一类连续和不连续分段线性系统的周期解研究

数学物理学报 ›› 2021, Vol. 41 ›› Issue (4): 1053-1065.

一类连续和不连续分段线性系统的周期解研究

杨静¹,柯昌成¹,魏周超^1,^2,*()

¹ 中国地质大学(武汉)数学与物理学院武汉 430074
² 中国地质大学(武汉)浙江研究院杭州 311305

收稿日期:2019-09-14 出版日期:2021-08-26 发布日期:2021-08-09
通讯作者: 魏周超 E-mail:weizhouchao@163.com
基金资助:
国家自然科学基金(11772306);浙江省自然科学基金(LY20A020001)

Study on Periodic Solutions of a Class of Continuous and Discontinuous Piece-Wise Linear Systems

Jing Yang¹,Changcheng Ke¹,Zhouchao Wei^1,^2,*()

¹ School of Mathematics and Physics, China University of Geosciences(Wuhan), Wuhan 430074
² Zhejiang Research Institute, China University of Geosciences(Wuhan), Hangzhou 311305

Received:2019-09-14 Online:2021-08-26 Published:2021-08-09
Contact: Zhouchao Wei E-mail:weizhouchao@163.com
Supported by:
the NSFC(11772306);the NSF of Zhejiang Province(LY20A020001)

摘要/Abstract

摘要：

近年来，非光滑系统的研究成为一个热点，其有关分段线性系统的定性分析成了必不可少的研究问题.该文研究了一个变换后的Michelson微分系统，利用平均法理论证明了变换后的连续和不连续分段线性系统的周期解的存在性.

关键词: 连续分段线性微分系统, 不连续分段线性微分系统, 周期解, 平均法理论

Abstract:

In recent years, the study of non-smooth systems has become a hot spot, and the qualitative analysis of piece-wise linear systems has become an indispensable research problem. In this paper, a transformed Michelson differential system is studied, and the existence of periodic solutions for continuous and discontinuous piece-wise linear systems is proved by means of average theory.

Key words: Continuous piece-wise linear differential system, Discontinuous piece-wise linear differential system, Periodic solution, Average theory

中图分类号:

O175

杨静,柯昌成,魏周超. 一类连续和不连续分段线性系统的周期解研究[J]. 数学物理学报, 2021, 41(4): 1053-1065.

Jing Yang,Changcheng Ke,Zhouchao Wei. Study on Periodic Solutions of a Class of Continuous and Discontinuous Piece-Wise Linear Systems[J]. Acta mathematica scientia,Series A, 2021, 41(4): 1053-1065.

图/表 1

参考文献 24

1	Andronov A , Vitt A , Khaikin S . Theory of Oscillations. Oxford: Pergamon Press, 1996
2	Bernardo M Di , Budd C , Champneys A R , et al. Piece-wise smooth dynamical systems: theory and applications. Appl Math Sci, 2007, 163 (4): 1072- 1118
3	Simpson D J W . Bifurcations in piece-wise smooth continuous systems. J Nonlinear Sci, 2010, 69 (1): 23- 36
4	Makarenkov O , Lamb J S W . Dynamics and bifurcations of nonsmooth systems: A survey. Physica D, 2012, 241 (22): 1826- 1844
5	Carvalho B D , Fernando M L . More than three limit cycles in discontinuous piece-wise linear differential systems with two zones in the plane. Int J Bifurcat Chaos, 2014, 24 (4): 1450056 doi: 10.1142/S0218127414500564
6	Euzebio R D , Llibre J . On the number of limit cycles in discontinuous piece-wise linear differential systems with two pieces separated by a straight line. J Integral Equ Appl, 2015, 424 (1): 475- 486
7	Llibre J , Teixeira M A . Piece-wise linear differential systems without equilibria produce limit cycles. Nonlinear Dyn, 2017, 88 (1): 157- 164 doi: 10.1007/s11071-016-3236-9
8	Wang J F , Huang C X , Huang L H . Discontinuity-induced limit cycles in a general planar piece-wise linear system of saddle-focus type. Nonlinear Anal-Hybri, 2019, 33, 162- 178 doi: 10.1016/j.nahs.2019.03.004
9	Li S M , Llibre J . On the limit cycles of planar discontinue piece-wise linear differential systems with a unique equilibrium. Discrete Cont Dyn-B, 2019, 24 (11): 5885- 5901
10	Fatou P . Sur le mouvement d'un systeme soumis a des forces a courte periode. Bull Soc Math, 1928, 56, 98- 139
11	Bogoliubov N , Krylov N . Application of methods of nonlinear mechanics in the theory of stationary oscillations. Časo Pěst Mate Fysiky, 1935, 64 (5): 107- 115
12	Hale J K . On the method of averaging. IRE Trans CT, 1960, 7 (4): 517- 519
13	Hlanay A . On the method of averaging for differential equations with retarded argument. J Integral Equ Appl, 1966, 14 (1): 70- 76
14	Sethna P R , Meyer K R , Bajaj A K . On the method of averaging, integral manifolds and systems with symmetry. Siam J Appl Math, 1985, 45 (1): 343- 359
15	Lehman B , Bass R M . Extensions of averaging theory for power electronic systems. IEEE Trans Power Electr, 1996, 11 (4): 542- 553 doi: 10.1109/63.506119
16	Llibre J . Averaging theory and limit cycles for quadratic systems. Rad Mat, 2002, 3 (2): 215- 228
17	Llibre J , Novaes D , Teixeira M A . Higher order averaging theory for finding periodic solutions via Brouwer degree. Nonlinearity, 2014, 27 (1): 563- 583
18	Llibre J , Oliveira R , Rodrigues C A B . On the periodic solutions of the Michelson continuous and discontinuous piece-wise linear differential system. Comp Appl Math, 2018, 37 (2): 1550- 1561 doi: 10.1007/s40314-016-0413-x
19	Michelson D . Steady solutions for the Kuramoto-Sivashinsky equation. Physics D, 1986, 19 (1): 89- 111
20	Wilczak D . The existence of Shilnikov homoclinic orbits in the Michelson system: A computer assisted proof. Found Comput Math, 2006, 6 (4): 495- 535 doi: 10.1007/s10208-005-0201-2
21	Llibre J . The Michelson system is neither global analytic, nor Darboux integrable. Physics D, 2010, 239 (8): 414- 419 doi: 10.1016/j.physd.2010.01.007
22	Llibre J , Makhlouf A . Zero-Hopf bifurcation in the generalized Michelson system. Chaos, 2015, 89, 228- 231
23	Verhulst F . Nonlinear Differential Equations and Dynamical Systems. Berlin: Springer-Verlag, 1996
24	Francoise B A , Francoise J P , Llibre J . Periodic solutions of nonlinear periodic differential systems with a small parameter. Comm Pure Appl Anal, 2007, 6 (1): 103- 111 doi: 10.3934/cpaa.2007.6.103

[1]	张清业,徐斌. 一类带局部非线性项的静态狄拉克方程的多重周期解[J]. 数学物理学报, 2021, 41(4): 1013-1023.
[2]	鲁世平,周诗乐,余星辰. 具有不确定奇性的Liénard方程周期正解的存在性[J]. 数学物理学报, 2021, 41(3): 686-701.
[3]	蒋婷婷,杜增吉. 带有脉冲和Holling-IV型功能反应函数的中立型捕食-食饵模型的周期解[J]. 数学物理学报, 2021, 41(1): 178-193.
[4]	曹忠威,文香丹,冯微,祖力. 一类具有随机扰动的非自治SIRI流行病模型的动力学行为[J]. 数学物理学报, 2020, 40(1): 221-233.
[5]	阳超,李润洁. 一类具有不连续捕获的Lasota-Wazewska模型周期解存在性及稳定性分析[J]. 数学物理学报, 2019, 39(4): 785-796.
[6]	王超. 一类带有界回复力非线性Hill型方程的周期解[J]. 数学物理学报, 2019, 39(4): 761-772.
[7]	童常青,郑静. 半线性Klein-Gordon方程的高频周期解[J]. 数学物理学报, 2019, 39(3): 484-500.
[8]	程志波,毕中华,姚绍文. 一类具有偏差变元的p-Laplacian Liénard型方程在吸引奇性条件下周期解的存在性[J]. 数学物理学报, 2019, 39(2): 277-285.
[9]	蓝桂杰,付盈洁,魏春金,张树文. 具有Holling Ⅲ功能性反应的随机捕食食饵模型的平稳分布和周期解[J]. 数学物理学报, 2018, 38(5): 984-1000.
[10]	姚绍文, 程志波. 一类带阻尼项的奇性Duffing方程周期解的存在性[J]. 数学物理学报, 2018, 38(3): 543-548.
[11]	梁志清, 曾夏萍, 周泽文, 庞国萍, 黄军华. 状态反馈脉冲控制Holling-Tanner系统周期解的存在性^*[J]. 数学物理学报, 2018, 38(1): 174-189.
[12]	李周红, 张丰硕, 曹进德, Alsaedi Ahmed, Alsaadi Fuad E. 时标上带有反馈控制的非自治两种群竞争系统的概周期解[J]. 数学物理学报, 2017, 37(4): 730-750.
[13]	傅金波, 陈兰荪. 基于生态环境和反馈控制的多种群竞争系统的正周期解[J]. 数学物理学报, 2017, 37(3): 553-561.
[14]	段双双, 黄守军. Keller-Segel生物学方程组周期解的爆破[J]. 数学物理学报, 2017, 37(2): 326-341.
[15]	魏含玉, 夏铁成. 广义Broer-Kaup-Kupershmidt孤子方程的拟周期解[J]. 数学物理学报, 2016, 36(2): 317-327.