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

For citation:

Kuznetsov A. P., Sataev I. R. Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der Pol oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2006, vol. 14, iss. 5, pp. 3-29. DOI: 10.18500/0869-6632-2006-14-5-3-29

Full text:
Language:
Russian
Article type:
Article
UDC:
517.9

Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der Pol oscillators

Autors:
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotelnikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Sataev Igor Rustamovich, Saratov Branch of Kotelnikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract:

We present a method and results of numerical computations on verification of hyperbolic nature for the chaotic attractor in a system of two coupled nonautonomous van der Pol oscillators (Kuznetsov, Phys. Rev. Lett., 95, 2005, 144101). At selected parameter values, we indicate a toroidal domain in four-dimensional phase space of Poincare map (topologically, a direct product of a circle and a three-dimensional ball), which is mapped into itself and contains the attractor we analyze. In accordance with our computations, in this absorbing domain the conditions are valid guaranteeing hyperbolicity, which are formulated in terms of contracting and expanding cones in the vector spaces of the small state perturbations (the tangent spaces). We discuss also some numerical results illustrating certain attributes of hyperbolic dynamics.

Key words:
Reference:
1. Sinai YG. Stochasticity of dynamic systems. Nonlinear waves. Ed. Gapanov-Grehov AV. Moscow: Nauka; 1979. 361 p. (In Russian).
2. Modern problems of mathematics: Fundamental areas. Results of science and technology. Ed. Gamkrelidze RV. Vol. 2. Moscow: VINITI AS USSR; 1985. 294 p. (In Russian).
3. Guckenheimer J, Holms P. Nonlinear oscillations, dynamical sytems, and ifurcations of vector fileds. New York: Springer; 2002. 559 p.
4. Devaney RL. An Introduction to Chaotic Dynamical Systems. New York: Addison-Wesley; 1989. 336 p.
5. Shilnikov L. Mathematical problems of nonlinear dynamics: A Tutorial. Int. J. of Bifurcation and Chaos. 1997;7(9):1353–2001.
6. Katok AB, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press; 1999. 768 p.
7. Afraimovich V, Hsu S-B. Lectures on chaotic dynamical systems. AMS/IP Studies in Advanced Mathematics, Vol. 28, American Mathematical Society, Providence, RI. Somerville, MA: International Press; 2003. 353 p.
8. Ott E. Chaos in Dynamical Systems. Cambridge: Cambridge University Press; 1993. 385 p.
9. Anishchenko VS, Astakhov VV, Vadivasova ТЕ, Neiman AB, Strelkova GI, Schimansky-Geier L. Nonlinear Dynamics Of Chaotic And Stochastic System. Moscow - Izhevsk: ICR, 2003. 544 p. (In Russian).
10. Bunimovich LA, Sinai YG. Stochasticity of the attractor in the Lorentz model. Nonlinear waves. Ed. AV. Gaponova-Grekhova. Moscow: Nauka; 1979. P. 212. (In Russian).
11. Afraimovich VS, Bykov VV, Shilnikov LP. On the origin and structure of the Lorenz attractor. Dokl. Akad. Nauk SSSR. 1977;234(2):336–339.
12. Turaev DV, Shilnikov LP. Blue sky catastrophes. Dokl. Akad. Nauk. 1995;342(5):596–599.
13. Hunt TJ, MacKay RS. Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor. Nonlinearity. 2003;16(4):1499–1510. DOI: 10.1088/0951-7715/16/4/318.
14. Hunt TJ. Low Dimensional Dynamics: Bifurcations of Cantori and Realisations of Uniform Hyperbolicity. PhD Thesis. Cambridge: Univ. of Cambridge; 2000. 121 p. (http://www.timhunt.me.uk/maths/thesis.ps.gz).
15. Belykh V, Belykh I, Mosekilde. The hyperbolic Plykin attractor can exist in neuron models. Int. J. of Bifurcation and Chaos. 2005;15(11):3567–3578. DOI: 10.1142/S0218127405014222.
16. Kuznetsov SP. Example of a physical system with a hyperbolic attractor of the Smale-Williams type. Phys Rev Lett. 2005;95(14):144101. DOI: 10.1103/PhysRevLett.95.144101.
17. Kuznetsov SP, Seleznev EP. A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. Journal of Experimental and Theoretical Physics. 2006;102(2):355–364. DOI: 10.1134/S1063776106020166.
18. Arnold VI. Ordinary differential equations. Мoscow: Nauka; 1975. 240 p. (In Russian).
19. Andronov AA, Witt AA, Haikin SE. Theory of oscillations. Moscow: Nauka; 1981. 568 p.
20. Neimark YuI. The method of point mappings in the theory of nonlinear oscillations. Moscow: Nauka; 1972. 472 p.
21. Hairer E, Norsett SP, Wanner J. Solving ordinary differential equations. Berlin: Springer; 1990. 528 p.
22. LAPACK – Linear Algebra PACKage, version 3.0. May, 2000. [Electronic resource]. Available from: http://www.netlib.org/lapack
23. Benettin G, Galgani L, Giorgilli A, Strelcyn J-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part {I}: Theory. Part {II}: Numerical application. Meccanica. 1980;15(1): 21–30. DOI: 10.1007/BF02128236.
24. Kuznetsov SP. Dynamic chaos. Moscow: Fizmatlit; 2001. 296 p. (In Russian).
25. Bunimovich LA, Sinai YaG. Statistical mechanics of coupled map lattices. Theory and application of coupled map lattices. Ed. by K.Kaneko. New York: John Wiley & Sons Ltd; 1993. P. 169.
26. Bunimovich LA, Sinai YaG. Spacetime chaos in coupled map lattices. Nonlinearity. 1988;1(4):491–516.