TRANSVERSE HOMOCLINIC ORBIT BIFURCATED FROM A HOMOCLINIC MANIFOLD BY THE HIGHER ORDER MELNIKOV INTEGRALS

Bin Long; Changrong Zhu

doi:10.11948/20190311

2020 Volume 10 Issue 4

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Bin Long, Changrong Zhu. TRANSVERSE HOMOCLINIC ORBIT BIFURCATED FROM A HOMOCLINIC MANIFOLD BY THE HIGHER ORDER MELNIKOV INTEGRALS[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1651-1665. doi: 10.11948/20190311

Citation:

Bin Long, Changrong Zhu. TRANSVERSE HOMOCLINIC ORBIT BIFURCATED FROM A HOMOCLINIC MANIFOLD BY THE HIGHER ORDER MELNIKOV INTEGRALS[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1651-1665. doi: 10.11948/20190311

TRANSVERSE HOMOCLINIC ORBIT BIFURCATED FROM A HOMOCLINIC MANIFOLD BY THE HIGHER ORDER MELNIKOV INTEGRALS

Bin Long^1, ,,
Changrong Zhu²

1.
Department of Mathematics, Shaanxi University of Science and Technology, Xi'an, 710021, China
2.
School of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

Corresponding author: Email address:longbin210@126.com(B. Long)
Fund Project: The authors were supported by National Natural Science Foundation of China (No. 11801343) and Natural Science Basic Research Plan in Shaanxi Province of China(No. 2018JQ1031)

Abstract

Abstract

Consider an autonomous ordinary differential equation in $ \mathbb{R}^n $ that has a $ d $ dimensional homoclinic solution manifold $ W^H $. Suppose the homoclinic manifold can be locally parametrized by $ (\alpha,\theta) \in \mathbb{R}^{d-1}\times \mathbb{R} $. We study the bifurcation of the homoclinic solution manifold $ W^H $ under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed $ (\alpha_0,\theta_0) $ on $ W^H $, if the Melnikov function have a simple zeros, then the perturbed system can have transverse homoclinic solutions near $ W^H $.
- Homoclinic manifold /
- Lyapunov-Schmidt reduction /
- exponential dichotomies /
- Melnikov integral /
- chaos
MSC: 34C23, 34C37, 34C45

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References

[1]	Y. An and M. Han, On the number of limit cycles near a homoclinic loop with a nilpotent singular point, J. Diff. Eqns., 2015, 258, 3194-3247. doi: 10.1016/j.jde.2015.01.006 CrossRef Google Scholar
[2]	F. Battelli and K. J. Palmer, Tangencies between stable and unstable manifolds, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1992, 121, 73-90. doi: 10.1017/S0308210500014153 CrossRef Google Scholar
[3]	S. N. Chow, J. K. Hale and J. Mallet-Parret, An example of bifurcation to homoclinic orbits, J. Diff. Equs., 1980, 37, 551-573. Google Scholar
[4]	F. Chen and Q. Wang, High order Melnikov method: Theory and application, J. Diff. Eqns., 2019, DOI:10.1016/j.jde.2019.02.003. CrossRef Google Scholar
[5]	F. Chen and Q. Wang, High order Melnikov method for time-periodic equations, Adv. Nonlinear Stud., 2017, 17, 793-818. doi: 10.1515/ans-2017-6017 CrossRef Google Scholar
[6]	M. Fečkan, Bifurcation from degenerate homoclinics in periodically forced systems, Discr. Cont. Dyn. Systems, 1999, 5, 359-374. doi: 10.3934/dcds.1999.5.359 CrossRef Google Scholar
[7]	M. Fečkan, Higher dimensional Melnikov mappings, Mathematica Slovaca, 1999, 1, 75-83. Google Scholar
[8]	J. R. Gruendler, Homoclinic solutions for autonomous systems in arbitrary dimension, SIAM J.Math. Anal., 1992, 23, 702-721. doi: 10.1137/0523036 CrossRef Google Scholar
[9]	J. R. Gruendler, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbation, J. Diff. Equs., 1995, 122, 1-26. doi: 10.1006/jdeq.1995.1136 CrossRef Google Scholar
[10]	J. R. Gruendler, The existence of transverse homoclinic solutions for higher order equations, J. Diff. Equs., 1996, 130, 307-320. doi: 10.1006/jdeq.1996.0145 CrossRef Google Scholar
[11]	J. R. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. Google Scholar
[12]	J. K. Hale, Introduction to dynamic bifurcation, in "Bifurcation Theory and Applications, " p. 106-151, Lecture Notes in Mathematics, V. 1057, Springer-Verlag, Berlin, 1984. Google Scholar
[13]	M. Han, Asymptotic expansions of melnikov functions and limit cycle bifurcations, International Journal of Bifurcation and Chaos, 2012, 22, 1250296. doi: 10.1142/S0218127412502963 CrossRef Google Scholar
[14]	M. Han and P. Yu, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, Springer-Verlag, New York 2012. Google Scholar
[15]	J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Eqns., 1997, 9, 427-444. doi: 10.1007/BF02227489 CrossRef Google Scholar
[16]	L. M. Lerman and I. L. Umanskii, On the existence of separatrix loops in four-dimensional systems similar to the integrable hamiltonian systems, Journal of Applied Mathematics and Mechanics, 1984, 47, 335-340. Google Scholar
[17]	V. K. Melnikov, On the stability of the center for time periodic perturbation, Trans. Moscow Math. Soc., 1963, 12, 1-57. Google Scholar
[18]	K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Equs., 1984, 55, 225-256. doi: 10.1016/0022-0396(84)90082-2 CrossRef Google Scholar
[19]	K. J. Palmer, Existence of transversal homoclinic points in a degenerative case, Rocky Mountain Journal of Mathematics, 1990, 20, 1099-1118. doi: 10.1216/rmjm/1181073065 CrossRef Google Scholar
[20]	C. Zhu and W. Zhang, Homoclinic finger-rings in $R.N$, J. Diff. Eqns., 2017, 263, 3460-3490. doi: 10.1016/j.jde.2017.04.026 CrossRef Google Scholar