Codimension 3 heteroclinic bifurcations with Orbit and Inclination Flips in Reversible Systems

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

[1]  Bifurcations of nontwisted heteroclinic loop , 2000 .

[2]  Jens D. M. Rademacher,et al.  Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit , 2005 .

[3]  Deming Zhu,et al.  Bifurcations of rough heteroclinic loop with two saddle points , 2003 .

[4]  J. Lamb,et al.  Time-reversal symmetry in dynamical systems: a survey , 1998 .

[5]  R. Churchill,et al.  Homoclinic and heteroclinic orbits of reversible vectorfields under perturbation , 1986, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Jeroen S. W. Lamb,et al.  Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R 3 , 2005 .

[7]  R. D. Driver,et al.  Ordinary and Delay Differential Equations , 1977 .

[8]  Xiao-Biao Lin,et al.  Using Melnikov's method to solve Silnikov's problems , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[9]  Deming Zhu,et al.  Bifurcations of Rough Heteroclinic Loops with Three Saddle Points , 2002 .

[10]  M. Krupa,et al.  Homoclinic Twist Bifurcation in a System of Two Coupled Oscillators , 2000 .

[11]  Björn Sandstede,et al.  Existence and stability of N -pulses on optical fibers with phase-sensitive amplifiers , 1997 .

[12]  J. Klaus,et al.  Bifurcation of homoclinic orbits to a saddle-Center in Reversible Systems , 2003, Int. J. Bifurc. Chaos.

[13]  T. Wagenknecht.  About a homoclinic pitchfork bifurcation in reversible systems with additional Z2-symmetry , 2002 .

[14]  Zhu Deming,et al.  Problems in homoclinic bifurcation with higher dimensions , 1998 .

[15]  Deming Zhu,et al.  Bifurcations of heteroclinic loops , 1998 .

[16]  Alan R. Champneys,et al.  A non-transverse homoclinic orbit to a saddle-node equilibrium , 1996 .

[17]  Alan R. Champneys,et al.  Homoclinic orbits in reversible systems and their applications in mechanics , 1998 .

[18]  P. Rabinowitz.  Heteroclinics for a reversible Hamiltonian system , 1994, Ergodic Theory and Dynamical Systems.

[19]  Jörg Härterich.  Cascades of reversible homoclinic orbits to a saddle-focus equilibrium , 1998 .

[20]  Heteroclinic cycles and periodic orbits in reversible systems , 1992 .

[21]  Tiansi Zhang,et al.  Codimension 3 homoclinic bifurcation of Orbit flip with Resonant eigenvalues Corresponding to the Tangent Directions , 2004, Int. J. Bifurc. Chaos.

[22]  A. J. Homburg,et al.  Multiple homoclinic orbits in conservative and reversible systems , 2005 .

[23]  Thomas Wagenknecht,et al.  Homoclinic snaking near a heteroclinic cycle in reversible systems , 2005 .