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Emerging attractors and the transition from dissipative to conservative dynamics

Christian S. Rodrigues*, Alessandro P. S. de Moura, and Celso Grebogi

  • Department of Physics, King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom and Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

  • *c.rodrigues@abdn.ac.uk
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Phys. Rev. E 80, 026205 – Published 18 August, 2009

DOI: https://doi.org/10.1103/PhysRevE.80.026205

Abstract

The topological structure of basin boundaries plays a fundamental role in the sensitivity to the final state in chaotic dynamical systems. Herewith we present a study on the dynamics of dissipative systems close to the Hamiltonian limit, emphasizing the increasing number of periodic attractors, and on the structural changes in their basin boundaries as the dissipation approaches zero. We show numerically that a power law with nontrivial exponent describes the growth of the total number of periodic attractors as the damping is decreased. We also establish that for small scales the dynamics is governed by effective dynamical invariants, whose measure depends not only on the region of the phase space but also on the scale under consideration. Therefore, our results show that the concept of effective invariants is also relevant for dissipative systems.

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References (25)

  1. N. Nagao, H. Nishimura, and N. Matsui, Neural Process. Lett. 12, 267 (2000).
  2. S. J. Schiff et al., Nature (London) 370, 615 (1994).
  3. O. Peters and K. Christensen, Phys. Rev. E 66, 036120 (2002).
  4. P. Bak, K. Christensen, L. Danon, and T. Scanlon, Phys. Rev. Lett. 88, 178501 (2002).
  5. M. Anghel, Chaos, Solitons Fractals 19, 399 (2004).
  6. W. F. Wang and P. P. Ong, Phys. Rev. A 55, 1522 (1997).
  7. A. E. Motter, A. P. S. de Moura, C. Grebogi, and H. Kantz, Phys. Rev. E 71, 036215 (2005).
  8. J. D. Meiss, J. R. Cary, J. D. Crawford, C. Grebogi, A. K. Kaufman, and H. D. I. Abarbanel, Physica D 6, 375 (1983).
  9. E. G. Altmann, A. E. Motter, and H. Kantz, Phys. Rev. E 73, 026207 (2006).
  10. U. Feudel, C. Grebogi, B. R. Hunt, and J. A. Yorke, Phys. Rev. E 54, 71 (1996).
  11. J. Palis, Ann. Inst. Henri Poincare, Anal. Non Lineaire 22, 485 (2005), see also references therein.
  12. J. Palis, Asterisque 261, 335 (2000).
  13. G. M. Zaslavsky, Phys. Lett. A 69, 145 (1978).
  14. B. Chirikov, Phys. Rep. 52, 263 (1979).
  15. S. E. Newhouse, Topology 13, 9 (1974).
  16. L. Tedeschini-Lalli and J. A. Yorke, Commun. Math. Phys. 106, 635 (1986).
  17. U. Feudel and C. Grebogi, Chaos 7, 597 (1997); Phys. Rev. Lett. 91, 134102 (2003).
  18. C. Grebogi, H. E. Nusse, E. Ott, and J. A. Yorke, in Lectures Notes in Mathematics, edited by J. C. Alexander (Springer-Verlag, New York, 1988), Vol. 1342, p. 220, see also references therein.
  19. C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, Phys. Lett. A 99, 415 (1983).
  20. The numerical simulations for ε≤10−13 were carried out with high precision by using the MPFR C library for arbitrary precision [21], which seemed to be the best choice for trigonometric functions. However, the computational cost is relatively high in comparison to standard C precision. For the exponent uncertainty, we have iterated a sufficient number of initial conditions, in order to reach 400 uncertain points. Using standard precision, we have fixed this limit up to 10 000.
  21. L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann, ACM Trans. Math. Softw. 33, 13 (2007).
  22. Effective fractal dimension has been defined in a similar way for Hamiltonian system [7].
  23. D. Ruelle, Commun. Math. Phys. 82, 137 (1981).
  24. In fact this definition is general for a Cr diffeomorphism f:M→M, where M is some smooth manifold.
  25. H. E. Nusse and J. A. Yorke, Ergod. Theory Dyn. Syst. 17, 463 (1997); Physica D 90, 242 (1996); Phys. Rev. Lett. 84, 626 (2000).

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Phys. Rev. E 80, 026205– Published 18 August, 2009
  • Received 28 August 2008

© 2009 American Physical Society

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