Planar Filippov Systems with Maximal Crossing Set and Piecewise Linear Focus Dynamics
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.
KeywordsPeriodic Orbit Equilibrium Point Hopf Bifurcation Canonical Form Boundary Equilibrium
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.
- 5.Freire, E., Ponce, E., Torres, F.: The discontinuous matching of two planar foci can have three nested limit cycles. Publications Matemàtiques, Volume EXTRA, (2013, to appear). Proceedings of the Conference “New Trends in Dynamical Systems” held in Salou (Tarragona), Spain, 1–5 Oct 2012Google Scholar
- 8.Iwatani, Y., Hara, S.: In: Stability Analysis and Stabilization for Bimodal Piecewise Linear Systems Based on Eigenvalue Loci. Mathematical Engineering Technical Reports (2004). Available on line at http://www.keisu.t.u-tokyo.ac.jp/research/techrep/data/2004/METR04-34.pdf
- 10.Ponce, E., Ros, J., Vela, E.: The focus-center limit cycle bifurcation in planar discontinuous piecewise linear systems without sliding. In: Ibáñez, S. et al. (eds) Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics, vol. 54 (this volume)Google Scholar