1Institute of Neuroscience, Fourth Military Medical University, Xi'an 710032 2Institute of Nonlinear Dynamics, Xi'an Jiaotong University, Xi'an 710049 3XAC Research and Development Center, Xi'an 710089
Exotic Homoclinic Surface of a Saddle-Node Limit Cycle in a Leech Neuron Model
1Institute of Neuroscience, Fourth Military Medical University, Xi'an 710032 2Institute of Nonlinear Dynamics, Xi'an Jiaotong University, Xi'an 710049 3XAC Research and Development Center, Xi'an 710089
摘要We carry out numerical and theoretical investigations on the global unstable invariant set (manifold) of a saddle-node limit cycle in a leech heart interneuron model. The corresponding global bifurcation is accompanied by an explosion of secondary bifurcations of limit cycles and the emergence of loop-shaped bifurcation structures. The dynamical behaviors of the trajectories of the invariant set are very complicated and can only be partially explained by existing theories.
Abstract：We carry out numerical and theoretical investigations on the global unstable invariant set (manifold) of a saddle-node limit cycle in a leech heart interneuron model. The corresponding global bifurcation is accompanied by an explosion of secondary bifurcations of limit cycles and the emergence of loop-shaped bifurcation structures. The dynamical behaviors of the trajectories of the invariant set are very complicated and can only be partially explained by existing theories.
[1] Shilnikov A and Cymbalyuk G 2005 Phys. Rev. Lett. 94 048101
[2] Shilnikov A, Calabrese R and Cymbalyuk G 2005 Phys. Rev. E 71 056214
[3] Yooer C, Xu J and Zhang X 2009 Chin. Phys. Lett. 26 080501
[4] Yooer C, Xu J and Zhang X 2009 Chin. Phys. Lett. 26 070504
[5] Turaev D V and Shilnikov L P 1995 Dokl. Math. 51 404
[6] Rinzel J 1985 Lect. Notes Math. 1151 304
[7] Izhikevich E M 2000 Int. J. Bifurcat. Chaos 10 1171
[8] Dellnitz M and Hohmann A 1996 Nonlinear Dynamical Systems and Chaos ed Broer H W, Van Gils S A, Hoveijn I and Takens F (Basel: Birkhauser) p 449
[9] Dellnitz M and Hohmann A 1997 Numer. Math. 75 293
[10] Dellnitz M and Junge O 2002 Handbook of Dynamical Systems II: Towards Applications ed Fiedler B, Iooss G and Kopell N (Singapore: World Scientific) p 221
[11] Hsu C S 1992 Int. J. Bifurcat. Chaos 2 727
[12] Krauskopf B, Osinga H M, Doedel E J, Henderson M E, Guckenheimer J, Vladimirsky A, Dellnitz M and Junge O 2005 Int. J. Bifurcat. Chaos 15 763
[13] Dellnitz M, Froyland G and Junge O 2001 Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems ed Fiedler B (Berlin: Springer-Verlag) p 145
[14] Doedel E J 2009 AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations
[15] Pontryagin L S 1957 Izv. Akad. Nauk SSSR Ser. Mater. 21 605
[16] Pontryagin L S and Rodygin L V 1960 Sov. Math. Dokl. 1 237
[17] Tikhonov A N 1948 Math. Sb. 22 193
[18] Tikhonov A N 1952 Math. Sb. 31 575
[19] Fenichel N 1979 J. Diff. Eq. 31 53
2011, Vol. 28 
