Periodic windows within windows within windows

Transcription

1 Periodic windows within windows within windows Madhura Joglekar Applied Mathematics & Statistics, and Scientific Computation University of Maryland College Park March 29, 2014 References C. Grebogi, S. McDonald, E. Ott and J. A. Yorke, Phys. Let. A 110, (1985), 1-4 B. R. Hunt and E. Ott, J. Phys. A 30 (1997), J. D. Farmer, Phys. Rev. Lett. 55, (1985),

2 Basins of attraction of the Forced Damped Pendulum Equation: X + 0.2X + sin X = ρ cos T ρ = 2.45 gives a chaotic attractor ρ = 2.55 gives periodic attractors. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 2/17

3 What is an ɛ-uncertain point? ɛ-uncertainty: A point (x, C) lying in the basin of a periodic attractor is ɛ-uncertain if there is a point within ɛ-distance that can result in chaos Can be defined in state space as well as in parameter space Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 3/17

4 Questions What fraction of the space consists of ɛ-uncertain points? Where are ɛ-uncertain points most likely to lie? In higher dimensional systems, difficult to predict asymptotic behavior given initial state Study the 1-dim quad map x n+1 = C x n 2 (start from x = 0) Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 4/17

5 Questions What fraction of the space consists of ɛ-uncertain points? Where are ɛ-uncertain points most likely to lie? In higher dimensional systems, difficult to predict asymptotic behavior given initial state Study the 1-dim quad map x n+1 = C x n 2 (start from x = 0) Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 4/17

6 Periodic Windows in the 1 Dim Quadratic Map x n+1 = C x n 2 where C [ 0.25, 2] Infinitely many windows Dense in parameter space, fractal structure Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 5/17

7 ɛ-uncertain C values in x n+1 = C x n 2 Given C results in a periodic attractor C is within ɛ of chaos Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 6/17

8 ɛ-uncertain C values in x n+1 = C x n 2 The fraction of ɛ-uncertain C values in a window depends on its width Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 7/17

9 Randomly choosing an ɛ-uncertain value of C C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Exterior dimension of fat fractals" Phys. Let. A 110, 1-4, 1985 fraction of ɛ-uncertain C values ɛ 0.41 Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 8/17

10 Study distribution of primary-window widths Using kneading theory, determine sequence of all windows Compute C width = C crisis C saddlenode Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 9/17

11 For x n+1 = C x n 2, what is distribution of C-widths? N(ɛ): No. of primary windows with C-width > ɛ Cluster computation in quadruple precision Computed windows of periods 25 : (No. of windows with C-width ɛ) vs ɛ Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 10/17

12 For x n+1 = C x n 2, what is distribution of C-widths? N(ɛ): No. of primary windows with C-width > ɛ Cluster computation in quadruple precision Computed windows of periods 25 : (No. of windows with C-width ɛ) vs ɛ Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 10/17

13 For x n+1 = C x n 2, what is distribution of C-widths? N(ɛ) = 0.133ɛ.51 Scaling exponent α 0.51 : (No. of windows with C-width ɛ) vs ɛ How does this relate to the fraction of ɛ-uncertain C values? Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 11/17

14 For x n+1 = C x n 2, what is distribution of C-widths? N(ɛ) = 0.133ɛ.51 Scaling exponent α 0.51 : (No. of windows with C-width ɛ) vs ɛ How does this relate to the fraction of ɛ-uncertain C values? Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 11/17

15 Relation between N(ɛ) and f P (ɛ) f P (ɛ): fraction of ɛ-uncertain C values in primary windows lim ɛ 0 log f P (ɛ) log ɛ 1 α Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 12/17

16 Relation between N(ɛ) and f P (ɛ) f P (ɛ): fraction of ɛ-uncertain C values in primary windows lim ɛ 0 log f P (ɛ) log ɛ 1 α As ɛ 0, most of the ɛ-uncertain C values lie in higher order windows! Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 13/17

17 Relation between N(ɛ) and f P (ɛ) f P (ɛ): fraction of ɛ-uncertain C values in primary windows lim ɛ 0 log f P (ɛ) log ɛ 1 α As ɛ 0, most of the ɛ-uncertain C values lie in higher order windows! Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 13/17

18 Where does a randomly chosen ɛ-uncertain value of C lie? Primary window width scaling N 1 (ɛ) = 0.133ɛ.51 : (No. of windows with C-width ɛ) vs ɛ Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 14/17

19 Where does a randomly chosen ɛ-uncertain value of C lie? Primary window width scaling N 1 (ɛ) = 0.133ɛ.51 Assume exact self-similarity of periodic windows N k (ɛ): No. of k th order windows with width > ɛ Derive a generalized formula for scaling of higher order windows, i.e., N k (ɛ) for all positive integers k Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 15/17

20 Where does a randomly chosen ɛ-uncertain value of C lie? Primary window width scaling N 1 (ɛ) = 0.133ɛ.51 Assume exact self-similarity of periodic windows N k (ɛ): No. of k th order windows with width > ɛ Derive a generalized formula for scaling of higher order windows, i.e., N k (ɛ) for all positive integers k Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 15/17

21 Where does a randomly chosen ɛ-uncertain value of C lie? Theorem: Choose an ɛ-uncertain point randomly. Say this point lies in a window of order r. As per the theorem, for all positive integers n, lim Probability(r > n) = 1 ɛ 0 As ɛ takes small values, Most ɛ-uncertain points lie in a window within a window within a window... N th order window for large N. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 16/17

22 Where does a randomly chosen ɛ-uncertain value of C lie? Theorem: Choose an ɛ-uncertain point randomly. Say this point lies in a window of order r. As per the theorem, for all positive integers n, lim Probability(r > n) = 1 ɛ 0 As ɛ takes small values, Most ɛ-uncertain points lie in a window within a window within a window... N th order window for large N. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 16/17

23 Where does a randomly chosen ɛ-uncertain value of C lie? Theorem: Choose an ɛ-uncertain point randomly. Say this point lies in a window of order r. As per the theorem, for all positive integers n, lim Probability(r > n) = 1 ɛ 0 As ɛ takes small values, Most ɛ-uncertain points lie in a window within a window within a window... N th order window for large N. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 16/17

24 And thus, As ɛ takes small values, Most ɛ-uncertain points lie in a window within a window within a window... N th order window for large N. Acknowledgements: I would like to thank my advisor Jim Yorke, and Ed Ott for their suggestions. Madhura Joglekar Uncertainty: Chaos vs Periodic Attractors 17/17