Robustness of periodic orbits in the presence of noise. (4th February 2015)

• Record Type: Journal Article
• Authors: Joglekar, Madhura; Yorke, James A
• Is Part Of: Nonlinearity. Volume 28:Number 3(2015:Mar.)
• Page Start: 697
• Page End: 711
• Publication Date: 2015-02-04
• Subjects: dynamical systems -- stability -- period-doubling cascades; 37E05 -- 37C75; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/28/3/697
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: For a smooth dynamical system x n +1 = f ( C, x n ) (depending on a parameter C ), there may be infinitely many periodic windows, that is, intervals in C having a region of stable periodic behaviour. However, the smaller of these windows are easily destroyed with tiny perturbations, so that only finitely many of the windows can be detected for a given level of noise. For a fixed perturbation size ϵ, we consider the system behaviour in the presence of noise. We look at the ' ϵ -robust windows', that is, those periodic windows such that for the superstable parameter value C in that window, the general periodic behaviour persists despite noise of amplitude ⩽ ϵ . We focus on the quadratic map, and numerically compute the number of periodic windows that are ϵ -robust. We obtain a robustness exponent α = 0.51 ± 0.03, which characterizes the robustness of periodic windows in the presence of noise.