About Chaotic Dynamics in the Twisted Horseshoe Map

The twisted horseshoe map was developed in order to study a class of density dependent Leslie population models with two age classes. From the beginning, scientists have tried to prove that this map presents chaotic dynamics. Some demonstrations that have appeared in mathematical literature present some difficulties or delicate issues. In this paper, we give a simple and rigorous proof based on a different approach. We also highlight the possibility of getting chaotic dynamics for a broader class of maps.

[1]  S. Smale.  Differentiable dynamical systems , 1967 .

[2]  D. Stoffer,et al.  On the Definition of Chaos , 1989 .

[3]  Semi-hyperbolicity and bi-shadowing in nonautonomous difference equations with Lipschitz mappings , 2008 .

[4]  R. Srzednicki,et al.  A GEOMETRIC METHOD FOR DETECTING CHAOTIC DYNAMICS , 1997 .

[5]  Eduardo Liz,et al.  Chaos in Discrete Structured Population Models , 2012, SIAM J. Appl. Dyn. Syst..

[6]  Roman Srzednicki.  A generalization of the Lefschetz fixed point theorem and detection of chaos , 1999 .

[7]  Keith Burns,et al.  The Sharkovsky Theorem: A Natural Direct Proof , 2011, Am. Math. Mon..

[8]  Peter E. Kloeden,et al.  Chaotic difference equations in Rn , 1981 .

[9]  James A. Yorke,et al.  A Chaos Lemma , 2001, The American mathematical monthly.

[10]  M. Pireddu.  Chaotic dynamics in three dimensions: a topological proof for a triopoly game model , 2013, 1301.7556.

[11]  R M May,et al.  Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos , 1974, Science.

[12]  S. Smale,et al.  Finding a horseshoe on the beaches of Rio , 1998 .

[13]  Piotr Zgliczyński,et al.  Fixed point index for iterations of maps, topological horseshoe and chaos , 1996 .

[14]  P. Kloeden,et al.  Beyond the Li-Yorke Definition of Chaos , 2006 .

[15]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[16]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[17]  J. Guckenheimer,et al.  The dynamics of density dependent population models , 1977, Journal of mathematical biology.

[18]  F. Zanolin,et al.  Horseshoes in 3D equations with applications to Lotka–Volterra systems , 2015 .

[19]  Marian Gidea,et al.  Covering relations for multidimensional dynamical systems , 2004 .

[20]  H. Weiss,et al.  A geometric criterion for positive topological entropy , 1995 .

[21]  Piotr Zgliczyński.  On periodic points for systems of weakly coupled 1-dim maps ☆ , 2001 .

[22]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[23]  H. Poincaré,et al.  Les méthodes nouvelles de la mécanique céleste , 1899 .

[24]  François Blanchard,et al.  On Li-Yorke pairs , 2002, Journal für die reine und angewandte Mathematik (Crelles Journal).

[25]  P. Zgliczyński.  Sharkovskii's theorem for multidimensional perturbations of one-dimensional maps , 1999, Ergodic Theory and Dynamical Systems.

[26]  Giuseppe Cian.  Some remarks on topological horseshoes and applications , 2014 .

[27]  Peter E. Kloeden,et al.  Li–Yorke chaos in higher dimensions: a review , 2006 .

[28]  S. Kolyada,et al.  LI-Yorke sensitivity and other concepts of chaos , 2004 .

[29]  Marina Pireddu,et al.  Chaotic Dynamics for Maps in One and Two Dimensions: a Geometrical Method and Applications to Economics , 2009, Int. J. Bifurc. Chaos.