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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

R. George, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.  Google Scholar

[4]

Y. Jin and D. Zhu, Bifurcations of rough heteroclinic loop with two saddle points, Sci. China Ser. A, 46 (2003), 459-468.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. doi: 10.1016/0040-9383(74)90034-2.  Google Scholar

[11]

J. Palis, A global view of dynamics and a conjecture of the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347.  Google Scholar

[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.  Google Scholar

[13]

Q. Tian and D. Zhu, Bifurcations of nontwisted heteroclinic loop, Sci. China Ser. A, 43 (2000), 818-828. doi: 10.1007/BF02884181.  Google Scholar

[14]

L. Wen, Generic diffemorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419-452.  Google Scholar

[15]

D. Zhu and Z. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-848. doi: 10.1007/BF02871667.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

R. George, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.  Google Scholar

[4]

Y. Jin and D. Zhu, Bifurcations of rough heteroclinic loop with two saddle points, Sci. China Ser. A, 46 (2003), 459-468.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. doi: 10.1016/0040-9383(74)90034-2.  Google Scholar

[11]

J. Palis, A global view of dynamics and a conjecture of the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347.  Google Scholar

[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.  Google Scholar

[13]

Q. Tian and D. Zhu, Bifurcations of nontwisted heteroclinic loop, Sci. China Ser. A, 43 (2000), 818-828. doi: 10.1007/BF02884181.  Google Scholar

[14]

L. Wen, Generic diffemorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419-452.  Google Scholar

[15]

D. Zhu and Z. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-848. doi: 10.1007/BF02871667.  Google Scholar

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