Metric Universality of Order in One-Dimensional Dynamics
International Journal of Bifurcation and Chaos
The orbit of the critical point of a nonlinear dynamical system defines a family of functions in the parameter space of the system. For unimodal maps a renormalization makes these functions indistinguishable over a wide range of parameter values. The universal representation of these functions leads directly to a number of interesting results: (1) the positions in the parameter space of the windows of order; (2) the sizes of the windows of order; (3) measures of distortion in the window structure; and (4) various generalized Feigenbaum numbers. We explicitly discuss the examples of the quadratic and sine maps.
M. Frame and D. Peak, "Metric Universality of Order in One-Dimensional Dynamics," International Journal of Bifurcation and Chaos 3, 567-572 (1993).