Recognizing the Hyperbolicity: Cone Criterion and Other Approaches

  • Sergey P. Kuznetsov


Physical and technical devices usually are not specially suited to admitting simple mathematical proofs. So, it is vital to employ numerical tools for verification of the hyperbolicity in systems, which potentially may possess uniformly hyperbolic chaotic attractors. Substantiation of hyperbolicity is essential to accounting relevant conclusions of the mathematical theory, like availability of description in terms of Markov partitions with a finite alphabet, or structural stability of the chaotic altraclors, which may be of natural practical significance. Section 7.1 is devoted to approaches based on verification of a fundamental property of the hyperbolic invariant sets that is transversality of mutual location of stable and unstable manifolds for all orbits belonging to the invariant set. In Sect. 7.2 we turn to a technique of visualization of natural invariant measures along filaments of attractors, which shows presence or absence of singularities in the distributions. For comparison, besides models with uniformly hyperbolic attractors, some systems not relating to this class are considered in terms of the same approaches. Final part of the chapter is devoted to the cone criterion based on a rigorous mathematical result and appropriate for validation of hyperbolicity in compulations. Here it is reexamined in some detail, with explanation of technical hints, and with concrete examples of its application.


Invariant Measure Unstable Manifold Chaotic Attractor Stable Manifold Cone Criterion 
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  1. Afraimovich, V., Hsu, S.-B.: Lectures on Chaotic Dynamical Systems. International Press, Somerville. MA (2003).zbMATHGoogle Scholar
  2. Anishchenko. V.S., Kopeikin. A.S., Kurths, I., Vadivasova, T.E., Strelkova G.I.: Studying hyper-bolicity in chaotic systems. Physics Letters A 270, 301–307 (2000).MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. Anosov, D.V.: On the contribution of N.N. Bogolyubov to the theory of dynamical systems. Russ. Math. Surv. 49, 1–18 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bowen R.: Equilibrium states and ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470, Springer, Heidelberg (1975).Google Scholar
  5. Ginelli, F., Poggi, P., Turchi, A., Chaté, H., Livi, R., Politi, A.: Characterizing dynamics with covariant Lyapunov vectors. Phys. Rev. Lett. 99, 130601 (2007).ADSCrossRefGoogle Scholar
  6. Guckenheimer. J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York (1983).zbMATHGoogle Scholar
  7. Hasselblalt, B., Katok, A.A.: First Course in Dynamics: with a Panorama of Recent Developments. Cambridge University Press, Cambridge (2003).Google Scholar
  8. Hassclblatt, B., Pesin, Y.: Hyperbolic dynamics. Scholarpedia 3, 2208 (2008).CrossRefGoogle Scholar
  9. Hénon, M.: On the numerical computation of Poincaré maps. Physica D 5, 412–414 (1982).MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. Hirata, Y., Nozaki, K., Konishi, T.: The intersection angles between N-dimensional stable and unstable manifolds in 2N-dimensionai symplectic mappings. Prog. Theor. Phys. 102,. 701–706 (1999).MathSciNetADSCrossRefGoogle Scholar
  11. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995).zbMATHGoogle Scholar
  12. Krauskopf. B., Osinga, H.M., Doedel E.J.. Henderson. M.E., Guckenlieimer, J., Vladimirsky, A.Google Scholar
  13. Dellmtz, M.. Junge, O.: A survey of methods for computing (un) stable manifolds of vector fields. Int. J. Bifurcation and Chaos 15, 763–791 (2005).CrossRefGoogle Scholar
  14. Kuptsov, P.V., Kuznetsov, S.P.: Violation of hyperbolicity in a diffusive medium with local hyper-bolic attractor. Phys. Rev. E 80, 016205 (2009).MathSciNetADSCrossRefGoogle Scholar
  15. Kuznetsov, A.S., Kuznetsov, S.P., Sataev. L.R.: Parametric generator of hyperbolic chaos based on two coupled oscillators with nonlinear dissipation. Technical Physics 55, 1707–1715 (2010).ADSCrossRefGoogle Scholar
  16. Kuznetsov, S.P.: Example of a physical system with a hyperbolic attractor of the Smale-Williams type, Phys. Rev. Lett. 95, 144101 (2005).ADSCrossRefGoogle Scholar
  17. Kuznetsov, S.P.: A non-autonomous flow system with Plykin type attractor. CNSNS 14, 3487–3491 (2009).ADSzbMATHGoogle Scholar
  18. Kuznetsov, S.P. and Pikovsky, A.: Autonomous coupled oscillators with hyperbolic strange attractors. Physica D 232, 87–102 (2007).MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. Kuznetsov, S.P., Sataev. I.R.: Hyperbolic attractor in a system of coupled non-autonomous van der Pol oscillators: Numerical test for expanding and contracting cones. Physics Letters A 365, 97–104 (2007).ADSzbMATHCrossRefGoogle Scholar
  20. Kuznetsov, S.P., Sataev, I.R.: Visualization of invariant measures and comparison of hyperbolic and non-hyperbolic chaotic attractors. Preprint. SB IRE RAS, Saratov (2011).Google Scholar
  21. Kuznetsov, S.P., Seleznev, E.P.: A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system. JETP 102, 355–364 (2006).MathSciNetADSCrossRefGoogle Scholar
  22. Kuznetsov, S.P., Pikovsky, A.S., Sataev, I.R.: Hyperbolic Smale-Williams attractor in Poincaré map of a four-dimensional autonomous system. In: Proc. of the III Int. Conf. “Frontiers of Nonlinear Physics”. Nizhny Novgorod-Saratov-Nizhny Novgorod, pp. 66–67 (2007)Google Scholar
  23. Lai, Y.-C., Grebogi, C., Yorke, J. A., Kan, I.: How often are chaotic saddles nonhyperbolic? Non-linearity 6, 779–798 (1993).Google Scholar
  24. Ruelle, D.: A measure associated with Axiom A attractors. Amer. J. Math. 98, 619–654 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  25. Shilnikov, L.: Mathematical problems of nonlinear dynamics: a tutorial. Int. J. Bifurcation and Chaos 7, 1353–2001 (1997).MathSciNetGoogle Scholar
  26. Sinai, Y.G.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–70 (1972).MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. Sinai, Y.G.: Stochasticity of dynamical systems. In: Gaponov-Grekhov, A.V. (ed.) Nolinear waves, pp. 192–212. Moscow, Nauka (1979).Google Scholar
  28. Sinai, J.G., Vul, E.B.: Hyperbolicity conditions for the Lorenz model. Physica D 2, 3–7 (1981).MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. Smale, S.: Differentiable dynamical systems. Bull. Amer. Math. Soc. (NS) 73, 747–817 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  30. Tél, T., Gruiz, M.: Chaotic dynamics: an Introduction Based on Classical Mechanics. Cambridge University Press, Cambridge (2006).zbMATHGoogle Scholar
  31. Williams, R.F.: Expanding attractors. Publications mathématiques de l’.I.H.É.S. 43, 169–203 (1974).ADSCrossRefGoogle Scholar

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© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sergey P. Kuznetsov
    • 1
  1. 1.Kotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratov BranchSaratovRussian Federation

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