 108. (in Russian).
  17. Richter H. The generalized Henon maps: Examples for higher-dimensional chaos. Int. J. of Bifurcation and Chaos. 2002. Vol. 12, No 6. P. 1371.
  18. Richter H. On a family of maps with multiple chaotic attractors. Chaos, Solitons & Fractals. 2008. Vol. 36, No 3. P. 559.
  19. Elhadj Z., Sprott J.C. Classification of three-dimensional quadratic diffeomorphisms with constant Jacobian. Frontiers of Physics in China. 2009. Vol. 4, No 1. P. 111.
  20. Broer H, Simo C., Vitolo R. Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regul. Chaotic Dyn. 2011. Vol. 16, No 1–2. P. 154.
  21. Kuznetsov A.P., Sataev I.R., Stankevich N.V. Tyuryukina L.V. Physics of Quasiperiodic Oscillations. Saratov: Publishing Center «Nauka», 2013. 252 p. (in Russian).
  22. Zaslavsky G.M. The Physics of Chaos in Hamiltonian Systems. World Scientific, 2007.
  23. Morozov A.D. Resonances, Cycles and Chaos in Quasiconservative Systems. Moscow; Izhevsk: Regular and Chaotic Dynamics, 2005. 424 p. (in Russian).
  24. Matsumoto T., Chua L., Tokunaga R. Chaos via torus breakdown. IEEE Transactions on Circuits and Systems. 1987. Vol. 34, No. 3. P. 240.
  25. 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. P. 046216.
  26. Anishchenko V., Nikolaev S. Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations. Technical Physics Letters. 2005. Vol. 31, No 10. P. 853.
  27. Anishchenko V.S., Nikolaev S.M . Stability, synchronization and destruction of quasiperiodic motions. Nelineinaya dinamika. 2006. Vol. 2, No 3. P. 267 (in Russian).
  28. 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. P. 1676.
  29. Kuznetsov A.P., Stankevich N.V. Autonomous systems with quasi-periodic dynamics. Examples and their properties: Review. Izvestiya VUZ. Applied Nonlinear Dynamics. 2015. Vol. 23, No 3. P. 71 (in Russian).
  30. Kuznetsov A.P., Sedova Yu.V. The simplest map with three-frequency quasi-periodicity and quasi-periodic bifurcations. Int. J. of Bifurcation and Chaos. 2016. Vol. 26, No 8. P. 1630019.
  31. Broer H., Simo C., Vitolo R. Hopf saddle-node bifurcation for fixed points of 3D- diffeomorphisms: Analysis of a resonance «bubble». Physica D. 2008. Vol. 237, No 13. P. 1773.
  32. Vitolo R., Broer H., Simo C. Routes to chaos in the Hopf-saddle-node bifurcation  for fixed points of 3D-diffeomorphisms. Nonlinearity. 2010. Vol. 23. P. 1919.
  33. Broer H., Simo C., Vitolo R. The Hopf-saddle-node bifurcation for fixed points  of 3D-diffeomorphisms: the Arnol’d resonance web. Reprint from the Belgian Mathematical Society. 2008. P. 769.
  34. Shil’nikov A., Nicolis G., Nicolis C. Bifurcation and predictability analysis of a low-order atmospheric circulation model. Int. J. of Bifurcation and Chaos. 1995. Vol. 5, No 6. P. 1701.
  35. Broer H., Simo C., Vitolo R. Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing. Nonlinearity. 2002. Vol. 15, No. 4. P. 1205.  
Received: 
07.06.2017
Accepted: 
07.07.2017
Published: 
31.08.2017
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