# American Institute of Mathematical Sciences

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.

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##### 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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