Nongeneric bifurcations near a nontransversal heterodimensional cycle

Abstract

In this paper bifurcations of heterodimensional cycles with highly degenerate conditions are studied in three dimensional vector fields, where a nontransversal intersection between the two-dimensional manifolds of the saddle equilibria occurs. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, the authors construct a Poincaré return map under the nongeneric conditions and further obtain the bifurcation equations. By means of the bifurcation equations, the authors show that different bifurcation surfaces exhibit variety and complexity of the bifurcation of degenerate heterodimensional cycles. Moreover, an example is given to show the existence of a nontransversal heterodimensional cycle with one orbit flip in three dimensional system.

Keywords

Local moving frame / Nontransversal heterodimensional cycle / Orbit flip / Poincaré return map

References

[1] Homburg A. J., Sandstede B.. Homoclinic and heteroclinic bifurcations in vector fields, Handbook of Dynamical Systems, Broer. Henk (ed.) et al., 2010, 3: 379-524

[2] Han M. A., Zhu H. P.. The loop quantities and bifurcations of homoclinic loops. J. Differ. Equ., 2007, 234: 339-359

[3] Chow S. N., Jiang M., Lin X. B.. Traveling wave solutions in coupled Chua’s circuits, part I: Periodic solutions. J. Applied. Analysis and Computation, 2013, 3: 213-237

[4] Yanagida E.. Branching of double pulse solutions from single pulse solutions in nerve axon equations. J. Differ. Equ., 1987, 66: 243-262

[5] Sandstede B.. Constructing dynamical systems having homoclinic bifurcation points of codimension two. J. Dyn. Differ. Equ., 1997, 9: 269-288

[6] Naudot V.. A strange attractor in the unfolding of an orbit-flip homoclinic orbit. Dyn. Syst., 2002, 17: 45-63

[7] Golmakani A., Homburg A. J.. Lorenz attractors in unfoldings of homoclinic-flip bifurcations. Dyn. Syst., 2011, 26: 61-76

[8] Liu X. B., Wang Z. Z., Zhu D. M.. Bifurcation of rough heteroclinic loop with orbit flips. Int. J. Bifurc. Chaos, 2012, 22: 1250278

[9] Newhouse S. E., Palis J.. Bifurcations of Morse-Smale dynamical systems, Dynamical Systems, 1973, New York: Aca- demic Press 303-366

[10] Fernández-Sánchez F., Freire E., Rodr´iguez-Luis A. J.. Bi-spiraling homoclinic curves around a T-point in Chua’s circuit. Int. J. Bifurc. Chaos, 2004, 14: 1789-1793

[11] Algaba A., Freire E., Gamero E., Rodr´iguez-Luis A. J.. A tame degenerate Hopf-pitchfork bifurcation in a modified van der Pol-Duffing oscillator. Nonlinear Dyn., 2000, 22: 249-269

[12] Bykov V. V.. The bifurcations of separatrix contours and chaos. Physica D, 1993, 62: 290-299

[13] Rademacher J. D. M.. Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit. J. Differ. Equ., 2005, 218: 390-443

[14] Bonatti C., Diaz L. J., Pujals E., Rocha J.. Robust transitivity and heterodimensional cycles. Asterisque, 2003, 286: 187-222

[15] Diaz L. J., Rocha J.. Heterodimensional cycles, partial hyperbolity and limit dynamics. Fund. Math., 2002, 174: 127-186

[16] Lamb J. S. W., Teixeira M. A., Webster K. N.. Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R3. J. Differ. Equ., 2005, 219: 78-115

[17] Liu D., Geng F. J., Zhu D. M.. Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3. Nonlinear Analysis, 2008, 68: 2813-2827

[18] Liu X. B.. Bifurcations near the weak type heterodimensional cycle. Int. J. Bifur. Chaos, 2014, 9: 1450112

[19] Lu Q. Y., Qiao Z. Q., Zhang T. S., Zhu D. M.. Heterodimensional cycle bifurcation with orbit-flip. Int. J. Bifur. Chaos, 2010, 20: 491-508

[20] Fernández-Sánchez F., Freire E., Rodr´iguez-Luis A. J.. Bi-spiraling homoclinic curves around a T-point in Chua’s circuit. Int. J. Bifur. Chaos, 2004, 14: 1789-1793

[21] Zhu D. M., Xia Z. H.. Bifurcations of heteroclinic loops. Sci. China Ser. A, 1998, 41: 837-848

[22] Liu X. B.. Homoclinic flip bifurcations accompanied by transcritical bifurcation, Chin. Ann. Math.. Ser. B, 2011, 6: 905-916

[23] Xu Y. C., Zhu D. M.. Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip. Nonlinear Dyn., 2010, 60: 1-13

[24] Shilnikov L. P., Shilnikov A. L., Turaev D. V., Chua L. O.. Methods of Qualitative Theory in Nonlinear Dynamics, Part I, 1998, New Jersey: World Scientific

[25] Homburg A. J., Knobloch J.. Multiple homoclinic orbits in conservative and reversible systems. Trans. Am. Math. Soc., 2006, 4: 1715-1740

[26] Deng B.. Sil’nikov problem, exponential expansion, strong λ-Lemma, C¹-linearization and homoclinic bifurcation. J. Differ. Equ., 1989, 79: 189-231