January  2013, 18(1): 133-145. doi: 10.3934/dcdsb.2013.18.133

Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria

1. 

College of Mathematics and Science, China University of Geosciences(Beijing), Beijing, 100083, China, China

2. 

Department of Mathematics, East China Normal University, Shanghai, 200241

3. 

Department of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China

Received  March 2011 Revised  July 2012 Published  September 2012

In this paper, using the local moving frame approach, we investigate bifurcations of nongeneric heteroclinic loop with a nonhyperbolic equilibrium $p_1$ and a hyperbolic saddle $p_2$, where $p_1$ is assumed to undergo a transcritical bifurcation. Firstly, we establish the persistence of a nongeneric heteroclinic loop, the existence of a homoclinic loop and a periodic orbit when the transcritical bifurcation does not occur. Secondly, bifurcations of a nongeneric heteroclinic loop accompanied with a transcritical bifurcation are discussed. We obtain the existence of heteroclinic orbits, a homoclinic loop, a heteroclinic loop and a periodic orbit. Some bifurcation patterns different from the case of the generic heteroclinic loop accompanied with transcritical bifurcation are revealed. The results achieved here can be extended to higher dimensional systems.
Citation: Fengjie Geng, Junfang Zhao, Deming Zhu, Weipeng Zhang. Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 133-145. doi: 10.3934/dcdsb.2013.18.133
References:
[1]

S. N. Chow, B. Deng and J. M. Friedman, Theory and applicationsof a nongeneric heteroclinic loop bifurcation, SIAM J. Appl. Math., 59 (1999), 1303-1321.

[2]

A. R. Champneys, Codimension-one persistence beyond allorders of homoclinic orbits to singular saddle centres in reversible systems, Nonlinearity, 14 (2001), 87-112.

[3]

F. J. Geng, D. Liu and D. M. Zhu, Bifurcations of generic heteroclinic loop accompanied by transcritical bifurcation, International J. Bifurcation and Chaos, 4 (2008), 1069-1083.

[4]

X. B. Liu, X. L. Fu and D. M. Zhu, Homoclinic Bifurcation with non hyperbolic equilibria, Nonlinear Analysis, 66 (2007), 2931-2939. doi: 10.1016/j.na.2006.04.014.

[5]

X. B. Liu and D. M. Zhu, Homoclinic snaking near a heteroclinic cycles inreversible systems, Appl. Math. J. Chinese Univ. Ser. A (in Chinese), 19 (2004), 401-408.

[6]

J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Differential Equations, 218 (2005), 390-443. doi: 10.1016/j.jde.2005.03.016.

[7]

J. H. Sun and D. J. Luo, Local and global bifurcations with nonhyperbolic equilibria, Science in China, Series A, 37 (1994), 523-534.

[8]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.

[9]

D. M. Zhu and Z. H. Xia, Bifurcations of Morse-Smale dynamical systems, Science in China, Series A, 41 (1998), 837-848.

show all references

References:
[1]

S. N. Chow, B. Deng and J. M. Friedman, Theory and applicationsof a nongeneric heteroclinic loop bifurcation, SIAM J. Appl. Math., 59 (1999), 1303-1321.

[2]

A. R. Champneys, Codimension-one persistence beyond allorders of homoclinic orbits to singular saddle centres in reversible systems, Nonlinearity, 14 (2001), 87-112.

[3]

F. J. Geng, D. Liu and D. M. Zhu, Bifurcations of generic heteroclinic loop accompanied by transcritical bifurcation, International J. Bifurcation and Chaos, 4 (2008), 1069-1083.

[4]

X. B. Liu, X. L. Fu and D. M. Zhu, Homoclinic Bifurcation with non hyperbolic equilibria, Nonlinear Analysis, 66 (2007), 2931-2939. doi: 10.1016/j.na.2006.04.014.

[5]

X. B. Liu and D. M. Zhu, Homoclinic snaking near a heteroclinic cycles inreversible systems, Appl. Math. J. Chinese Univ. Ser. A (in Chinese), 19 (2004), 401-408.

[6]

J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Differential Equations, 218 (2005), 390-443. doi: 10.1016/j.jde.2005.03.016.

[7]

J. H. Sun and D. J. Luo, Local and global bifurcations with nonhyperbolic equilibria, Science in China, Series A, 37 (1994), 523-534.

[8]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.

[9]

D. M. Zhu and Z. H. Xia, Bifurcations of Morse-Smale dynamical systems, Science in China, Series A, 41 (1998), 837-848.

[1]

Hong Li. Bifurcation of limit cycles from a Li$ \acute{E} $nard system with asymmetric figure eight-loop case. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022033

[2]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173

[3]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233

[4]

Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209

[5]

Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969

[6]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175

[7]

Nicolai Sætran, Antonella Zanna. Chains of rigid bodies and their numerical simulation by local frame methods. Journal of Computational Dynamics, 2019, 6 (2) : 409-427. doi: 10.3934/jcd.2019021

[8]

Ling-Hao Zhang, Wei Wang. Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 591-604. doi: 10.3934/dcds.2017024

[9]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

[10]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153

[11]

André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations and Control Theory, 2022, 11 (3) : 749-779. doi: 10.3934/eect.2021024

[12]

Rong Zhang. Nonexistence of Positive Solutions for high-order Hardy-H$ \acute{e} $non Systems on $ \mathbb{R}^{n} $. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022078

[13]

Jianbin Li, Mengcheng Guan, Zhiyuan Chen. Optimal inventory policy for fast-moving consumer goods under e-commerce environment. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1769-1781. doi: 10.3934/jimo.2019028

[14]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295

[15]

Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031

[16]

Kota Kumazaki, Adrian Muntean. Local weak solvability of a moving boundary problem describing swelling along a halfline. Networks and Heterogeneous Media, 2019, 14 (3) : 445-469. doi: 10.3934/nhm.2019018

[17]

Arnab Roy, Takéo Takahashi. Local null controllability of a rigid body moving into a Boussinesq flow. Mathematical Control and Related Fields, 2019, 9 (4) : 793-836. doi: 10.3934/mcrf.2019050

[18]

Yurong Li, Zhengdong Du. Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6025-6052. doi: 10.3934/dcdsb.2019119

[19]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065

[20]

Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium. Conference Publications, 2011, 2011 (Special) : 931-940. doi: 10.3934/proc.2011.2011.931

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

[Back to Top]