Bifurcations of resonant double homoclinic loops for higher dimensional systems
-
2656
Downloads
-
3562
Views
Authors
Yinlai Jin
- School of Science, Linyi University, Linyi, Shandong, 276005, P. R. China.
Han Xu
- School of Science, Linyi University, Linyi, Shandong, 276005, P. R. China.
Yuerang Gao
- School of Science, Linyi University, Linyi, Shandong, 276005, P. R. China.
Xue Zhao
- School of Science, Linyi University, Linyi, Shandong, 276005, P. R. China.
Dandan Xie
- School of Science, Linyi University, Linyi, Shandong, 276005, P. R. China.
Abstract
In this work, we study the bifurcation problems of double homoclinic loops with resonant condition
for higher dimensional systems. The Poincaré maps are constructed by using the foundational solutions
of the linear variational systems as the local coordinate systems in the small tubular neighborhoods of the
homoclinic orbits. We obtain the existence, number and existence regions of the small homoclinic loops,
small periodic orbits, and the large homoclinic loops, large periodic orbits, respectively. Moreover, the
complete bifurcation diagrams are given.
Share and Cite
ISRP Style
Yinlai Jin, Han Xu, Yuerang Gao, Xue Zhao, Dandan Xie, Bifurcations of resonant double homoclinic loops for higher dimensional systems, Journal of Mathematics and Computer Science, 16 (2016), no. 2, 165-177
AMA Style
Jin Yinlai, Xu Han, Gao Yuerang, Zhao Xue, Xie Dandan, Bifurcations of resonant double homoclinic loops for higher dimensional systems. J Math Comput SCI-JM. (2016); 16(2):165-177
Chicago/Turabian Style
Jin, Yinlai, Xu, Han, Gao, Yuerang, Zhao, Xue, Xie, Dandan. "Bifurcations of resonant double homoclinic loops for higher dimensional systems." Journal of Mathematics and Computer Science, 16, no. 2 (2016): 165-177
Keywords
- Double homoclinic loops
- resonance
- bifurcation
- higher dimensional system.
MSC
References
-
[1]
S.-N. Chow, B. Deng, B. Fiedler, Homoclinic bifurcation at resonant eigenvalues, J. Dynam. Differential Equations, 2 (1990), 177-244.
-
[2]
M. Han, Y. Wu, The stability of double homoclinic loops, Appl. Math. Lett., 17 (2004), 1291-1298.
-
[3]
Y. Jin, F. Li, H. Xu, J. Li, L. Zhang, B. Ding, Bifurcations and stability of nondegenerated homoclinic loops for higher dimensional systems, Comput. Math. Methods Med., 2013 (2013 ), 9 pages.
-
[4]
Y. Jin, D. Zhu, Degenerated homoclinic bifurcations with higher dimensions, Chinese Ann. Math. Ser., 21 (2000), 201-210.
-
[5]
Y. L. Jin, D. M. Zhu, Twisted bifurcations and stability of homoclinic loop with higher dimensions, Appl. Math. Mech., 25 (2004), 1176-1183.
-
[6]
X. Liu, D. Zhu, On the stability of homoclinic loops with higher dimension, Discrete Contin. Dyn. Syst. Ser., 17 (2012), 915-932.
-
[7]
Q. Lu, Codimension 2 bifurcation of twisted double homoclinic loops, Comput. Math. Appl. , 57 (2009), 1127-1141.
-
[8]
D. Luo, M. Han, D. Zhu, The uniqueness of limit cycles bifurcating from a singular closed orbit (I), Acta Math. Sinica, 35 (1992), 407-417.
-
[9]
K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.
-
[10]
C. G. Ragazzo, On the Stability of Double Homoclinic Loops, Comm. Math. Phys., 184 (1997), 251-272.
-
[11]
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos: Second Edition, Springer-Verlag, New York (2003)
-
[12]
W. Zhang, D. Zhu , Codimension 2 bifurcations of double homoclinic loops, Chaos Solitons Fractals, 39 (2009), 295-303.
-
[13]
D. Zhu , Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica, 14 (1998), 341-352.