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Predictability

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Predictability of Chaotic Dynamics

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

Dynamical systems describe magnitudes evolving in time according to deterministic rules. These magnitudes may evolve in time towards some final state, depending on the initial conditions and on the specific choice of parameters.

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Notes

  1. 1.

    We see another zero crossing at around Δt = 9.0. This value is slightly below the Tcross range of values. But as Δt increases, the distribution returns to a flat shape again. As a consequence, the peaks appear because we are still in the local regime.

References

  1. Aguirre, J., Viana R.L., Sanjuan M.A.F.: Fractal structures in nonlinear dynamics. Rev. Mod. Phys. 81, 333 (1999)

    Article  ADS  Google Scholar 

  2. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos. An introduction to dynamical systems, p. 383. Springer, New York (1996)

    Google Scholar 

  3. Athanassoula, E., Romero-Gómez, M., Bosma, A., Masdemont, J.J.: Rings and spirals in barred galaxies – III. Further comparisons and links to observations. Mon. Not. R. Astron. Soc. 407, 1433 (2010)

    Google Scholar 

  4. Buljan, H., Paar, V.: Many-hole interactions and the average lifetimes of chaotic transients that precede controlled periodic motion. Phys. Rev. E 63, 066205 (2001)

    Article  ADS  Google Scholar 

  5. Contopoulos, G.: Orbits in highly perturbed dynamical systems. I. Periodic orbits. Astron. J. 75, 96 (1970)

    ADS  MathSciNet  Google Scholar 

  6. Contopoulos, G., Grousousakou, E., Voglis, N.: Invariant spectra in Hamiltonian systems. Astron. Astrophys. 304, 374 (1995)

    ADS  Google Scholar 

  7. Davidchack, R.L., Lai, Y.C.: Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbit. Phys. Lett. A 270, 308 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  8. Daza, A., Wagemakers, A., Sanjuan, M.A.F., Yorke, J.A.: Testing for Basins of Wada. Basin entropy: a new tool to analyze uncertainty in dynamical systems. Sci. Rep. 6, 31416 (2016)

    Google Scholar 

  9. Do, Y., Lai, Y.C.: Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability. Phys. Rev. E 69, 16213 (2004)

    Article  ADS  Google Scholar 

  10. Grassberger, P., Badii, R., Politi, A.: Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135 (1988)

    Article  ADS  Google Scholar 

  11. Grebogi, C., Kostelich, E., Ott, E., Yorke, J.A.: Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor. Phys. D 25, 347 (1987)

    Article  MathSciNet  Google Scholar 

  12. Hunt B.R., Ott E., Rosa, E.: Sporadically fractal basin boundaries of chaotic systems. Phys. Rev. Lett. 82, 3597 (1999)

    Article  ADS  Google Scholar 

  13. Kapitaniak, T.: Distribution of transient Lyapunov exponents of quasiperiodically forced systems. Prog. Theor. Phys. 93, 831 (1995)

    Article  ADS  Google Scholar 

  14. Kottos, T., Politi, A., Izrailev F.M., Ruffo S.: Scaling properties of Lyapunov spectra for the band random matrix model. Phys. Rev. E. 53, 6 (1996)

    Article  Google Scholar 

  15. McDonald, S.W., Grebogi, C., Ott E., Yorke, J.A.: Fractal basin boundaries. Phys. D 17, 125 (1985)

    Article  MathSciNet  Google Scholar 

  16. Maffione, N.P., Darriba, L.A., Cincotta, P.M., Giordano, C.M.: Chaos detection tools: application to a self-consistent triaxial model. Mon. Not. R. Astron. Soc. 429, 2700 (2013)

    Article  ADS  Google Scholar 

  17. Mandelbrot, B.B.: Les objets fractals: forme, hasard et dimension. Flammarion, Paris (1975)

    MATH  Google Scholar 

  18. Manos, T., Athanassoula, E.: Regular and chaotic orbits in barred galaxies – I. Applying the SALI/GALI method to explore their distribution in several models. Mon. Not. R. Astron. Soc. 415, 629 (2011)

    Google Scholar 

  19. Manos, T., Machado, R.E.G.: Chaos and dynamical trends in barred galaxies: bridging the gap between N-body simulations and time-dependent analytical models. Mon. Not. R. Astron. Soc. 438, 2201 (2014)

    Article  ADS  Google Scholar 

  20. Ott E., Alexander, J.C., Kan, I., Sommerer, J.C., Yorke, J.A.: The transition to chaotic attractors with riddled basins. Phys. D 76, 384 (1994)

    Article  MathSciNet  Google Scholar 

  21. Prasad, A., Ramaswany, R.: Characteristic distributions of finite-time Lyapunov exponents. Phys. Rev. E 60, 2761 (1999)

    Article  ADS  Google Scholar 

  22. Sauer, T.: Shadowing breakdown and large errors in dynamical simulations of physical systems. Phys. Rev. E. 65, 036220 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  23. Sauer, T., Grebogi, C., Yorke, J.A.: How long do numerical chaotic solutions remain valid? Phys. Lett. A 79, 59 (1997)

    Article  Google Scholar 

  24. Sepulveda, M.A., Badii, R., Pollak, E.: Spectral analysis of conservative dynamical systems. Phys. Lett. 63, 1226 (1989)

    Article  Google Scholar 

  25. Tomsovic, S., Lakshminarayan A.: Fluctuations of finite-time stability exponents in the standard map and the detection of small islands. Phys. Rev. E 76, 036207 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  26. Vallejo, J.C., Aguirre, J., Sanjuan, M.A.F.: Characterization of the local instability in the Henon-Heiles Hamiltonian. Phys. Lett. A 311, 26 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  27. Vallejo, J.C., Viana, R., Sanjuan, M.A.F.: Local predictibility and non hyperbolicity through finite Lyapunov Exponents distributions in two-degrees-of-freedom Hamiltonian systems. Phys. Rev. E 78, 066204 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  28. Vallejo, J.C., Sanjuan, M.A.F.: Predictability of orbits in coupled systems through finite-time Lyapunov exponents. New J. Phys. 15, 113064 (2013)

    Article  ADS  Google Scholar 

  29. Vallejo, J.C., Sanjuan, M.A.F.: The forecast of predictability for computed orbits in galactic models. Mon. Not. R. Astron. Soc. 447, 3797 (2015)

    Article  ADS  Google Scholar 

  30. Viana, R.L., Pinto, S.E., Barbosa, J.R., Grebogi, C.: Pseudo-deterministic chaotic systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, 1 (2003)

    Google Scholar 

  31. Viana, R.L., Barbosa, J.R., Grebogi, C., Batista, C.M.: Simulating a Chaotic Process. Braz. J. Phys. 35, 1 (2005)

    Article  ADS  Google Scholar 

  32. Werndl, C.: What are the new implications of Chaos for unpredictability. Br. J. Philos. Sci. 60, 195–220 (2009)

    Article  MathSciNet  Google Scholar 

  33. Westfall, P.H.: Kurtosis as Peakedness. 1905–2014, R.I.P. Am. Stat. 68, 191 (2014)

    Google Scholar 

  34. Yanchuk, S., Kapitaniak, T.: Chaos-hyperchaos transition in coupled Rössler systems. Phys. Lett. A 290, 139 (2001)

    Article  ADS  Google Scholar 

  35. Yanchuk, S., Kapitaniak, T.: Symmetry increasing bifurcation as a predictor of chaos-hyperchaos transition in coupled systems. Phys. Rev. E 64, 056235 (2001)

    Article  ADS  Google Scholar 

  36. Yoneyama, K.: Theory of continuous sets of points. Tohoku Math. J. 11, 43 (1917)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Fisica, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain

    Juan C. Vallejo & Miguel A. F. Sanjuan

Authors
  1. Juan C. Vallejo
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  2. Miguel A. F. Sanjuan
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Vallejo, J.C., Sanjuan, M.A.F. (2019). Predictability. In: Predictability of Chaotic Dynamics . Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-030-28630-9_4

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