Systematic search for wide periodic windows and bounds for the set of regular parameters for the quadratic map

Abstract

An efficient method to find positions of periodic windows for the quadratic map f(x)=ax(1-x) and a heuristic algorithm to locate the majority of wide periodic windows are proposed. Accurate rigorous bounds of positions of all periodic windows with periods below 37 and the majority of wide periodic windows with longer periods are found. Based on these results, we prove that the measure of the set of regular parameters in the interval [3,4] is above 0.613960137. The properties of periodic windows are studied numerically. The results of the analysis are used to estimate that the true value of the measure of the set of regular parameters is close to 0.6139603.

Continuation-Based Method to Find Periodic Windows in Bifurcation Diagrams With Applications to the Chua’s Circuit With a Cubic Nonlinearity

“…This relation is also satisfied approximately for 12 ≤ p ≤ 17. A similar phenomenon is observed in bifurcation diagrams of the logistic map (compare [31]), which indicates that periodic windows emerge in a similar fashion for these two systems.…”

On the Validity of Linear Response Theory in High-Dimensional Deterministic Dynamical Systems

“…Since, as we have shown above, Vq → 0 as N → ∞, we conclude that N q → ∞ in probability as N → ∞. Hence, if F is the cumulative distribution function of the χ 2 K−2 distribution, the p-value obtained using the χ 2 -test, p = 1 − F (χ 2 ) = 1 − F (K − 2 + N q), (23) converges quickly in probability to zero as N → ∞ [10]. This implies that the probability of falsely accepting the null hypothesis of linear response at any significance level can be made arbitrarily small for sufficiently large data length N .…”

“…For regular values of α, when the logistic map x n with parameter α has a stable periodic orbit, calculating the stable periodic orbit allows for an accurate evaluation of the expectation. We use the database of periodic windows given in [23] to identify regular points and stable periodic orbits.…”

Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps

“…We note as well the more expository account by Wang and Young ([48]), which we found remarkably helpful in preparing this work. There are also by now several works attempting to quantify the set of parameters for the quadratic map family at which various dynamical regimes are observed ( [20,23,36,46]). Also related to our finite-time checkable criteria are frameworks attempt-ing to understand dynamical properties at "finite resolution" ( [19,35]) or along finite, bounded timescales ( [9]).…”