American Institute of Mathematical Sciences

Advanced
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

[1]

Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218
[2]

Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1511-1532. doi: 10.3934/dcdss.2011.4.1511
[3]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[4]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[5]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[6]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[7]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[8]

Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
[9]

Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162
[10]

Shiqiu Liu, Frédérique Oggier. On applications of orbit codes to storage. Advances in Mathematics of Communications, 2016, 10 (1) : 113-130. doi: 10.3934/amc.2016.10.113
[11]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080
[12]

Xueting Tian, Shirou Wang, Xiaodong Wang. Intermediate Lyapunov exponents for systems with periodic orbit gluing property. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1019-1032. doi: 10.3934/dcds.2019042
[13]

Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010
[14]

Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315
[15]

Peter Giesl. On a matrix-valued PDE characterizing a contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4839-4865. doi: 10.3934/dcdsb.2020315
[16]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015
[17]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177
[18]

Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221
[19]

Wade Hindes. Orbit counting in polarized dynamical systems. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 189-210. doi: 10.3934/dcds.2021112
[20]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, 2021, 29 (5) : 2987-3015. doi: 10.3934/era.2021023

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy