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Planar Filippov Systems with Maximal Crossing Set and Piecewise Linear Focus Dynamics

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Progress and Challenges in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 54))

Abstract

This paper includes some new results and a survey of known bifurcations for a family of Filippov systems. Such a family is constituted by planar piecewise linear systems with a discontinuity line where the crossing set is maximal and it has two dynamics of focus type. From the natural 12 parameters needed we obtain, under some generic conditions, a Liénard canonical form topologically equivalent to the original system with only four parameters. We describe, taking into account the number of equilibria inside each zone of linearity: zero, one or two, the qualitatively different phase portraits that can occur and the bifurcations connecting them.

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Acknowledgements

Authors are partially supported by the Ministerio de Ciencia y Tecnología, Plan Nacional \(I + D + I\), in the frame of projects MTM2009-07849, MTM2012-31821 and by the Consejería de Educación y Ciencia de la Junta de Andalucía under the gants TIC-0130 and P08-FQM-03770.

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Authors and Affiliations

  1. Departamento Matemática Aplicada II, E.T.S. Ingeniería, 41092, Sevilla, Spain

    Emilio Freire, Enrique Ponce & Francisco Torres

Authors
  1. Emilio Freire
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  2. Enrique Ponce
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  3. Francisco Torres
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Corresponding author

Correspondence to Emilio Freire .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Santiago Ibáñez

  2. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Jesús S. Pérez del Río

  3. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Antonio Pumariño

  4. Department of Mathematics, University of Oviedo, Oviedo, Spain

    J. Ángel Rodríguez

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Freire, E., Ponce, E., Torres, F. (2013). Planar Filippov Systems with Maximal Crossing Set and Piecewise Linear Focus Dynamics. In: Ibáñez, S., Pérez del Río, J., Pumariño, A., Rodríguez, J. (eds) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38830-9_13

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