Fractal basin boundaries in coupled map lattices

Ying-Cheng Lai; Raimind L. Winslow

doi:10.1103/PhysRevE.50.3470

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Fractal basin boundaries in coupled map lattices

Ying-Cheng Lai and Raimind L. Winslow

Department of Biomedical Engineering, The Johns Hopkins University School of Medicine, Baltimore, Maryland 21205

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Phys. Rev. E 50, 3470 – Published 1 November, 1994

DOI: https://doi.org/10.1103/PhysRevE.50.3470

Abstract

It has been suggested that spatiotemporal dynamical systems cannot exhibit fractal basin boundaries, as interactions among chaotic elements at different spatial sites may destroy fine scale phase-space structures. We present evidence of an extreme type of fractal basin boundary in spatiotemporal chaotic systems modeled by globally coupled, two-dimensional maps. The existence of fractal basin boundaries for these systems indicates an extreme sensitive dependence of asymptotic attractors on both initial conditions and parameters.

References (11)

C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Phys. Lett. 99A, 415 (1983); C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. Lett. 50, 935 (1983); S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, Physica D 17, 125 (1985); E. G. Gwinn and R. M. Westervelt, Phys. Rev. A 33, 4143 (1986); C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. Lett. 56, 1011 (1986); Physica D 24, 243 (1987) C. Grebogi, E. Kostelich, E. Ott and J. A. Yorke, ibid. 25, 347 (1987).
F. C. Moon, Phys. Rev. Lett. 53, 962 (1984); F. C. Moon and G.-X. Li, ibid. 55, 1439 (1985).
J. C. Alexander, J. A. Yorke, Z. You and I. Kan, Int. J. Bif. Chaos 2, 795 (1992); E. Ott, J. C. Sommerer, J. C. Alexander, I. Kan and J. A. Yorke, Phys. Rev. Lett. 71, 4134 (1993).
K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); ibid. 74, 1033 (1985); J. P. Crutchfield and K. Kaneko, in Directions in Chaos (World Scientific, Singapore, 1987); K. Kaneko, Phys. Rev. Lett. 60, 2715 (1988) Physica D 34, 1 (1989); ibid. 37, 60 (1989); Chaos 2, (3) (1992), special issue on coupled map lattices, references therein, edited by K. Kaneko.
D. K. Umberger, C. Grebogi, E. Ott and B. Afeyan, Phys. Rev. A 39, 4835 (1989).
Y. C. Lai and R. L. Winslow, Phys. Rev. Lett. 72, 1640 (1994); Physica D 74, 353 (1994).
K. Kaneko, Phys. Rev. Lett. 65, 1391 (1990); Physica D 54, 5 (1991) ibid. 55, 368 (1992); S. Sinha, Phys. Rev. Lett. 69, 3306 (1992); S. Sinha, D. Biswas, M. Azam and S. V. Lawande, Phys. Rev. A 46, 3193 (1992); ibid. 46, 6242 (1992); M. Ding and L. T. Wille, Phys. Rev. E 48, 1605 (1993).
M. Hénon, Commun. Math. Phys. 50, 69 (1976). Locally coupled Hénon map lattices have been studied by Politi and Torcini, Chaos 2, 293 (1992).
To assure that the positive peak $λ_{1}$ approx 0.18 corresponds to a chaotic attractor rather than a long chaotic transient, we take a small grid of 32 times 32 initial conditions on the same two-dimensional region and evolve them under Eq. (1) for 5 times $10^{5}$ iterations. It is found that the histograms obtained after $10^{4}$ iterations are essentially the same. In particular, the number of chaotic trajectories with $λ_{1}$ approx 0.18 does not decrease. As a further check, we take ten initial conditions that result in $λ_{1}$ approx 0.18 at n= $10^{4}$ , and compute $λ_{1}$ using n= $10^{7}$ interactions. The values of $λ_{1}$ obtained at n= $10^{4}$ and n= $10^{7}$ are the same. These tests thus suggest that the positive peak in Fig. 1 represents a chaotic attractor. On the other hand, the peaks at $λ_{1}$ =0 and $λ_{1}$ approx -0.32 correspond actually to quasiperiodic and periodic attractors. This is so because (1) the basins of these nonchaotic attractors contain closed regions in the phase space (Fig. 2), and (2) these zero and negative $λ_{1}$ values remain invariant as Eq. (1) is iterated in time.
Near-zero uncertainty exponent means that most values of $λ_{1}$ computed are likely to be wrong, i.e., positive value could be negative. Nonetheless, the computation of the uncertainty fraction f( ε ) and the subsequent estimation of the uncertainty exponent alpha are reliable. This can be argued heuristically as follows. Note that the uncertainty fraction f( ε ) can be expressed as f( ε )= $N_{u}$ / $N_{t}$ = $N_{u}$ /( $N_{u}$ + $N_{c}$ ), where $N_{u}$ is the number of uncertain parameter values, and $N_{c}$ is the number of certain parameter values (parameter values that result in $λ_{1}$ with the same signs upon small perturbations). Let the uncertain probability at epsilon be p, i.e., the probability that $λ_{1}$ <=0 (or >0) is p when it is numerically determined that $λ_{1}$ >0 (or <= 0). The errors in $N_{u}$ and $N_{c}$ are thus Δ $N_{u}$ =2p(1-p) $N_{u}$ and Δ $N_{c}$ =2p(1-p) $N_{c}$ , respectively. In other words, Δ $N_{c}$ = Δ $N_{u}$ , since a parameter value that is not uncertain is certain. Therefore, the error in f( ε ) is Δ f( ε ) = ( $N_{c}$ Δ $N_{u}$ - $N_{u}$ Δ $N_{c}$ )/( $N_{c}$ + $N_{u}$ $)^{2}$ =0.
Fractal basis boundaries and near-zero uncertainty exponents not only exist for Eq. (1), which corresponds to a mean field like spatially extended system and has a translation symmetry with respect to map sites, but also exist in systems where the symmetry is broken. For example, when the parameter a is randomly chosen to be different for each map, similar fractal basin boundaries are observed [Y. C. Lai, C. Grebogi, and E. J. Kostelich (unpublished)].

[c1] C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Phys. Lett. 99A, 415 (1983); C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. Lett. 50, 935 (1983); S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke, Physica D 17, 125 (1985); E. G. Gwinn and R. M. Westervelt, Phys. Rev. A 33, 4143 (1986); C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. Lett. 56, 1011 (1986); Physica D 24, 243 (1987) C. Grebogi, E. Kostelich, E. Ott and J. A. Yorke, ibid. 25, 347 (1987).

[c2] F. C. Moon, Phys. Rev. Lett. 53, 962 (1984); F. C. Moon and G.-X. Li, ibid. 55, 1439 (1985).

[c3] J. C. Alexander, J. A. Yorke, Z. You and I. Kan, Int. J. Bif. Chaos 2, 795 (1992); E. Ott, J. C. Sommerer, J. C. Alexander, I. Kan and J. A. Yorke, Phys. Rev. Lett. 71, 4134 (1993).

[c4] K. Kaneko, Prog. Theor. Phys. 72, 480 (1984); ibid. 74, 1033 (1985); J. P. Crutchfield and K. Kaneko, in Directions in Chaos (World Scientific, Singapore, 1987); K. Kaneko, Phys. Rev. Lett. 60, 2715 (1988) Physica D 34, 1 (1989); ibid. 37, 60 (1989); Chaos 2, (3) (1992), special issue on coupled map lattices, references therein, edited by K. Kaneko.

[c5] D. K. Umberger, C. Grebogi, E. Ott and B. Afeyan, Phys. Rev. A 39, 4835 (1989).

[c6] Y. C. Lai and R. L. Winslow, Phys. Rev. Lett. 72, 1640 (1994); Physica D 74, 353 (1994).

[c7] K. Kaneko, Phys. Rev. Lett. 65, 1391 (1990); Physica D 54, 5 (1991) ibid. 55, 368 (1992); S. Sinha, Phys. Rev. Lett. 69, 3306 (1992); S. Sinha, D. Biswas, M. Azam and S. V. Lawande, Phys. Rev. A 46, 3193 (1992); ibid. 46, 6242 (1992); M. Ding and L. T. Wille, Phys. Rev. E 48, 1605 (1993).

[c8] M. Hénon, Commun. Math. Phys. 50, 69 (1976). Locally coupled Hénon map lattices have been studied by Politi and Torcini, Chaos 2, 293 (1992).

[c9] To assure that the positive peak $λ_{1}$ approx 0.18 corresponds to a chaotic attractor rather than a long chaotic transient, we take a small grid of 32 times 32 initial conditions on the same two-dimensional region and evolve them under Eq. (1) for 5 times $10^{5}$ iterations. It is found that the histograms obtained after $10^{4}$ iterations are essentially the same. In particular, the number of chaotic trajectories with $λ_{1}$ approx 0.18 does not decrease. As a further check, we take ten initial conditions that result in $λ_{1}$ approx 0.18 at n= $10^{4}$ , and compute $λ_{1}$ using n= $10^{7}$ interactions. The values of $λ_{1}$ obtained at n= $10^{4}$ and n= $10^{7}$ are the same. These tests thus suggest that the positive peak in Fig. 1 represents a chaotic attractor. On the other hand, the peaks at $λ_{1}$ =0 and $λ_{1}$ approx -0.32 correspond actually to quasiperiodic and periodic attractors. This is so because (1) the basins of these nonchaotic attractors contain closed regions in the phase space (Fig. 2), and (2) these zero and negative $λ_{1}$ values remain invariant as Eq. (1) is iterated in time.

[c10] Near-zero uncertainty exponent means that most values of $λ_{1}$ computed are likely to be wrong, i.e., positive value could be negative. Nonetheless, the computation of the uncertainty fraction f( ε ) and the subsequent estimation of the uncertainty exponent alpha are reliable. This can be argued heuristically as follows. Note that the uncertainty fraction f( ε ) can be expressed as f( ε )= $N_{u}$ / $N_{t}$ = $N_{u}$ /( $N_{u}$ + $N_{c}$ ), where $N_{u}$ is the number of uncertain parameter values, and $N_{c}$ is the number of certain parameter values (parameter values that result in $λ_{1}$ with the same signs upon small perturbations). Let the uncertain probability at epsilon be p, i.e., the probability that $λ_{1}$ <=0 (or >0) is p when it is numerically determined that $λ_{1}$ >0 (or <= 0). The errors in $N_{u}$ and $N_{c}$ are thus Δ $N_{u}$ =2p(1-p) $N_{u}$ and Δ $N_{c}$ =2p(1-p) $N_{c}$ , respectively. In other words, Δ $N_{c}$ = Δ $N_{u}$ , since a parameter value that is not uncertain is certain. Therefore, the error in f( ε ) is Δ f( ε ) = ( $N_{c}$ Δ $N_{u}$ - $N_{u}$ Δ $N_{c}$ )/( $N_{c}$ + $N_{u}$ $)^{2}$ =0.

[c11] Fractal basis boundaries and near-zero uncertainty exponents not only exist for Eq. (1), which corresponds to a mean field like spatially extended system and has a translation symmetry with respect to map sites, but also exist in systems where the symmetry is broken. For example, when the parameter a is randomly chosen to be different for each map, similar fractal basin boundaries are observed [Y. C. Lai, C. Grebogi, and E. J. Kostelich (unpublished)].

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Fractal basin boundaries in coupled map lattices

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References (11)

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