Limit cycles in piecewise smooth differential systems of focus-focus and saddle-saddle dynamics

Document Type : Research Paper

Authors

1 Fergusson College (Autonomous)‎, ‎Pune‎, ‎India.

2 Abasaheb Garware College, Pune, India.

3 Department of Mathematics, Savitribai Phule Pune University, Pune, India.

Abstract

In this paper, we obtained the Poincare return maps for the planar piecewise linear differential systems of the type focus-focus. Normal forms for planar piecewise smooth systems with two zones of the type focus-focus and saddle-saddle, separated by a straight line and with a center at the origin, are obtained. Upper bounds for the number of limit cycles bifurcated from the period annulus of these normal forms due to perturbation by polynomial functions of any degree are established.

Keywords

Main Subjects

References

  • [1] F. Basem A and A. MK, New subclasses of bi-univalent functions, Applied Mathematics Letters, 24(9) (2011), 1569-1573.
  • [2] N. Douglas D and T. Joan, On extended Chebyshev systems with positive accuracy, Journal of mathematical analysis and applications, 448(1) (2017), 171-186.
  • [3] F. Emilio, P. Enrique, and T. Francisco, Planar Filippov systems with maximal crossing set and piecewise linear focus dynamics, Progress and Challenges in Dynamical Systems, (2013), 221-232.
  • [4] F. Emilio, P. Enrique, and T. Francisco, The discontinuous matching of two planar linear foci can have three nested crossing limit cycles, Publicacions matematiques, (2014), 221-253.
  • [5] P. Enrique, R. Javier, and V. El´ısabet, The boundary focus–saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators, Nonlinear Analysis: Real World Applications, 43 (2018), 495-514.
  • [6] L. Haihua, L. Shimin, and Z. Xiang, Limit cycles and global dynamics of planar piecewise linear refracting systems of focus-focus type, Nonlinear Analysis: Real World Applications, 58 (2021), 103228.
  • [7] L. Jaume, T. Marco Antonio, and T. Joan, Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation, International Journal of Bifurcation and Chaos,23(4) (2013), 1350066.
  • [8] L. Jaume and Z. Xiang, Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center, Journal of Mathematical Analysis and Applications, 467(1) (2018), 537-549.
  • [9] W. Jiafu, H. Chuangxia, and H. Lihong, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle–focus type, Nonlinear Analysis: Hybrid Systems, 33 (2019), 162-178.
  • [10] W. Jiafu, C. Xiaoyan, and H. Lihong, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, Journal of Mathematical Analysis and Applications, 469(1) (2019), 405-427.
  • [11] Y. Jihua, Z. Erli, and L. Mei, Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp, Communications on Pure & Applied Analysis, 16(6) (2017), 2321.
  • [12] Y. Jihua and Z. Erli, On the number of limit cycles for a class of piecewise smooth Hamiltonian systems with discontinuous perturbations, Nonlinear Analysis: Real World Applications, 52 (2020), 103046.
  • [13] M. Joaqu´ın, L. Miguel A, and M. Raquel, Explicit Construction of the Inverse of an Analytic Real Function: Some Applications, Mathematics,8(12) (2020), 2154
  • [14] W. Lijun and Z. Xiang, Normal form and limit cycle bifurcation of piecewise smooth differential systems with a center, Journal of Differential Equations, 261(2) (2016), 1399-1428.
  • [15] L. Liping, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electronic Journal of Qualitative Theory of Differential Equations, 70 (2014), 1-14.
  • [16] J. C. Medrado and J. Torregrosa, Uniqueness of limit cycles for sewing planar piecewise linear systems, Journal of Mathematical Analysis and Applications, 431(1) (2015), 529-544.
  • [17] J. Sanders, An easy way to determine the sense of rotation of phase plane trajectories, Pi Mu Epsilon Journal, 13(8) (2013),491-494.
  • [18] L. Shimin, L. Changjian, and L. Jaume, The planar discontinuous piecewise linear refracting systems have at most one limit cycle, Nonlinear Analysis: Hybrid Systems, 41 (2021), 101045.
  • [19] H. Song-Mei and Y. Xiao-Song, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics, Nonlinear Analysis: Theory, Methods & Applications, 92 (2013), 82-95.
  • [20] H. Song-Mei and Y. Xiao-Song, On the number of limit cycles in general planar piecewise linear systems of node-node types, Journal of Mathematical Analysis and Applications, 411(1) (2014), 340-353.
  • [21] C. Victoriano, F. Emilio, P. Enrique, and T. Francisco, On simplifying and classifying piecewise-linear systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory, and Applications, 49(5) (2002), 609-620.
  • [22] C. Victoriano, F. Fernando, and N. Douglas D, A new simple proof for Lum–Chuas conjecture, Nonlinear Analysis: Hybrid Systems, 40 (2021), 100992.
  • [23] L. Xia and H. Maoan, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, International Journal of Bifurcation and Chaos, 20(5) (2010), 1379-1390.

 

Computational Methods for Differential Equations
Volume 13, Issue 2
March 2025
Pages 395-419

Files

History

  • Receive Date: 08 December 2023
  • Revise Date: 01 May 2024
  • Accept Date: 08 June 2024

Share

How to cite

Statistics