一类连续和不连续分段线性系统的周期解研究

Abstract

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

CLC Number:

O175

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.

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References

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
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
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
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. ?aso 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
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
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
21	Llibre J . The Michelson system is neither global analytic, nor Darboux integrable. Physics D, 2010, 239 (8): 414- 419
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

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

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