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Periodic Orbits for Planar Piecewise Smooth Systems with a Line of Discontinuity

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Abstract

In this work we examine the existence of periodic orbits for planar piecewise smooth dynamical systems with a line of discontinuity. Unlike existing works, we consider the case where the line does not contain the equilibrium point. Most of the analysis is for a family of piecewise linear systems, and we discover new phenomena which produce the birth of periodic orbits, as well as new bifurcation phenomena of the periodic orbits themselves. A model nonlinear piecewise smooth systems is examined as well.

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References

  1. Dieci, L., Lopez, L.: Fundamental matrix solutions of piecewise smooth differential systems. Math. Comput. Simul. 81, 932–953 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Du, Z., Li, Y., Zhang, W.: Bifurcation of periodic orbits in a class of planar Filippov systems. Nonlinear Anal. 69(10), 3610–3628 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and Its Applications. Kluwer, Dordrecht (1988)

    Book  Google Scholar 

  4. Guardia, M., Seara, T.M., Teixeira, M.A.: Generic bifurcations of low codimension of planar filippov systems. J. Differ. Equ. 250(4), 1967–2023 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Küpper, T., Moritz, S.: Generalized Hopf bifurcation for non-smooth planar systems. R. Soc. Lond. Philos. Trans. Ser. A 359(1789), 2483–2496 (2001). Non-smooth mechanics

    Article  MATH  MathSciNet  Google Scholar 

  6. Kuznetsov, Y.A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13(8), 2157–2188 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Leine, R.I.: Bifurcations in Discontinuous Mechanical Systems of Filippov’s type. PhD thesis, Techn. Univ. Eindhoven, The Netherlands (2000)

  8. Leine, R.I.: Bifurcations of equilibria in mechanical systems. Physica D 223, 121–137 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mueller, P.C.: Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos, Solitons Fractals 5, 167–1681 (1995)

    Google Scholar 

  10. Pi, D., Yu, J.: On the sliding bifurcation of a class of planar Filippov systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 23(3), 1350040 (2013)

    Article  MathSciNet  Google Scholar 

  11. Pi, D., Zhang, X.: The sliding bifurcations in planar piecewise smooth differential systems. J. Dyn. Differ. Equ. 25(4), 1001–1026 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zou, Y., Kuepper, T., Beyn, W.-J.: Generalized hopf bifurcation for planar Filippov systems continuous at the origin. J. Nonlinear Sci. 16, 159–177 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work was perfomed while the second author was on leave from the University of Bari, Bari, Italy, and in visit to the School of Mathematics of the Georgia Institute of Technology, whose support is gratefully acknowledged. The first author is grateful to the University of Jilin, Changchun, China, where he spent part of the Summer 2013 as Tang Aoqing Professor.

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  1. School of Mathematics, Georgia Tech, Atlanta, GA , 30332, USA

    L. Dieci & C. Elia

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  1. L. Dieci
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  2. C. Elia
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Correspondence to C. Elia.

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Dieci, L., Elia, C. Periodic Orbits for Planar Piecewise Smooth Systems with a Line of Discontinuity. J Dyn Diff Equat 26, 1049–1078 (2014). https://doi.org/10.1007/s10884-014-9380-3

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  • DOI: https://doi.org/10.1007/s10884-014-9380-3

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