Analysis of the T-Point-Hopf bifurcation with Z2-Symmetry: Application to Chua's equation

The aim of this work is twofold — on the one hand, to perform a theoretical analysis of the global behavior organized by a T-point–Hopf in ℤ2-symmetric systems; on the other hand, to apply the obtained results for a numerical study of Chua's equation, where for the first time this bifurcation is considered. In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-two heteroclinic cycle occurs. A more degenerate scenario appears when one of the equilibria involved in such a cycle undergoes a Hopf bifurcation. This degeneration, which corresponds to a codimension-three bifurcation, is called T-point–Hopf and has been recently studied for a generic system. However, the presence of ℤ2-symmetry may lead to the existence of a double T-point–Hopf heteroclinic cycle, which is responsible for the appearance of interesting global behavior that we will study in this paper. The theoretical models proposed for two different situations are based on the construction of a Poincare map. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved and their organization close to the T-point–Hopf bifurcation is described. The numerical phenomena found in Chua's equation strongly agree with the results deduced from the models.

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

[2]  Antonio Algaba,et al.  Open-to-Closed Curves of saddle-Node bifurcations of Periodic orbits Near a Nontransversal T-Point in Chua's equation , 2006, Int. J. Bifurc. Chaos.

[3]  Colin Sparrow,et al.  T-points: A codimension two heteroclinic bifurcation , 1986 .

[4]  Alejandro J. Rodríguez-Luis,et al.  Nontransversal curves of T-points: a source of closed curves of global bifurcations , 2002 .

[5]  Bernd Krauskopf,et al.  Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity , 2003 .

[6]  Mario A. Natiello,et al.  Accumulations of T-points in a model for solitary pulses in an excitable reaction-diffusion medium , 2005 .

[7]  Alejandro J. Rodríguez-Luis,et al.  Closed Curves of Global bifurcations in Chua's equation: a Mechanism for their Formation , 2003, Int. J. Bifurc. Chaos.

[8]  V. V. Bykov,et al.  The bifurcations of separatrix contours and chaos , 1993 .

[9]  Chai Wah Wu,et al.  LORENZ EQUATION AND CHUA’S EQUATION , 1996 .

[10]  Bifurcations near homoclinic orbits with symmetry , 1984 .

[11]  V. V. Bykov,et al.  On systems with separatrix contour containing two saddle-foci , 1999 .

[12]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[13]  F. Dumortier,et al.  Chaotic dynamics in ${\mathbb Z}_2$-equivariant unfoldings of codimension three singularities of vector fields in ${\mathbb R}^3$ , 2000, Ergodic Theory and Dynamical Systems.

[14]  F. Dumortier,et al.  Cocoon bifurcation in three-dimensional reversible vector fields , 2006 .

[15]  Freddy Dumortier,et al.  New aspects in the unfolding of the nilpotent singularity of codimension three , 2001 .

[16]  Alejandro J. Rodríguez-Luis,et al.  Analysis of the T-point-Hopf bifurcation , 2008 .

[17]  Alejandro J. Rodríguez-Luis,et al.  A model for the analysis of the dynamical consequences of a nontransversal intersection of the two-dimensional manifolds involved in a T-point , 2003 .

[18]  FERNANDO FERNÁNDEZ-SÁNCHEZ,et al.  Bi-spiraling homoclinic Curves around a T-Point in Chua's equation , 2004, Int. J. Bifurc. Chaos.

[19]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[20]  G. Belitskii.  Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class , 1973 .

[21]  E. Freire,et al.  T-Points in a Z2-Symmetric Electronic Oscillator. (I) Analysis , 2002 .

[22]  Alejandro J. Rodríguez-Luis,et al.  The non-transverse Shil'nikov-Hopf bifurcation: uncoupling of homoclinic orbits and homoclinic tangencies , 1999 .

[23]  Antonio Algaba,et al.  Some Results on Chua's equation Near a Triple-Zero Linear Degeneracy , 2003, Int. J. Bifurc. Chaos.