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Bifurcation Analysis of Planar Piecewise Smooth Systems with a Line of Discontinuity

Dingheng Pi

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China

E-mail Address: pidh@hqu.edu.cn

Corresponding author.

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and

Shihong Xu

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China

E-mail Address: 605523214@qq.com

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https://doi.org/10.1142/S0218127416501042Cited by:0 (Source: Crossref)

Abstract

In this paper, we consider bifurcations of a class of planar piecewise smooth differential systems constituted by a general linear system and a quadratic Hamiltonian system. The linear system has four parameters. When the parameters vary in different regions, the left linear system can have a saddle, a node or a focus. For each case, we provide a completely qualitative analysis of the dynamical behavior for this piecewise smooth system. Our results generalize and improve the results in this direction.

Keywords:

References

Artés, J. C. & Llibre, J. [1994] “ Quadratic Hamiltonian vector fields,” J. Diff. Eqs. 107, 80–95. Crossref, Web of Science, Google Scholar
Buzzi, C. A., da Silva, P. R. & Teixeira, M. A. [2006] “ A singular approach to discontinuous vector fields on the plane,” J. Diff. Eqs. 231, 633–655. Crossref, Web of Science, Google Scholar
di Bernado, M., Budd, C. J., Champneys, A. R. & Kowalczyk, P. [2008] Piecewise-Smooth Dynamical Systems, Theory and Applications, Applied Mathematical Sciences, Vol. 163 (Springer-Verlag, London). Google Scholar
Dieci, L. & Elia, C. [2014] “ Periodic orbits for planar piecewise linear systems with a line of discontinuity,” J. Dyn. Diff. Eqs. 26, 1049–1078. Crossref, Web of Science, Google Scholar
Filippov, A. F. [1988] Differential Equations with Discontinuous Right-Hand Sides (Kluwer Academic Publishers, Dordrecht). Crossref, Google Scholar
Freire, E., Ponce, E. & Torres, F. [2012] “ Canonical discontinuous planar piecewise linear systems,” SIAM J. Appl. Dyn. Syst. 11, 181–211. Crossref, Web of Science, Google Scholar
Freire, E., Ponce, E. & Torres, F. [2014] “ A general mechanism to generate three limit cycles in planar Filippov systems with two zones,” Nonlin. Dyn. 78, 251–263. Crossref, Web of Science, Google Scholar
Freire, E., Ponce, E. & Torres, F. [2015] “ On the critical crossing cycle bifurcation in planar Filippov systems,” J. Diff. Eqs. 259, 7086–7107. Crossref, Web of Science, Google Scholar
Gannakopoulos, F. & Pliete, K. [2001] “ Planar systems of piecewise linear differential equations with a line of discontinuity,” Nonlinearity 14, 1611–1632. Crossref, Web of Science, Google Scholar
Guardia, M., Seara, T. & Teixeira, M. A. [2011] “ Generic bifurcations of low codimension of planar Filippov systems,” J. Diff. Eqs. 250, 1967–2023. Crossref, Web of Science, Google Scholar
Han, M. & Zhang, W. [2010] “ On Hopf bifurcation in non-smooth planar systems,” J. Diff. Eqs. 248, 2399–2416. Crossref, Web of Science, Google Scholar
Huan, S. & Yang, X. [2013] “ Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics,” Nonlin. Anal. 92, 82–95. Crossref, Web of Science, Google Scholar
Huan, S. & Yang, X. [2014] “ On the number of limit cycles in general planar piecewise linear systems of node–node types,” J. Math. Anal. Appl. 411, 340–353. Crossref, Web of Science, Google Scholar
Kuznetsov, Yu. A., Rinaldi, S. & Gragnani, A. [2003] “ One-parametric bifurcations in planar Filippov systems,” Int. J. Bifurcation and Chaos 13, 2157–2188. Link, Web of Science, Google Scholar
Li, J. [2003] “ Hilbert’s 16th problem and bifurcation of planar polynomial vector fields,” Int. J. Bifurcation and Chaos 13, 47–106. Link, Web of Science, Google Scholar
Li, L. & Huang, L. [2014] “ Concurrent homoclinic bifurcation and Hopf bifurcation for a class of planar Filippov systems,” J. Math. Anal. Appl. 411, 83–94. Crossref, Web of Science, Google Scholar
Llibre, J., Ponce, E. & Torres, F. [2008] “ On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities,” Nonlinearity 21, 2121–2142. Crossref, Web of Science, Google Scholar
Pi, D. & Zhang, X. [2013] “ The sliding bifurcations in planar piecewise smooth differential systems,” J. Dyn. Diff. Eqs. 25, 1001–1026. Crossref, Web of Science, Google Scholar
Pi, D., Yu, J. & Zhang, X. [2013] “ On the sliding bifurcation of a class of planar Filippov systems,” Int. J. Bifurcation and Chaos 23, 1350040-1–18. Link, Web of Science, Google Scholar
Simpson, D. [2010] Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science, Vol. 69 (World Scientific, Singapore). Link, Google Scholar
Tang, S., Liang, J., Xiao, Y. & Robert, A. [2012] “ Sliding bifurcations of Filippov two stage pest control models with economic thresholds,” SIAM J. Appl. Math. 72, 1061–1080. Crossref, Web of Science, Google Scholar