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 and Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
References:
[1]

F. Battelli and M. Feckan, Subharmonic solutions in singular systems, J. Diff. Eqns., 132 (1996), 21-45. doi: 10.1006/jdeq.1996.0169.

[2]

C. M. Blazquez, Bifurcations from a homoclinic orbit in parabolic differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 103 (1986), 265-274. doi: 10.1017/S0308210500018916.

[3]

C. M. Blazquez, Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal., 10 (1986), 1277-1291. doi: 10.1016/0362-546X(86)90066-0.

[4]

S.-N. Chow and B. Deng, Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems, Trans. Amer. Math. Soc., 312 (1989), 539-587. doi: 10.1090/S0002-9947-1989-0988882-6.

[5]

S.-N. Chow, J. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Diff. Eqns., 37 (1980), 351-373. doi: 10.1016/0022-0396(80)90104-7.

[6]

M. Feckan and J. Gruendler, Bifurcation from homoclinic to periodic solutions in singular ordinary differential equations, J. Math. Anal. Appl., 246 (2000), 245-264. doi: 10.1006/jmaa.2000.6791.

[7]

J. Gruendler, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Diff. Eqns., 122 (1995), 1-26. doi: 10.1006/jdeq.1995.1136.

[8]

J. Gruendler, Homoclinic solutions and chaos in ordinarry differential equations with singular perturbations, Trans. Amer. Math. Soc., 350 (1998), 3797-3814. doi: 10.1090/S0002-9947-98-02211-9.

[9]

J. K. Hale and X. B. Lin, Heteroclinic orbits for retarded functional differential equations, J. Diff. Eqns., 65 (1986), 175-202. doi: 10.1016/0022-0396(86)90032-X.

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Berlin, Springer, 1981.

[11]

M. Kamenskii, B. Mikhaylenko and P. Nistri, A bifurcation problem for a class of periodically perturbed autonomous parabolic equations, Boundary Value Problems, 2013 (2013), 1-18. doi: 10.1186/1687-2770-2013-101.

[12]

J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.

[13]

X. B. Lin, Exponential dichotomies and homoclinic orbits in functional defferential equations, J. Diff. Eqns., 63 (1986), 227-254. doi: 10.1016/0022-0396(86)90048-3.

[14]

X. B. Lin, Using Melnikov's method to solve Silnikov's problem, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295-325. doi: 10.1017/S0308210500031528.

[15]

X. B. Lin, Lin's method Scholarpedia, 3 (2008), 6972, http://www.scholarpedia.org/article/Lin

[16]

K. Matthies, Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing, Disc. Cont. Dyn. Sys., 9 (2003), 585-602. doi: 10.3934/dcds.2003.9.585.

[17]

K. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Eqns., 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.

[18]

J. D. M. Rademacher, Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies, J. Diff. Eqns., 249 (2010), 305-348. doi: 10.1016/j.jde.2010.04.007.

[19]

S. Ruan, J. Wei and J. Wu, Bifurcation from a homoclinic orbit in partial functional differential equations, Disc. Cont. Dyn. Sys., 9 (2003), 1293-1322. doi: 10.3934/dcds.2003.9.1293.

[20]

L. P. Silnikov, A case of the existence of a countable number of periodic motions, (Russian) Dokl. Akad. Nauk SSSR , 160 (1965), 558-561.

[21]

L. P. Silnikov, The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus, Soviet Math. Dokl., 172 (1967), 54-57.

[22]

L. P. Silnikov, On the generalization of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle, Math. USSR Sb., 77 (1968), 461-472.

[23]

L. P. Silnikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focue type, Math. USSR Sb., 81 (1970), 92-103.

[24]

H. O. Walther, Bifurcation from saddle connection in functional differential equations: An approach with inclination lemmas, Dissertationes Math. (Rozprawy Mat.), 291 (1990), 74 pp, http://eudml.org/doc/268509

[25]

C. Zhu, The coexistence of subharmonics bifurcated from homoclinic orbits in singular systems, Nonlinearity, 21 (2008), 285-303. doi: 10.1088/0951-7715/21/2/005.

show all references

References:
[1]

F. Battelli and M. Feckan, Subharmonic solutions in singular systems, J. Diff. Eqns., 132 (1996), 21-45. doi: 10.1006/jdeq.1996.0169.

[2]

C. M. Blazquez, Bifurcations from a homoclinic orbit in parabolic differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 103 (1986), 265-274. doi: 10.1017/S0308210500018916.

[3]

C. M. Blazquez, Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal., 10 (1986), 1277-1291. doi: 10.1016/0362-546X(86)90066-0.

[4]

S.-N. Chow and B. Deng, Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems, Trans. Amer. Math. Soc., 312 (1989), 539-587. doi: 10.1090/S0002-9947-1989-0988882-6.

[5]

S.-N. Chow, J. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Diff. Eqns., 37 (1980), 351-373. doi: 10.1016/0022-0396(80)90104-7.

[6]

M. Feckan and J. Gruendler, Bifurcation from homoclinic to periodic solutions in singular ordinary differential equations, J. Math. Anal. Appl., 246 (2000), 245-264. doi: 10.1006/jmaa.2000.6791.

[7]

J. Gruendler, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Diff. Eqns., 122 (1995), 1-26. doi: 10.1006/jdeq.1995.1136.

[8]

J. Gruendler, Homoclinic solutions and chaos in ordinarry differential equations with singular perturbations, Trans. Amer. Math. Soc., 350 (1998), 3797-3814. doi: 10.1090/S0002-9947-98-02211-9.

[9]

J. K. Hale and X. B. Lin, Heteroclinic orbits for retarded functional differential equations, J. Diff. Eqns., 65 (1986), 175-202. doi: 10.1016/0022-0396(86)90032-X.

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Berlin, Springer, 1981.

[11]

M. Kamenskii, B. Mikhaylenko and P. Nistri, A bifurcation problem for a class of periodically perturbed autonomous parabolic equations, Boundary Value Problems, 2013 (2013), 1-18. doi: 10.1186/1687-2770-2013-101.

[12]

J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.

[13]

X. B. Lin, Exponential dichotomies and homoclinic orbits in functional defferential equations, J. Diff. Eqns., 63 (1986), 227-254. doi: 10.1016/0022-0396(86)90048-3.

[14]

X. B. Lin, Using Melnikov's method to solve Silnikov's problem, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295-325. doi: 10.1017/S0308210500031528.

[15]

X. B. Lin, Lin's method Scholarpedia, 3 (2008), 6972, http://www.scholarpedia.org/article/Lin

[16]

K. Matthies, Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing, Disc. Cont. Dyn. Sys., 9 (2003), 585-602. doi: 10.3934/dcds.2003.9.585.

[17]

K. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Eqns., 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.

[18]

J. D. M. Rademacher, Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies, J. Diff. Eqns., 249 (2010), 305-348. doi: 10.1016/j.jde.2010.04.007.

[19]

S. Ruan, J. Wei and J. Wu, Bifurcation from a homoclinic orbit in partial functional differential equations, Disc. Cont. Dyn. Sys., 9 (2003), 1293-1322. doi: 10.3934/dcds.2003.9.1293.

[20]

L. P. Silnikov, A case of the existence of a countable number of periodic motions, (Russian) Dokl. Akad. Nauk SSSR , 160 (1965), 558-561.

[21]

L. P. Silnikov, The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus, Soviet Math. Dokl., 172 (1967), 54-57.

[22]

L. P. Silnikov, On the generalization of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle, Math. USSR Sb., 77 (1968), 461-472.

[23]

L. P. Silnikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focue type, Math. USSR Sb., 81 (1970), 92-103.

[24]

H. O. Walther, Bifurcation from saddle connection in functional differential equations: An approach with inclination lemmas, Dissertationes Math. (Rozprawy Mat.), 291 (1990), 74 pp, http://eudml.org/doc/268509

[25]

C. Zhu, The coexistence of subharmonics bifurcated from homoclinic orbits in singular systems, Nonlinearity, 21 (2008), 285-303. doi: 10.1088/0951-7715/21/2/005.

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