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

For citation:

Kruglov V. P., Khadzhieva L. M. Uniformly hyperbolic attractor in a system based on coupled oscillators with «figure-eight» separatrix. Izvestiya VUZ. Applied Nonlinear Dynamics, 2016, vol. 24, iss. 6, pp. 54-64. DOI: 10.18500/0869-6632-2016-24-6-54-64

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 103)
Article type: 

Uniformly hyperbolic attractor in a system based on coupled oscillators with «figure-eight» separatrix

Kruglov Vjacheslav Pavlovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Khadzhieva Lejla Muhamed-Buharaevna, Saratov State University

A new autonomous system with chaotic dynamics corresponding to Smale–Williams attractor in Poincare map is introduced. The system is constructed on the basis of the model with «figure-eight» separatrix on the phase plane discussed in former times by Y.I. Neimark. Our system is composed of two Neimark subsystems with generalized coordinates x and y. It is described by the equations with additional terms due to which the system becomes self-oscillating. Furthermore, a special coupling between subsystems provides the tripling of the angle of vector (x, y) rotation when returning to the neighborhood of the origin in successive rounds on separatrix. Study is based on the numerical solution of the dynamical equations with the construction of the Poincare map.  Results of numerical simulation (iteration diagram for the angular variable, Lyapunov exponents) demonstrate that the angular variable undergoes expanding circle map, while in the other directions there is a strong compression of the phase volume element. Distribution of angles between stable and unstable manifolds of the attractor is obtained and it confirms the property of transversal manifolds of the attractor. Structural stability of the attractor is confirmed by the smooth dependence of the highest Lyapunov exponent on the parameters. With this we conclude that the attractor of the Smale–Williams type is observed in the phase space of the proposed system in a certain range of parameters.

  1. Kuznetsov S.P. Hyperbolic Chaos: A Physicist’s View, Higher Education Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg, 2012. 320 p.
  2. Kuznetsov S.P. Example of a physical system with a hyperbolic attractor of the Smale–Williams type. Phys. Rev. Lett. 2005. Vol. 95. P. 144101.
  3. Kuznetsov S.P. Some mechanical systems manifesting robust chaos. Nonlinear Dynamics and Mobile Robotics. 2013. Vol. 1, N1. P. 3.
  4. Kuznetsov S.P., Pikovsky A. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D: Nonlinear Phenomena. 2007. Vol. 232. N2. P. 87–102.
  5. Kruglov V.P., Kuznetsov S.P. An autonomous system with attractor of Smale–Williams type with resonance transfer of excitation in a ring array of van der Pol oscillators. Communications in Nonlinear Science and Numerical Simulation. 2011. Vol. 16. N8. P. 3219–3223.
  6. Kruglov V.P., Kuznetsov S.P., Pikovsky A. Attractor of Smale–Williams type in an autonomous distributed system // Regular and Chaotic Dynamics. 2014. Vol. 19, N4. P. 483.
  7. Neimark Y.I. The Point Mapping Method in the Theory of Nonlinear Oscillation. Moscow: Nauka, 1972. P. 129–135 (in Russian).
  8. Butenin N.V., Neimark Y.I., Fufaev N.A. Introduction to the Theory of Nonlinear Oscillation. Moscow: Nauka, 1987. P. 303 (in Russian).
  9. 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 1: Theory. Meccanica. 1980. Vol. 15. N1. P. 9–20.
  10. Shimada I., Nagashima T. A numerical approach to ergodic problem of dissipative  dynamical systems. Progress of Theoretical Physics. 1979. Vol. 61. N6. P. 1605–1616.
  11. Lai Y.-C., Grebogi C., Yorke J.A., Kan I. How often are chaotic saddles nonhyperbolic? Nonlinearity. 1993. Vol. 6. P. 779–798.
  12. Anishchenko V.S., Kopeikin A.S., Kurths J., Vadivasova T.E., Strelkova G.I. Studying hyperbolicity in chaotic systems. Physics Letters A. 2000. Vol. 270. P. 301.
  13. Kuptsov P.V. Fast numerical test of hyperbolic chaos. Phys. Rev. E. 2012. Vol. 85, N1. P. 015203.
  14. Kuznetsov S.P., Kruglov V.P. Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics. Regular and Chaotic Dynamics. 2016. Vol. 21, N2. P. 160–174.
  15. Kruglov V.P. Technique and results of numerical test for hyperbolic nature of attractors for reduced models of distributed systems. Izvestija VUZ. Applied Nonlinear Dynamics. 2014. Vol. 22. N6. P. 79–93 (in Russian).
Short text (in English):
(downloads: 120)