Homoclinic tangles associated with closed invariant curves in families of 2-D maps

In this paper we describe some sequences of global bifurcations of attracting and repelling closed invariant curves of two-dimensional maps that have a fixed point which may lose stability both via a supercritical Neimark bifurcation and a supercritical flip bifurcation. These bifurcations, characterized by the creation of heteroclinic and homoclinic connections or homoclinic tangles, are first described through qualitative phase diagrams and then by numerical examples.

[1]  Christian Mira,et al.  On Some Properties of Invariant Sets of Two-Dimensional Noninvertible Maps , 1997 .

[2]  Hao Bai-lin.  Elementary Symbolic Dynamics , 1988 .

[3]  Raymond A. Adomaitis,et al.  RESONANCE PHENOMENA IN AN ADAPTIVELY-CONTROLLED SYSTEM , 1991 .

[4]  Laura Gardini,et al.  Some global bifurcations related to the appearance of closed invariant curves , 2005, Math. Comput. Simul..

[5]  Nathalie Gicquel.  BIFURCATION STRUCTURE IN A TRANSMISSION SYSTEM MODELLED BY A TWO-DIMENSIONAL ENDOMORPHISM , 1996 .

[6]  David K. Arrowsmith,et al.  THE BOGDANOV MAP: BIFURCATIONS, MODE LOCKING, AND CHAOS IN A DISSIPATIVE SYSTEM , 1993 .

[7]  D. Aronson,et al.  Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study , 1982 .

[8]  Laura Gardini,et al.  Global bifurcations of Closed Invariant Curves in Two-Dimensional Maps: a Computer Assisted Study , 2005, Int. J. Bifurc. Chaos.

[9]  Leon O. Chua,et al.  BIFURCATIONS OF ATTRACTING CYCLES FROM TIME-DELAYED CHUA’S CIRCUIT , 1995 .

[10]  Stephen Wiggins.  Global Bifurcations and Chaos: Analytical Methods , 1988 .

[11]  C Mira,et al.  Chaotic Dynamics: From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism , 1987 .

[12]  N. MacDonald.  Nonlinear dynamics , 1980, Nature.

[13]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[14]  E Mosekilde,et al.  Torus breakdown in noninvertible maps. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Laura Gardini,et al.  The Hicksian floor-roof model for two regions linked by interregional trade , 2003 .

[16]  Laura Gardini,et al.  Global bifurcations in duopoly when the Cournot Point is Destabilized via a Subcritical Neimark bifurcation , 2006, IGTR.