Structure in the Parameter Dependence of Order and Chaos for the Quadratic Map

Brian R. Hunt and Edward Ott

J. Phys. A 30 (1997), 7067-7076.

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Many dynamical systems are thought to exhibit windows of attracting periodic behavior for arbitrarily small perturbations from parameter values yielding chaotic attractors. This structural instability of chaos is particularly well-documented and understood for the case of the one-dimensional quadratic map. In this paper we attempt to numerically characterize the global parameter-space structure of the dense set of periodic ``windows'' occurring in the chaotic regime of the quadratic map. In particular, we use scaling techniques to extract information on the probability distribution of window parameter widths as a function of period and location of the window in parameter space. We also use this information to obtain the uncertainty exponent which is a quantity that globally characterizes one's ability to identify chaos in the presence of small parameter uncertainties.

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