Stability and bifurcations of heteroclinic cycles of type Z

Dynamical systems that are invariant under the action of a non-trivial symmetry group can possess structurally stable heteroclinic cycles. In this paper, we study stability properties of a class of structurally stable heteroclinic cycles in which we call heteroclinic cycles of type Z. It is well known that a heteroclinic cycle that is not asymptotically stable can nevertheless attract a positive measure set from its neighbourhood. We say that an invariant set X is fragmentarily asymptotically stable, if for any δ > 0 the measure of its local basin of attraction is positive. A local basin of attraction is the set of such points that trajectories starting there remain in the δ-neighbourhood of X for all t > 0, and are attracted by X as t → ∞. Necessary and sufficient conditions for fragmentary asymptotic stability are expressed in terms of eigenvalues and eigenvectors of transition matrices. If all transverse eigenvalues of linearizations near steady states involved in the cycle are negative, then fragmentary asymptotic stability implies asymptotic stability. In the latter case the condition for asymptotic stability is that the transition matrices have an eigenvalue larger than one in absolute value. Finally, we discuss bifurcations occurring when the conditions for asymptotic stability or for fragmentary asymptotic stability are broken.

[1]  Sofia B. S. D. Castro,et al.  Chaotic switching in a two-person game , 2010 .

[2]  Peter Ashwin,et al.  Group theoretic conditions for existence of robust relative homoclinic trajectories , 2002, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Philip Holmes,et al.  Heteroclinic cycles and modulated travelling waves in a system with D 4 symmetry , 1992 .

[4]  Melbourne,et al.  Asymptotic stability of heteroclinic cycles in systems with symmetry , 1995, Ergodic Theory and Dynamical Systems.

[5]  Werner Brannath,et al.  Heteroclinic networks on the tetrahedron , 1994 .

[6]  Peter Ashwin,et al.  On local attraction properties and a stability index for heteroclinic connections , 2010, 1008.3063.

[7]  Arnd Scheel,et al.  Transverse bifurcations of homoclinic cycles , 1997 .

[8]  J. Milnor.  On the concept of attractor , 1985 .

[9]  A. J. Homburg,et al.  Essentially asymptotically stable homoclinic networks , 2009 .

[10]  John Guckenheimer,et al.  Heteroclinic cycles and modulated traveling waves in systems with O(2) symmetry , 1988 .

[11]  On riddling and weak attractors , 2000 .

[12]  Ian Melbourne.  An example of a non-asymptotically stable attractor , 1991 .

[13]  M. Field.  Lectures on bifurcations, dynamics and symmetry , 1996 .

[14]  Nicola Sottocornola,et al.  Robust homoclinic cycles in Bbb R4 , 2003 .

[15]  M. Krupa.  Robust heteroclinic cycles , 1997 .

[16]  T. Ura.  On the flow outside a closed invariant set, stability, relative stability and saddle sets. , 1964 .

[17]  C. Postlethwaite,et al.  Resonance bifurcations from robust homoclinic cycles , 2009, 0906.4591.

[18]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[19]  A new mechanism for stability loss from a heteroclinic cycle , 2010 .

[20]  Martin Krupa,et al.  Asymptotic stability of heteroclinic cycles in systems with symmetry. II , 2004, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

[22]  Michael Field,et al.  Stationary bifurcation to limit cycles and heteroclinic cycles , 1991 .

[23]  Mark R. Proctor,et al.  The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance , 1988, Journal of Fluid Mechanics.

[24]  John Milnor,et al.  On the concept of attractor: Correction and remarks , 1985 .

[25]  Vivien Kirk,et al.  A competition between heteroclinic cycles , 1994 .

[26]  A. J. Homburg,et al.  Resonance bifurcation from homoclinic cycles , 2009 .