Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 69, Issue 1 2, 2004, Pages 162131-1621310

  Do, Younghae ^a   Lai, Ying Cheng ^a,b  

^a ARIZONA STATE UNIVERSITY   (United States)

^b Arizona State University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; COMPUTER SIMULATION; EIGENVALUES AND EIGENFUNCTIONS; LYAPUNOV METHODS; PARAMETER ESTIMATION; PROBABILITY DISTRIBUTIONS; PROBLEM SOLVING; SET THEORY; STATISTICAL METHODS;

CHAOTIC SYSTEMS; UNSTABLE DIMENSIONS;

CHAOS THEORY;

EID: 18344404059     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (39)

1. Because of the sensitivity of a chaotic system to initial conditions and parameters, the existence of the computer roundoff error means that two trajectories starting from exactly the same initial condition will diverge exponentially from each other in time. Shadowing of a trajectory by another one is possible only when they start from slightly different initial conditions.
2. The first paper, to our knowledge, addressing shadowing of statistical averages is T. Sauer, Phys. Rev. E 65, 036220 (2002), which deals with nonhyperbolic chaotic systems with unstable dimension variability.
3. A common situation where statistical averages depend on noise is when the system is in a periodic window, where a periodic attractor and a nonattracting chaotic invariant set (chaotic saddle) coexist in the phase space, For small noise, a trajectory can remain in the neighborhood of the periodic attractor indefinitely. As the noise amplitude exceeds a critical value, the trajectory on the periodic attractor can be perturbed away from it to visit the chaotic saddle. As the saddle is nonattracting, the trajectory will go back to the original periodic attractor, be kicked away again, and so on. Noise thus induces an intermittent behavior. It has been shown recently that statistical averages associated with the intermittency scale with the noise amplitude algebraically [Y.-C. Lai, Z. Liu. G. Wei, and C.-H. Lai, Phys. Rev. Lett. 89, 184101 (2002)].
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19. While Lyapunov exponents are asymptotic quantities defined with respect to the natural measure of the chaotic attractor, the relevant entities that determine the shadowing dynamics are the statistical characteristics of the distribution of the finite-time Lyapunov exponents. Say one chooses an ensemble of random initial conditions, computes the exponents in a finite time, and then constructs histograms of these exponents. The mean values of the histograms are the asymptotic exponents.
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28. Theoretical and numerical investigations on the characterization of the transition to high-dimensional chaos by unstable periodic orbits indicate that the transition is typically accompanied by severe unstable dimension variability [R.L. Davtdchack and Y.-C. Lai, Phys. Lett. A 270, 308 (2000)], which also leads to a smooth variation of the Lyapunov exponents (except the largest one) when they cross zero from the negative side [M.A. Harrison and Y.-C. Lai, Phys. Rev. E 59, R3799 (1999)]. In fact, a quantitative measure for the degree of unstable dimension variability can be defined based on unstable periodic orbits, demonstrating that the variability is most severe at the transition [Y.-C. Lai, ibid. 59, R3807 (1999)].
29. Theoretical and numerical investigations on the characterization of the transition to high-dimensional chaos by unstable periodic orbits indicate that the transition is typically accompanied by severe unstable dimension variability [R.L. Davtdchack and Y.-C. Lai, Phys. Lett. A 270, 308 (2000)], which also leads to a smooth variation of the Lyapunov exponents (except the largest one) when they cross zero from the negative side [M.A. Harrison and Y.-C. Lai, Phys. Rev. E 59, R3799 (1999)]. In fact, a quantitative measure for the degree of unstable dimension variability can be defined based on unstable periodic orbits, demonstrating that the variability is most severe at the transition [Y.-C. Lai, ibid. 59, R3807 (1999)].
30. Theoretical and numerical investigations on the characterization of the transition to high-dimensional chaos by unstable periodic orbits indicate that the transition is typically accompanied by severe unstable dimension variability [R.L. Davtdchack and Y.-C. Lai, Phys. Lett. A 270, 308 (2000)], which also leads to a smooth variation of the Lyapunov exponents (except the largest one) when they cross zero from the negative side [M.A. Harrison and Y.-C. Lai, Phys. Rev. E 59, R3799 (1999)]. In fact, a quantitative measure for the degree of unstable dimension variability can be defined based on unstable periodic orbits, demonstrating that the variability is most severe at the transition [Y.-C. Lai, ibid. 59, R3807 (1999)].
31. Strictly, the random-walk model can be solved by the Fokker-Planck equation when Z is a zero-mean, Gaussian random variable. For our shadowing problem, numerically we find that the distribution of Z is approximately Gaussian (by definition Z has a zero mean).
32. C.W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, New York, 1997); H. Risken, The Fokker-Plank Equation (Springer-Verlag, Berlin, 1989).
33. C.W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, New York, 1997); H. Risken, The Fokker-Plank Equation (Springer-Verlag, Berlin, 1989).
34. u, and (iv) the angle between stable subspace and unstable subspace is bounded away from zero. The key feature associated with nonuniform hyperbolicity is that the positive number K for the definition of hyperbolicity is replaced by a positive function. The shadowing lemma in Ref. [17] guarantees the existence of long shadowing trajectories for nonuniformly hyperbolic dynamical systems. The nonhyperbolic systems studied here, i.e., dynamical systems with unstable-dimension variability, violate one of the essential conditions for hyperbolicity: The continuous splitting of the tangent space between the stable and unstable subspaces. Thus the shadowing lemma in Ref. [17] does not hold for these severely nonhyperbolic systems. For them, shadowing of numerical trajectories, even of relatively short lengths, cannot be expected. Our discovery of the combination of algebraic (for short time) and exponential (for long time) behaviors in the statistical distribution of the shadowing time answers the question "for how long a numerical trajectory can be expected to be valid?" in a quantitative way.
35. See, for example; J.F. Heagy, N. Platt, and S.M. Hammel, Phys. Rev. E 49, 1140 (1994); D. Marthaler, D. Armbruster, Y.-C. Lai, and E.J. Kostelich, ibid. 64, 016220 (2001).
36. See, for example; J.F. Heagy, N. Platt, and S.M. Hammel, Phys. Rev. E 49, 1140 (1994); D. Marthaler, D. Armbruster, Y.-C. Lai, and E.J. Kostelich, ibid. 64, 016220 (2001).
37. A. Čenys, A.N. Anagnostopoulos, and G.L. Bleris, Phys. Lett. A 224, 346 (1997).
38. This insight was first conceived by J.A. Yorke (private communication). The modeling problem was investigated in detail in the context of coupled chaotic oscillators in Ref. [10].
39. While model solutions may not be valid for long time, the model may still be useful for yielding statistical or ergodic averages of physical quantities of interest [14]. An interesting question is how to identify situations in which models do not even yield useful statistical averages of physically relevant quantities [2,3].