Recognizing the Hyperbolicity: Cone Criterion and Other Approaches
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.
KeywordsInvariant Measure Unstable Manifold Chaotic Attractor Stable Manifold Cone Criterion
Unable to display preview. Download preview PDF.
- Bowen R.: Equilibrium states and ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470, Springer, Heidelberg (1975).Google Scholar
- Hasselblalt, B., Katok, A.A.: First Course in Dynamics: with a Panorama of Recent Developments. Cambridge University Press, Cambridge (2003).Google Scholar
- Krauskopf. B., Osinga, H.M., Doedel E.J.. Henderson. M.E., Guckenlieimer, J., Vladimirsky, A.Google Scholar
- 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
- 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
- Lai, Y.-C., Grebogi, C., Yorke, J. A., Kan, I.: How often are chaotic saddles nonhyperbolic? Non-linearity 6, 779–798 (1993).Google Scholar
- Sinai, Y.G.: Stochasticity of dynamical systems. In: Gaponov-Grekhov, A.V. (ed.) Nolinear waves, pp. 192–212. Moscow, Nauka (1979).Google Scholar