Invariant Graphs and Dynamics of a Family of Continuous Piecewise Linear Planar Maps

• Anna Cima ^[1] ; Armengol Gasull ^[1] ; Víctor Mañosa ^[2] ; Francesc Mañosas ^[1]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  2. [2] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
• Idioma: inglés
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• Dialnet Métricas: 1 Cita
• Resumen

  • We consider the family of piecewise linear maps Fa,b(x, y) = (|x| − y + a, x − |y| + b), where (a, b) ∈ R2. This family belongs to a wider one that has deserved some interest in the recent years as it provides a framework for generalized Lozi-type maps. Among our results, we prove that for a 0 all the orbits are eventually periodic and moreover that there are at most three different periodic behaviors formed by at most seven points.

    For a < 0 we prove that for each b ∈ R there exists a compact graph , which is invariant under the map F, such that for each (x, y) ∈ R2 there exists n ∈ N (that may depend on x) such that Fn a,b(x, y) ∈ . We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all (a, b) ∈ R2 obtaining, among other results, a full characterization of when Fa,b| has positive or zero entropy

• Referencias bibliográficas
  • 1. Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)
  • 2. Aiewcharoen, B., Boonklurb, R., Konglawan, N.: Global and local behavior of the system of piecewise linear difference equations xn+1...
  • 3. Alsedà, L., Llibre, J., Misiurewicz, M.: Combinatorial Dynamics and Entropy in Dimension One. Advanced Series in Nonlinear Dynamics, vol....
  • 4. Banerjee, S., Verghese, G.C.: Nonlinear Phenomena in Power Electronics. IEEE, Piscataway Township (1999)
  • 5. Block, L., Guckenheimer, J., Misiurewicz, M., Young, L.S.: Periodic points and topological entropy of one-dimensional maps. In: Global...
  • 6. Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)
  • 7. Brogliato, B.: Nonsmooth mechanics. Models, dynamics and control. Communications and Control Engineering Series., 3rd edn. Springer, Berlin...
  • 8. Brucks, K.M., Misiurewicz, M., Tresser, C.: Monotonicity properties of the family of trapezoidal maps. Commun. Math. Phys. 137, 1–12 (1991)
  • 9. Bula, I., S¯ıle, A.: About a System of Piecewise Linear Difference Equations with Many Periodic Solutions. In: Olaru, S. et al. (eds.)...
  • 10. Bula, I., Kristone, J.L.: Behavior of systems of piecewise linear difference equations with many periodic solutions. In: Communication...
  • 11. Cima, A., Gasull, A., Mañosa, V., Mañosas, F.: Supplementary results for a family of continuous piecewise linear planar maps. Preprint...
  • 12. Devaney, R.L.: A piecewise linear model for the zones of instability of an area-preserving map. Physica D 10(3), 387–393 (1984)
  • 13. Gantmacher, F.R.: The theory of matrices. Volumes 1, 2. Translated by K. A. Hirsch. Chelsea Publishing Co., New York (1959)
  • 14. Gardini, L., Sushko, I., Matsuyama, K.: 2D discontinuous piecewise linear map: emergence of fashion cycles. Chaos 28(5), 055917–05 (2018)
  • 15. Gardini, L., Tikjha, W.: Dynamics in the transition case invertible/non-invertible in a 2D piecewise linear map. Chaos Solitons Fractals...
  • 16. Goetz, A., Quas, A.: Global properties of a family of piecewise isometries. Ergod. Theory Dyn. Syst. 29(2), 545–568 (2009)
  • 17. Grove, E.A., Ladas, G.: Periodicities in Nonlinear Difference Equations. Advances in Discrete Mathematics and Applications, vol. 4. Chapman...
  • 18. Herman, M.R.: Sur les courbes invariantes par les difféomorphismes de l’anneau. Astérisque 144, 256 (1986)
  • 19. Li, T., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 10(82), 985–992 (1975)
  • 20. Linero Bas, A., Nieves Roldán, D.: On the relationship between Lozi maps and max-type difference equations. J. Differ. Equ. Appl. 23,...
  • 21. Lozi, R.: Un attracteur étrange (?) du type attracteur de Hénon. J. Phys. Colloq. 39, C5-9-C5-10 (1978)
  • 22. Lozi, R.: Survey of recent applications of the chaotic lozi map. Algorithms 16, 491, 63 (2023)
  • 23. Misiurewicz, M., Ziemian, K.: Horseshoes and entropy for piecewise continuous piecewise monotone maps. In: From Phase Transitions to Chaos,...
  • 24. Nicolas, J.L.: On Highly Composite Numbers. In: Andrews, G.E., et al. (eds.) Ramanujan revisited, pp. 216–244. Academic Press, London...
  • 25. Nicolas, J.L., Robin, G.: Majorations explicites pour le nombre de diviseurs de N. Can. Math. Bull. 26, 485–492 (1983)
  • 26. Nitecki, Z.: Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms. The MIT Press, Cambridge (1971)
  • 27. Simpson, D.J.W., Meiss, J.D.: Neimark–Sacker bifurcations in planar, piecewise-smooth, continuous maps. SIAM J. Appl. Dyn. Syst. 7(3),...
  • 28. Smith, L.D., Umbanhowar, P.B., Lueptow, R.M., Ottino, J.M.: The geometry of cutting and shuffling: an outline of possibilities for piecewise...
  • 29. Strelcyn, J.-M.: The “coexistence problem” for conservative dynamical systems: a review. Colloq. Math. 62, 331–345 (1991)
  • 30. Tikjha, W., Gardini, L.: Bifurcation sequences and multistability in a two-dimensional piecewise linear map. Int. J. Bifur. Chaos Appl....
  • 31. Tikjha, W., Lenbury, Y., Lapierre, E.: Equilibriums and periodic solutions of related systems of piecewise linear difference equations....
  • 32. Tikjha, W., Lapierre, E., Sitthiwirattham, T.: The stable equilibrium of a system of piecewise linear difference equations. Adv. Differ....
  • 33. Zhusubaliyev, Z.T., Mosekilde, E.: Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems, vol. 44. World Scientific, London (2003)