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

Bifurcation of a heterodimensional cycle with weak inclination flip

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

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.
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 $\mathbb{R}^3$, 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 $\mathbb{R}^3$, 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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