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Bifurcations of rough heteroclinic loop with two saddle points. (English) Zbl 1215.37039
Summary: The bifurcation problems of rough 2-point-loop are studied for the case $\rho _{1} ^{1} >\lambda _{1} ^{1} ,\rho _{2} ^{1} <\lambda _{2} ^{1} ,\rho _{1} ^{1} \rho _{2} ^{1} <\lambda _{1} ^{1} \lambda _{2} ^{1}$, where - $\rho _{ i } ^{1}$ <0 and $\lambda_{ i } ^{1}$ >0 are the pair of principal eigenvalues of unperturbed system at saddle point $p_{i}$, $i = 1,2$. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.

37G20Hyperbolic singular points with homoclinic trajectories
34C37Homoclinic and heteroclinic solutions of ODE
34C23Bifurcation (ODE)
Full Text: DOI
[1] Reyn, J. W., Generation of limit cycles from separatrix polygons in the phase plane, Lecture Notes in Math., 1980, 810: 264--289. · Zbl 0437.34025
[2] Roussarie, R., On the number of limit cycles which appear by perturbation of separatrix loop of planar fields, Bol. Soc. Brasil Mat., 1986, 17:67--1011. · Zbl 0628.34032 · doi:10.1007/BF02584827
[3] Feng, B., The stability of heteroclinic loop under the critical condition, Science in China, Ser. A, 1991, 34(6): 673--684.
[4] Feng, B., Xiao, D., Homoclinic and heteroclinic bifurcations of heteroclinic loops, Acta Math. Sinica, 1992, 35(6): 815--830. · Zbl 0782.34043
[5] Han, M., Luo, D., Zhu, D., The uniqueness of limit cycles bifurcating from a singular closed orbit, Acta Math. Sinica, 1992, 35(3): 407--417. · Zbl 0763.34018
[6] Han, M., Luo, D., Zhu, D., The uniqueness of limit cycles bifurcating from a singular closed orbit, Acta Math. Sinica, 1992, 35(4): 541--548. · Zbl 0772.34027
[7] Han, M., Luo, D., Zhu, D., The uniqueness of limit cycles bifurcating from a singular closed orbit, Acta Math. Sinica, 1992, 35(5): 673--684. · Zbl 0772.34028
[8] Mourtada, A., Degenerate and nontrivial hyperbolic polycycles with vertices, J. Diff. Equs., 1994, 113(1): 68--83. · Zbl 0810.58027 · doi:10.1006/jdeq.1994.1114
[9] Zhang, Z., Li, C., Zheng, Z. et al., The Foundation of Bifurcation Theory for Vector Field (in Chinese), Beijing: Advanced Education Press, 1997.
[10] Luo, D., Wang, X., Zhu, D. et al., Bifurcation Theory and Methods of Dynamical Systems, Singapore:World Scientific, 1997. · Zbl 0961.37015
[11] Chow, S. N., Deng, B., Fiedler, B., Homoclinic bifurcation at resonant eigenvalues, J. Dyna. Syst. and Diff. Equs., 1990, 2(2): 177--244. · Zbl 0703.34050
[12] Sun, J., Bifurcations of heteroclinic loop with nonhyperbolic critical points in $\mathbb{R}$n, Science in China, Ser. A, 1994, 24(11): 1145--1151.
[13] Zhu, D., Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica, New Series, 1998, 14(3): 341--352. · Zbl 0932.37032 · doi:10.1007/BF02580437
[14] Jin, Y., Zhu, D., Degenerated homoclinic bifurcations with higher dimensions, Chin. Ann. Math., Ser. B, 2000, 21(2): 201--210. · Zbl 0974.37014 · doi:10.1142/S0252959900000224
[15] Jin, Y., Zhu, D., Bifurcations of rough heteroclinic loops with three saddle points, Acta Math. Sinica, English Series, 2002, 18(1): 199--208. · Zbl 1010.34037 · doi:10.1007/s101140100139
[16] Zhu, D., Exponential trichotomy and heteroclinic bifurcation, Non1. Anal. TMA, 1997, 28(3): 547--557. · Zbl 0877.34037 · doi:10.1016/0362-546X(95)00164-Q
[17] Zhu, D., Xia, Z., Bifurcations of heteroclinic loops, Science in China, Ser. A, 1998, 41(8): 837--848. · Zbl 0993.34040 · doi:10.1007/BF02871667
[18] Tian, Q., Zhu, D., Bifurcations of nontwisted heteroclinic loops, Science in China, Ser. A, 2000, 30(3): 193--202.
[19] Fenichel, N., Persistence and smoothness of invariant manifold for flows, Indiana Univ. Math. J., 1971, 21: 193--226. · Zbl 0246.58015 · doi:10.1512/iumj.1972.21.21017
[20] Wiggins, S., Introduction to Applied Nonlinear Dynamical System and Chaos, New York: Springer-Verleg, 1990. · Zbl 0701.58001
[21] Abraham, R., Marsden, J. E., Manifolds, Tensor Analysis and Applications, London: Addison Wesley, 1983. · Zbl 0508.58001