Advertisement

Weak Transient Chaos

Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 20)

Abstract

A phenomenon of weak transient chaos is discussed that is caused by sub-exponential divergence of trajectories in the basin of a non-chaotic attractor. Such a regime is not easy to detect, because conventional characteristics such as the largest Lyapunov exponent are non-positive. Here we study how such a divergence can be exposed and detected. First, we show that weak transient chaos can be exposed if a small random perturbation is added to the system, leading to positive values of the largest Lyapunov exponent. Second, we introduce an alternative definition of the Lyapunov exponent, which allows us to detect weak transient chaos in the deterministic unperturbed system. We show that this novel characteristic becomes positive, reflecting transient chaos. We demonstrate this phenomenon and its detection using a master–slave system where the master possesses a heteroclinic cycle attractor, while the slave is the Van-der-Pol–Duffing oscillator possessing a stable limit cycle.

Keywords

Weak transient chaos Lyapunov exponent Heteroclinic cycle Random perturbation 

Notes

Acknowledgements

The authors dedicate this paper to the 75th anniversary of Mikhail Rabinovich. We wish him great health, and maintenance of his remarkable enthusiasm and energy, which generated and will generate many exciting ideas and fundamental works. The authors thank T. Young for useful discussions and Ya. Pesin for allowing them to read the manuscript of his chapter [19] in this volume. The authors acknowledge support by the RSF grant 14-41-00044 of the Russian Science Foundation during their stay at Nizhny Novgorod University.

References

  1. 1.
    Afraimovich, V., Zaslavsky, G.: Space-time complexity in Hamiltonian dynamics. Chaos: Interdiscip. J. Nonlinear Sci. 13 (2), 519 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Afraimovich, V., Zhigulin, V., Rabinovich, M.: On the origin of reproducible sequential activity in neural circuits. Chaos: Interdiscip. J. Nonlinear Sci. 14 (4), 1123 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Afraimovich, V., Tristan, I., Huerta, R., Rabinovich, M.I.: Winnerless competition principle and prediction of the transient dynamics in a Lotka-Volterra model. Chaos: Interdiscip. J. Nonlinear Sci. 18 (4), 043103 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Afraimovich, V., Ashwin, P., Kirk, V.: Robust heteroclinic and switching dynamics. Dyn. Syst. 25 (3), 285 (2010)CrossRefGoogle Scholar
  5. 5.
    Afraimovich, V., Cuevas, D., Young, T.: Sequential dynamics of master-slave systems. Dyn. Syst. 28 (2), 154 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Afraimovich, V., Tristan, I., Varona, P., Rabinovich, M.: Transient dynamics in complex systems: heteroclinic sequences with multidimensional unstable manifolds. Discontinuity, Nonlinearity and Complexity 2 (1), 21 (2013)Google Scholar
  7. 7.
    Ashwin, P., Coombes, S., Nicks, R.: Mathematical frameworks for oscillatory network dynamics in neuroscience. J. Math. Neurosci. 6 (1), 1 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Canovas, J.: A guide to topological sequence entropy. In: Progress in Mathematical Biology Research, pp. 101–139. Nova Science Publishers, New York (2008)Google Scholar
  9. 9.
    Goldobin, D.S., Pikovsky, A.: Synchronization and desynchronization of self-sustained oscillators by common noise. Phys. Rev. E 71 (4), 045201 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Goodman, T.: Topological sequence entropy. Proc. Lond. Math. Soc. 29 (3), 331 (1974)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grebogi, C., Ott, E., Yorke, J.A.: Crises, sudden changes in chaotic attractors, and transient chaos. Physica D: Nonlinear Phenom. 7 (1), 181 (1983)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grebogi, C., Ott, E., Yorke, J.A.: Critical exponent of chaotic transients in nonlinear dynamical systems. Phys. Rev. Lett. 57 (11), 1284 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klages, R.: From Hamiltonian Chaos to Complex Systems, pp. 3–42. Springer, Berlin (2013)Google Scholar
  14. 14.
    Kushnirenko, A.G.: On metric invariants of entropy type. Russ. Math. Surv. 22 (5), 53 (1967)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lai, Y.C., Tél, T.: Transient Chaos: Complex Dynamics on Finite Time Scales, vol. 173. Springer Science & Business Media, Dordrecht (2011)Google Scholar
  16. 16.
    Leoncini, X.: Hamiltonian Chaos Beyond the KAM Theory, pp. 143–192. Springer, Berlin/Heidelberg (2010)Google Scholar
  17. 17.
    Leoncini, X., Zaslavsky, G.M.: Jets, stickiness, and anomalous transport. Phys. Rev. E 65 (4), 046216 (2002)Google Scholar
  18. 18.
    Leoncini, X., Agullo, O., Benkadda, S., Zaslavsky, G.M.: Anomalous transport in Charney-Hasegawa-Mima flows. Phys. Rev. E 72 (2), 026218 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pesin, Y., Zelerowicz, A., Zhao, Y.: Time rescaling of Lyapunov exponents. In: Aranson, I.S., Pikovsky, A., Rulkov, N.F., Tsimring, L. (eds.) Advances in Dynamics, Patterns, Cognition. Springer, Cham (2017)Google Scholar
  20. 20.
    Rabinovich, M.I., Simmons, A.N., Varona, P.: Dynamical bridge between brain and mind. Trends Cogn. Sci. 19 (8), 453 (2015)CrossRefGoogle Scholar
  21. 21.
    Rabinovich, M.I., Tristan, I., Varona, P.: Hierarchical nonlinear dynamics of human attention. Neurosci. Biobehav. Rev. 55, 18 (2015)CrossRefGoogle Scholar
  22. 22.
    Rabinovich, M., Volkovskii, A., Lecanda, P., Huerta, R., Abarbanel, H., Laurent, G.: Dynamical encoding by networks of competing neuron groups: winnerless competition. Phys. Rev. Lett. 87 (6), 068102 (2001)CrossRefGoogle Scholar
  23. 23.
    Szlenk, W.: On weakly conditionally compact dynamical systems. Stud. Math. 66 (1), 25 (1979)MathSciNetMATHGoogle Scholar
  24. 24.
    Zaslavsky, G., Edelman, M.: Polynomial dispersion of trajectories in sticky dynamics. Phys. Rev. E 72 (3), 036204 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Valentin S. Afraimovich
    • 1
  • Alexander B. Neiman
    • 2
  1. 1.Institute de Investigacion Communicasion OpticaUniversidad Autonoma de San Luis PotosiSan Luis PotosiMexico
  2. 2.Department of Physics and Astronomy and Neuroscience ProgramOhio UniversityAthensUSA

Personalised recommendations