从退化的双同宿环附近分支出周期解

Abstract

Abstract:

This paper investigates the bifurcation problem of autonomous differential equations with double homoclinic loops in high-dimensional systems under periodic perturbations. The double homoclinic loop consist of two degenerate homoclinic orbits connecting to the same hyperbolic equilibrium. Applying Lin's method to the double homoclinic loops, we derived the bifurcation function. Under certain conditions, the existence of zeros for these bifurcation functions is proven. Consequently, the perturbed system possesses periodic solutions near the unperturbed double homoclinic loop.

Key words: Lin's method, exponential dichotomy, periodic solutions, bifurcation

CLC Number:

O175.1

Bin Long, Ying Liu. Periodic Solutions Bifurcated from a Degenerate Double Homoclinic Loop[J].Acta mathematica scientia,Series A, 2025, 45(5): 1729-1744.

Trendmd

References

[1]	Wang Q. Periodically forced double homoclinic loops to a dissipative saddle. Journal of Differential Equations, 2016, 260(5): 4366-4392
[2]	Giannakopoulos F, Küpper T, Zou Y. Homoclinic bifurcations in a planar dynamical system. International Journal of Bifurcation and Chaos, 2001, 11(4): 1183-1191
[3]	Ragazzo C G. On the stability of double homoclinic loops. Communications in Mathematical Physics, 1997, 184: 251-272
[4]	Morales C A, Pacifico M J, Martin B S. Contracting Lorenz attractors through resonant double homoclinic loops. SIAM Journal on Mathematical Analysis, 2006, 38(1): 309-332
[5]	Kooij R E, Giannakopoulos F. Periodic orbits in planar systems modelling neural activity. Quarterly of Applied Mathematics, 2000, 58(3): 437-457
[6]	Chen S, Huang J. Destabilization of synchronous periodic solutions for patch models. Journal of Differential Equations, 2023, 364: 378-411
[7]	Xiong Y. Limit cycle bifurcations by perturbing a piecewise Hamiltonian system with a double homoclinic loop. International Journal of Bifurcation and Chaos, 2016, 26(6): 150-165
[8]	Zhang T, Zhu D. Bifurcations of double homoclinic flip orbits with resonant eigenvalues. Applied Mathematics and Mechanics, 2007, 28(11): 1517-1526
[9]	Zhang W, Zhu D. Codimension 2 bifurcations of double homoclinic loops. Chaos, Solitons and Fractals, 2009, 39(1): 295-303
[10]	Zhang W, Zhu D, Liu D. Codimension 3 nontwisted double homoclinic loops bifurcations with resonant eigenvalues. Journal of Dynamics and Differential Equations, 2008, 20(4): 893-908
[11]	吴家伟. 具有轨道翻转的对称双同宿环分支问题. 华东师范大学, 2021
[11]	Wu J W. The symmetric double homoclinic orbit bifurcation with orbit flipping. East China Normal University, 2021
[12]	张利群. 高维系统退化双同宿环分支问题. 济南: 山东师范大学, 2014
[12]	Zhang L Q. The Degenerate Double Homoclinic Loops Bofurcation in High-Dimensional Sysytems. Jinan: Shandong Normal University, 2014
[13]	Lu Q. Codimension 2 bifurcation of twisted double homoclinic loops. Computers and Mathematics with Applications, 2009, 57(7): 1127-1141
[14]	Jin Y, Zhu M, Li F, et al. Bifurcations of twisted double homoclinic loops with resonant condition. Journal of Nonlinear Sciences and Applications, 2016, 9(10): 165-177
[15]	朱红英, 邹荣, 李成群. 一类具有三个双同宿环的多项式李纳系统的极限环 (英文). 数学进展, 2022, 51(4): 598-608
[15]	Zhu H Y, Zou R, Li C Q. Limit cycles in a polynomial Liénard system with three double homoclinic loops. Advances in Mathematics, 2022, 51(4): 598-608
[16]	Liang F, Han M. Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems. Chaos, Solitons and Fractals, 2012, 45(4): 454-464
[17]	Bai Y, Liu X. Bifurcations of double homoclinic loops in reversible systems. International Journal of Bifurcation and Chaos, 2020, 30(16): 250-263
[18]	Jia Q, Zhang W, Lu Q. Bifurcations of double homoclinic loops with inclination flip and nonresonant eigenvalues. Journal of Nonlinear Modeling and Analysis, 2020, 2(1): 25-44
[19]	Lin X B. Lin's method. Scholarpedia, 2008, 3(9): 69-72
[20]	Vanderbauwhede A. Bifurcation of degenerate homoclinics. Results in Mathematics, 1992, 21(1): 211-223
[21]	Chow S N, Leiva H. Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces. Journal of Differential Equations, 1995, 120(2): 429-477
[22]	Fe?kan M. Bifurcation from homoclinic to periodic solutions in ordinary differential equations with multivalued perturbations. Journal of Differential Equations, 1996, 130(2): 415-450

Periodic Solutions Bifurcated from a Degenerate Double Homoclinic Loop

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Chao Zhu, Qingming Hao, Zhigang Pan, Yanhua Wang. Steady-State Bifurcation to a Class of Extended Fisher-Kolmogorov System with Cooperative and Self-Limiting Effects [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1432-1443.
[2]	Yushan Xi, Jicheng Duan, Rongsan Chen, Haijun Xiao. Sliding Bifurcation and Complex Dynamics Analysis of a Class of Filippov-Type Faraday Model [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1616-1631.
[3]	Tu Kun, Ding Huisheng. Shadowing Properties of Semilinear Nonautonomous Evolution Equations on Banach Spaces [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1144-1160.
[4]	Xu Junwen, Wu Hongxing, Sun Yangjian. The Poincaré Bifurcation of a Class of Pendulum Equations [J]. Acta mathematica scientia,Series A, 2025, 45(3): 843-849.
[5]	Liang Juan, Guo Zunguang, Zhang Hongtao. Steady-State Bifurcation for a Vegetation Model with Shading Effect [J]. Acta mathematica scientia,Series A, 2025, 45(3): 875-887.
[6]	Ren Chenchen, Yang Sudan. Existence and Uniqueness of Solutions for Sub-Linear Heat Equations with Almost Periodic Coefficients [J]. Acta mathematica scientia,Series A, 2025, 45(1): 31-43.
[7]	Xiao Jianglong, Song Yongli, Xia Yonghui. Spatiotemporal Dynamics Induced by the Interaction Between Fear and Schooling Behavior in a Diffusive Model [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1577-1594.
[8]	Yang Hong, Zhang Xiaoguang. A Stochastic SIS Epidemic Model on Simplex Complexes [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1392-1399.
[9]	Yang Jiaopeng, Liang Yong. Solitary and Periodic Solutions of the Generalized b-equation [J]. Acta mathematica scientia,Series A, 2024, 44(3): 670-686.
[10]	Ma Yani, Yuan Hailong. Bifurcation Analysis of a Class of Gierer-Meinhardt Activation Inhibition Model with Time Delay [J]. Acta mathematica scientia,Series A, 2023, 43(6): 1774-1788.
[11]	Dai Guowei,Gao Siyu,Ma Ruyun. Global Bifurcation for the Yamabe Equation on the Unit Sphere [J]. Acta mathematica scientia,Series A, 2023, 43(5): 1391-1396.
[12]	Wang Zihuan,Wang Chao. Symmetric and Periodic Solutions for a Class of Weakly Coupled Systems Composed of Two Particles with Obstacles [J]. Acta mathematica scientia,Series A, 2023, 43(5): 1427-1439.
[13]	Chen Min,Hu Biyan,Luo Hong. Boundary Layer Separation of 2-D Incompressible Navier-Stokes-Allen-Cahn System [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1123-1132.
[14]	Rui Xu,Yan Yang. Dynamics of an HTLV-I Infection Model with Delayed and Saturated CTL Immune Response and Immune Impairment [J]. Acta mathematica scientia,Series A, 2022, 42(6): 1836-1848.
[15]	Xueli Jiang,Xuan Deng,Qiuhao Wen,Yanqin Xiong. Limit Circle Bifurcations of Polynomial Differential System with Two Parallel Switch Straight Lines [J]. Acta mathematica scientia,Series A, 2022, 42(2): 353-364.