Abstract
Stickiness is a well known phenomenon in which chaotic orbits expend an expressive amount of time in specific regions of the chaotic sea. This phenomenon becomes important when dealing with area-preserving open systems because, in this case, it leads to a temporary trapping of orbits in certain regions of phase space. In this work, we propose that the different scenarios of dynamical trapping can be explained by analyzing the crossings between invariant manifolds. In order to corroborate this assertion, we use an adaptive refinement procedure to approximately obtain the sets of homoclinic and heteroclinic intersections for the area-preserving Hénon map, an archetype of open systems, for a generic parameter interval. We show that these sets have very different statistical properties when the system is highly influenced by dynamical trapping, whereas they present similar properties when stickiness is almost absent. We explain these different scenarios by taking into consideration various effects that occur simultaneously in the system, all of which are connected with the topology of the invariant manifolds.
Similar content being viewed by others
References
S. Wiggins, in Introduction to applied nonlinear dynamical systems and chaos (Springer Science Business & Media, 2003), pp. 107–110
J.D. Meiss, Chaos: Interdiscip, J. Nonlinear Sci. 7, 139 (1997)
E. Petrisor, Chaos Solitons Fractals 17, 651 (2003)
T. Tél, M. Gruiz, inChaotic dynamics: an introduction based on classical mechanics (Cambridge University Press, 2006), pp. 264–278
G. Contopoulos, M. Harsoula, Celest. Mech. Dyn. Astron. 107, 77 (2010)
G.M. Zaslavsky, in Hamiltonian chaos and fractional dynamics (OUP Oxford, 2004), pp. 139–158
E.G. Altmann, A.E. Motter, H. Kantz, Phys. Rev. E 73, 026207 (2006)
R.L. Devaney, J. Differ. Equ. 51, 254 (1984)
U. Kirchgraber, D. Stoffer, Ann. di Mat. Pura ed Appl. 185, S187 (2006)
M. Hénon, in The theory of chaotic attractors (Springer, New York, 1976), pp. 94–102.
K. Alligood, T. Sauer, J. Yorke, in Chaos (Springer, 1996), pp. 471–478.
D. Hobson, J. Comput. Phys. 104, 14 (1993)
D. Ciro, I. Caldas, R. Viana, T. Evans, Chaos: Interdiscip. J. Nonlinear Sci. 28, 093106 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Martins de Oliveira, V., Ciro, D. & Caldas, I.L. Dynamical trapping in the area-preserving Hénon map. Eur. Phys. J. Spec. Top. 229, 1507–1516 (2020). https://doi.org/10.1140/epjst/e2020-900155-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2020-900155-8