December  2016, 21(10): 3793-3808. doi: 10.3934/dcdsb.2016121

The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China, China

Received  August 2015 Revised  September 2016 Published  November 2016

Assume that the unperturbed parabolic equation has a degenerate homoclinic orbit. Under $T$-periodic perturbations, the periodic solutions bifurcated from the homoclinic solution are studied. By Fredholm alternative and Lyapunov-Schmidt reduction, the bifurcation functions defined between two finite-dimensional spaces are obtained. Some solvable conditions for the bifurcation functions are given. It is shown that, for any large $n>0$, the perturbed parabolic differential equation has a periodic solution with period $nT$.
Citation: Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
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show all references

References:
[1]

J. Diff. Eqns., 132 (1996), 21-45. doi: 10.1006/jdeq.1996.0169.  Google Scholar

[2]

Proc. Roy. Soc. Edinburgh Sect. A, 103 (1986), 265-274. doi: 10.1017/S0308210500018916.  Google Scholar

[3]

Nonlinear Anal., 10 (1986), 1277-1291. doi: 10.1016/0362-546X(86)90066-0.  Google Scholar

[4]

Trans. Amer. Math. Soc., 312 (1989), 539-587. doi: 10.1090/S0002-9947-1989-0988882-6.  Google Scholar

[5]

J. Diff. Eqns., 37 (1980), 351-373. doi: 10.1016/0022-0396(80)90104-7.  Google Scholar

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J. Math. Anal. Appl., 246 (2000), 245-264. doi: 10.1006/jmaa.2000.6791.  Google Scholar

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J. Diff. Eqns., 122 (1995), 1-26. doi: 10.1006/jdeq.1995.1136.  Google Scholar

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Trans. Amer. Math. Soc., 350 (1998), 3797-3814. doi: 10.1090/S0002-9947-98-02211-9.  Google Scholar

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J. Diff. Eqns., 65 (1986), 175-202. doi: 10.1016/0022-0396(86)90032-X.  Google Scholar

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Berlin, Springer, 1981.  Google Scholar

[11]

Boundary Value Problems, 2013 (2013), 1-18. doi: 10.1186/1687-2770-2013-101.  Google Scholar

[12]

Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.  Google Scholar

[13]

J. Diff. Eqns., 63 (1986), 227-254. doi: 10.1016/0022-0396(86)90048-3.  Google Scholar

[14]

Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295-325. doi: 10.1017/S0308210500031528.  Google Scholar

[15]

3 (2008), 6972, http://www.scholarpedia.org/article/Lin Google Scholar

[16]

Disc. Cont. Dyn. Sys., 9 (2003), 585-602. doi: 10.3934/dcds.2003.9.585.  Google Scholar

[17]

J. Diff. Eqns., 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[18]

J. Diff. Eqns., 249 (2010), 305-348. doi: 10.1016/j.jde.2010.04.007.  Google Scholar

[19]

Disc. Cont. Dyn. Sys., 9 (2003), 1293-1322. doi: 10.3934/dcds.2003.9.1293.  Google Scholar

[20]

(Russian) Dokl. Akad. Nauk SSSR , 160 (1965), 558-561.  Google Scholar

[21]

Soviet Math. Dokl., 172 (1967), 54-57.  Google Scholar

[22]

Math. USSR Sb., 77 (1968), 461-472.  Google Scholar

[23]

Math. USSR Sb., 81 (1970), 92-103.  Google Scholar

[24]

Dissertationes Math. (Rozprawy Mat.), 291 (1990), 74 pp, http://eudml.org/doc/268509  Google Scholar

[25]

Nonlinearity, 21 (2008), 285-303. doi: 10.1088/0951-7715/21/2/005.  Google Scholar

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