Computing the topological entropy of continuous maps with at most three different kneading sequences with applications to Parrondo’s paradox

Abstract We introduce an algorithm to compute the topological entropy of piecewise monotone maps with at most three different kneading sequences, with prescribed accuracy. As an application, we compute the topological entropy of 3-periodic sequences of logistic maps, disproving a commutativity formula for topological entropy with three maps, and analyzing the dynamics Parrondo’s paradox in this setting.

[1]  James Keesling,et al.  An improved algorithm for computing topological entropy , 1989 .

[2]  W. Thurston,et al.  On iterated maps of the interval , 1988 .

[3]  José María Amigó,et al.  A Simplified Algorithm for the Topological Entropy of Multimodal Maps , 2014, Entropy.

[4]  Computing the Topological Entropy for Piecewise Monotonic Maps on the Interval , 1999 .

[5]  H. Weiss,et al.  Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows , 1991 .

[6]  Jose S. Cánovas,et al.  Dynamic Parrondo’s paradox , 2006 .

[7]  Akio Matsumoto,et al.  Statistical dynamics in a chaotic Cournot model with complementary goods , 2006 .

[8]  M. Misiurewicz,et al.  Entropy of piecewise monotone mappings , 1980 .

[9]  Derek Abbott,et al.  Asymmetry and disorder: A decade of Parrondo's Paradox , 2010 .

[10]  Jose S. Cánovas,et al.  Computing topological entropy for periodic sequences of unimodal maps , 2014, Commun. Nonlinear Sci. Numer. Simul..

[11]  J. Almeida,et al.  Can two chaotic systems give rise to order , 2004, nlin/0406010.

[12]  J. Cánovas,et al.  REVISITING PARRONDO'S PARADOX FOR THE LOGISTIC FAMILY , 2013 .

[13]  Stewart Baldwin,et al.  Calculating Topological Entropy , 1997 .

[14]  Saber Elaydi,et al.  Periodic difference equations, population biology and the Cushing-Henson conjectures. , 2006, Mathematical biosciences.

[15]  Michael Kopel,et al.  Simple and complex adjustment dynamics in Cournot duopoly models , 1996 .

[16]  James Keesling,et al.  Computing the topological entropy of maps of the interval with three monotone pieces , 1992 .

[17]  Computing the topological entropy of general one-dimensional maps , 1991 .

[18]  D. Jillson.  Insect populations respond to fluctuating environments , 1980, Nature.

[19]  José María Amigó,et al.  Computing the Topological Entropy of Multimodal Maps via Min-Max Sequences , 2012, Entropy.