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
References:
[1]

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

[2]

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

[3]

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

[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. doi: 10.1090/S0002-9947-1989-0988882-6. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics,, Berlin, (1981). Google Scholar

[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. doi: 10.1186/1687-2770-2013-101. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

X. B. Lin, Lin's method Scholarpedia,, 3 (2008), 3 (2008). Google Scholar

[16]

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

[17]

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

[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. doi: 10.1016/j.jde.2010.04.007. Google Scholar

[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. doi: 10.3934/dcds.2003.9.1293. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

H. O. Walther, Bifurcation from saddle connection in functional differential equations: An approach with inclination lemmas,, Dissertationes Math. (Rozprawy Mat.), 291 (1990). Google Scholar

[25]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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. doi: 10.1090/S0002-9947-1989-0988882-6. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics,, Berlin, (1981). Google Scholar

[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. doi: 10.1186/1687-2770-2013-101. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

X. B. Lin, Lin's method Scholarpedia,, 3 (2008), 3 (2008). Google Scholar

[16]

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

[17]

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

[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. doi: 10.1016/j.jde.2010.04.007. Google Scholar

[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. doi: 10.3934/dcds.2003.9.1293. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

H. O. Walther, Bifurcation from saddle connection in functional differential equations: An approach with inclination lemmas,, Dissertationes Math. (Rozprawy Mat.), 291 (1990). Google Scholar

[25]

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

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