# American Institute of Mathematical Sciences

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 & 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. Google Scholar [2] A. R. Champneys, Codimension-one persistence beyond allorders of homoclinic orbits to singular saddle centres in reversible systems,, Nonlinearity, 14 (2001), 87. Google Scholar [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. Google Scholar [4] X. B. Liu, X. L. Fu and D. M. Zhu, Homoclinic Bifurcation with non hyperbolic equilibria,, Nonlinear Analysis, 66 (2007), 2931. doi: 10.1016/j.na.2006.04.014. Google Scholar [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. Google Scholar [6] J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit,, J. Differential Equations, 218 (2005), 390. doi: 10.1016/j.jde.2005.03.016. Google Scholar [7] J. H. Sun and D. J. Luo, Local and global bifurcations with nonhyperbolic equilibria,, Science in China, 37 (1994), 523. Google Scholar [8] S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Springer-Verlag, (1990). Google Scholar [9] D. M. Zhu and Z. H. Xia, Bifurcations of Morse-Smale dynamical systems,, Science in China, 41 (1998), 837. Google Scholar

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. Google Scholar [2] A. R. Champneys, Codimension-one persistence beyond allorders of homoclinic orbits to singular saddle centres in reversible systems,, Nonlinearity, 14 (2001), 87. Google Scholar [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. Google Scholar [4] X. B. Liu, X. L. Fu and D. M. Zhu, Homoclinic Bifurcation with non hyperbolic equilibria,, Nonlinear Analysis, 66 (2007), 2931. doi: 10.1016/j.na.2006.04.014. Google Scholar [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. Google Scholar [6] J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit,, J. Differential Equations, 218 (2005), 390. doi: 10.1016/j.jde.2005.03.016. Google Scholar [7] J. H. Sun and D. J. Luo, Local and global bifurcations with nonhyperbolic equilibria,, Science in China, 37 (1994), 523. Google Scholar [8] S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Springer-Verlag, (1990). Google Scholar [9] D. M. Zhu and Z. H. Xia, Bifurcations of Morse-Smale dynamical systems,, Science in China, 41 (1998), 837. Google Scholar
 [1] 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 & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233 [2] Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209 [3] Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969 [4] Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175 [5] 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 [6] 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 & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114 [7] Ling-Hao Zhang, Wei Wang. Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 591-604. doi: 10.3934/dcds.2017024 [8] Jianbin Li, Mengcheng Guan, Zhiyuan Chen. Optimal inventory policy for fast-moving consumer goods under e-commerce environment. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019028 [9] 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 [10] 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 [11] Wenbin Wang, Peng Zhang, Junfei Ding, Jian Li, Hao Sun, Lingyun He. Closed-loop supply chain network equilibrium model with retailer-collection under legislation. Journal of Industrial & Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039 [12] Kota Kumazaki, Adrian Muntean. Local weak solvability of a moving boundary problem describing swelling along a halfline. Networks & Heterogeneous Media, 2019, 14 (3) : 445-469. doi: 10.3934/nhm.2019018 [13] Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031 [14] 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 [15] Yurong Li, Zhengdong Du. Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6025-6052. doi: 10.3934/dcdsb.2019119 [16] Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761 [17] Renaud Leplaideur. From local to global equilibrium states: Thermodynamic formalism via an inducing scheme. Electronic Research Announcements, 2014, 21: 72-79. doi: 10.3934/era.2014.21.72 [18] Qiao-Fang Lian, Yun-Zhang Li. Reducing subspace frame multiresolution analysis and frame wavelets. Communications on Pure & Applied Analysis, 2007, 6 (3) : 741-756. doi: 10.3934/cpaa.2007.6.741 [19] Samuel Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3241-3285. doi: 10.3934/dcds.2014.34.3241 [20] Hunseok Kang. Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 939-959. doi: 10.3934/dcds.2008.20.939

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (6)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]