Bifurcation of a heterodimensional cycle with weak inclination flip

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May 2012, 17(3): 1009-1025. doi: 10.3934/dcdsb.2012.17.1009

Bifurcation of a heterodimensional cycle with weak inclination flip

Zhiqin Qiao ^1, , Deming Zhu ^2, and Qiuying Lu ^3,

1.	Department of Mathematics, North University of China, Taiyuan, 030051, China
2.	Department of Mathematics, East China Normal University, Shanghai, 200241
3.	Institute of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

Received January 2011 Revised August 2011 Published January 2012

Abstract
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Local moving frame is constructed to analyze the bifurcation of a heterodimensional cycle with weak inclination flip in $\mathbb{R}^4$. Under some generic hypotheses, the existence conditions for the heteroclinic orbit, $1$-homoclinic orbit, $1$-periodic orbit and two-fold or three-fold $1$-periodic orbit are given, respectively.

Keywords: homoclinic orbit, Heterodimensional cycle, weak inclination flip, heteroclinic orbit, periodic orbit..

Mathematics Subject Classification: Primary: 34C23, 37G10; Secondary: 34D0.

Citation: Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009

References:

[1]	L. Diaz, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcation, Nonlinearity, 8 (1995), 693-713. doi: 10.1088/0951-7715/8/5/003.
[2]	F. Geng, D. Zhu and Y. Xu, Bifurcation of heterodimensional cycles with two saddle points, Chaos Solitons Fractals, 39 (2009), 2063-2075. doi: 10.1016/j.chaos.2007.06.077.
[3]	R. George, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.
[4]	Y. Jin and D. Zhu, Bifurcations of rough heteroclinic loop with two saddle points, Sci. China Ser. A, 46 (2003), 459-468.
[5]	J. Knobloch and T. Wagenknecht, Homoclinic snaking near a heteroclinic cycle in reversible systems, Phys. D, 206 (2005), 82-93. doi: 10.1016/j.physd.2005.04.018.
[6]	J. Lamb, M.-A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $\mathbbR^3$, J. Differential Equations, 219 (2005), 78-115. doi: 10.1016/j.jde.2005.02.019.
[7]	D. Liu, F. Geng and D. Zhu, Degenerate bifurcations of nontwisted heterodimensional cycles with codimension $3$, Nonlinear Anal., 68 (2008), 2813-2827. doi: 10.1016/j.na.2007.02.028.
[8]	D. Liu, S. Ruan and D. Zhu, Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbbR^4$, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1511-1532. doi: 10.3934/dcdss.2011.4.1511.
[9]	S. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in "Dynamical Systems,'' (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, (1973), 303-366.
[10]	S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. doi: 10.1016/0040-9383(74)90034-2.
[11]	J. Palis, A global view of dynamics and a conjecture of the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347.
[12]	S. Shui and D. Zhu, Codimension 3 non-reasonant bifurcations of rough heteroclinic loops with one orbit flip, Chinese Ann. Math. Ser. B, 27 (2006), 657-674. doi: 10.1007/s11401-005-0472-6.
[13]	Q. Tian and D. Zhu, Bifurcations of nontwisted heteroclinic loop, Sci. China Ser. A, 43 (2000), 818-828. doi: 10.1007/BF02884181.
[14]	L. Wen, Generic diffemorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419-452.
[15]	D. Zhu and Z. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-848. doi: 10.1007/BF02871667.

show all references

References:

[1]	L. Diaz, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcation, Nonlinearity, 8 (1995), 693-713. doi: 10.1088/0951-7715/8/5/003.
[2]	F. Geng, D. Zhu and Y. Xu, Bifurcation of heterodimensional cycles with two saddle points, Chaos Solitons Fractals, 39 (2009), 2063-2075. doi: 10.1016/j.chaos.2007.06.077.
[3]	R. George, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.
[4]	Y. Jin and D. Zhu, Bifurcations of rough heteroclinic loop with two saddle points, Sci. China Ser. A, 46 (2003), 459-468.
[5]	J. Knobloch and T. Wagenknecht, Homoclinic snaking near a heteroclinic cycle in reversible systems, Phys. D, 206 (2005), 82-93. doi: 10.1016/j.physd.2005.04.018.
[6]	J. Lamb, M.-A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $\mathbbR^3$, J. Differential Equations, 219 (2005), 78-115. doi: 10.1016/j.jde.2005.02.019.
[7]	D. Liu, F. Geng and D. Zhu, Degenerate bifurcations of nontwisted heterodimensional cycles with codimension $3$, Nonlinear Anal., 68 (2008), 2813-2827. doi: 10.1016/j.na.2007.02.028.
[8]	D. Liu, S. Ruan and D. Zhu, Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbbR^4$, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1511-1532. doi: 10.3934/dcdss.2011.4.1511.
[9]	S. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in "Dynamical Systems,'' (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, (1973), 303-366.
[10]	S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. doi: 10.1016/0040-9383(74)90034-2.
[11]	J. Palis, A global view of dynamics and a conjecture of the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347.
[12]	S. Shui and D. Zhu, Codimension 3 non-reasonant bifurcations of rough heteroclinic loops with one orbit flip, Chinese Ann. Math. Ser. B, 27 (2006), 657-674. doi: 10.1007/s11401-005-0472-6.
[13]	Q. Tian and D. Zhu, Bifurcations of nontwisted heteroclinic loop, Sci. China Ser. A, 43 (2000), 818-828. doi: 10.1007/BF02884181.
[14]	L. Wen, Generic diffemorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419-452.
[15]	D. Zhu and Z. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-848. doi: 10.1007/BF02871667.

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