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Statistical dynamics of sine-square map

Author

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  • Philominathan, P.
  • Neelamegam, P.
  • Rajasekar, S.

Abstract

In this paper we study (i) the variation of local Lyapunov exponent, (ii) characterization of chaotic attractor at bifurcations using the variance σn(q) of fluctuations of coarse-grained local expansion rates of nearby orbits and (iii) characterization of weak and strong chaos in a sine-square map which describes the dynamics of the liquid crystal hybrid optical bistable device. The standard deviation of local Lyapunov exponent λ(X,L) calculated after every L time steps and Allan variance are found to approach zero in the limit L → ∞ as L−α. For all chaotic attractors of the map the σn(q) versus q plot exhibits a peak at q = qα. We show that additional peaks, however, occur only for the attractors just before and just after the bifurcations. We investigate the characteristics of the probability distributions of a k-step difference quantity ΔXk=Xi+k−Xi. We show that a nonstationary probability distribution occurs for weak chaos and a stationary distribution occurs for strong chaos.

Suggested Citation

  • Philominathan, P. & Neelamegam, P. & Rajasekar, S., 1997. "Statistical dynamics of sine-square map," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 242(3), pages 391-408.
    • Handle: RePEc:eee:phsmap:v:242:y:1997:i:3:p:391-408
    DOI: 10.1016/S0378-4371(97)00259-8
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    References listed on IDEAS

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    1. Philominathan, P. & Rajasekar, S., 1996. "Dynamic behaviours of 2∞ attractor and q-phase transitions at bifurcations in logistic map," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 229(2), pages 244-254.
    2. Victor Chow, K. & Denning, Karen C. & Ferris, Stephen & Noronha, Gregory, 1995. "Long-term and short-term price memory in the stock market," Economics Letters, Elsevier, vol. 49(3), pages 287-293, September.
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