ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS

  • Maoan HAN ,
  • Yan YE
Expand
  • School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321000, China
E-mail: yanyemengyi@163.com

Received date: 2022-10-08

  Online published: 2024-08-30

Supported by

This work was supported by the NSFC (11931016).

Abstract

In this paper, we address the stability of periodic solutions of piecewise smooth periodic differential equations. By studying the Poincaré map, we give a sufficient condition to judge the stability of a periodic solution. We also present examples of some applications.

Cite this article

Maoan HAN , Yan YE . ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia, Series B, 2024 , 44(4) : 1524 -1535 . DOI: 10.1007/s10473-024-0418-2

References

[1] Di Bernardo M, Budd C J, Champneys A R, et al.Piecewise-Smooth Dynamical Systems: Theory and Applications. London: Springer-Verlag, 2008
[2] Shahshahani S. Periodic solutions of polynomial first order differential equations. Nonlinear Analysis: Theory Methods & Applications, 1981, 5(2): 157-165
[3] Neto A L. On the number of solutions of the equation $\frac{{\rm d}x}{{\rm d}t} = \sum\limits_{j = 0}^n a_j (t) x^j, 0 \leq t \leq 1, $ for which $x (0)= x (1)$. Inventiones Mathematicae, 1980, 59(1): 67-76
[4] Lloyd N G. The number of periodic solutions of the equation $\frac{{\rm d}z}{{\rm d}t}=z^N+p_1(t)z^{N-1}+\cdots+p_{N}(t)$. Proceedings of the London Mathematical Society, 1973, 27(3): 667-700
[5] Sheng L, Han M.Periodic solutions of one dimensional periodic differential equations
(in Chinese). Sci Sin Math, 2017, 47(1): 171-186
[6] Buica A, Llibre J. Averaging methods for finding periodic orbits via Brouwer degree. Bulletin Des Sciences Mathematiques, 2004, 128(1): 7-22
[7] Buica A, Francoise J P, Llibre J. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure and Applied Analysis, 2007, 6(1): 103-111
[8] Llibre J, Novaes D, Rodrigues C. Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones. Physica D: Nonlinear Phenomena, 2017, 353/354: 1-10
[9] Sheng L, Wang S, Li X, et al. Bifurcation of periodic orbits of periodic equations with multiple parameters by averaging method. Journal of Nonlinear Modeling and Analysis, 2020, 490(2): 124311
[10] Han M, Yang J. The maximum nonumber of zero of functions with parameters and application to differential equations. Journal of Nonlinear Modeling and Analysis, 2021, 3(1): 13-34
[11] Han M. On the maximum number of periodic solutions of piecewise smooth periodic equations by average method. Journal of Applied Analysis & Computation, 2017, 7(2): 788-794
[12] Lloyd N G. A note on the number of limit cycles in certain two-dimensional systems. Journal of the London Mathematical Society, 1979, 20(2): 277-286
[13] Hou W, Liu S. Melnikov functions for a class of piecewise Hamiltonian systems. Journal of Nonlinear Modeling and Analysis, 2023, 5(1): 123-145
Options
Outlines

/