The discontinuous matching of two planar linear foci can have three nested crossing limit cycles

The existence and stability of limit cycles in discontinuous piecewise linear systems obtained by the aggregation of two linear systems of focus type and having only one equilibrium point is considered. By using an adequate canonical form with five parameters, a thorough study of some Poincar´e maps is performed. Different bifurcations which are responsible for the appearance of crossing limit cycles are detected and parameter regions with none, one, two and three crossing limit cycles are found. In particular, a first analytical proof of the existence, for certain differential systems in the considered family, of at least three homotopic crossing limit cycles surrounding the equilibrium point, is included. This fact has recently been numerically discovered in a particular example by S. -M. Huan and X. -S. Yang.

[1]  Enrique Ponce,et al.  Canonical Discontinuous Planar Piecewise Linear Systems , 2012, SIAM J. Appl. Dyn. Syst..

[2]  Wolf-Jürgen Beyn,et al.  Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin , 2006, J. Nonlinear Sci..

[3]  Xiao-Song Yang,et al.  On the number of limit cycles in general planar piecewise linear systems , 2012 .

[4]  Enrique Ponce,et al.  Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones , 1998 .

[5]  Fabio Dercole,et al.  Bifurcation analysis of piecewise smooth ecological models. , 2007, Theoretical population biology.

[6]  J. J. Stoker.  Nonlinear Vibrations in Mechanical and Electrical Systems , 1950 .

[7]  Fotios Giannakopoulos,et al.  Planar systems of piecewise linear differential equations with a line of discontinuity , 2001 .

[8]  Shuliang Shui,et al.  The qualitative analysis of a class of planar Filippov systems , 2010 .

[9]  S. E. Khaikin,et al.  Theory of Oscillators , 1966 .

[10]  Jaume Llibre,et al.  THREE NESTED LIMIT CYCLES IN DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH TWO ZONES , 2012 .

[11]  Fotios Giannakopoulos,et al.  Closed trajectories in planar relay feedback systems , 2002 .

[12]  Maoan Han,et al.  On Hopf bifurcation in non-smooth planar systems , 2010 .

[13]  Yuri A. Kuznetsov,et al.  One-Parameter bifurcations in Planar Filippov Systems , 2003, Int. J. Bifurc. Chaos.

[14]  Stephen John Hogan,et al.  Local Analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems , 1999 .

[15]  Jaume Llibre,et al.  On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities , 2008 .

[16]  Luis Fernando Mello,et al.  Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane , 2013 .

[17]  Mario di Bernardo,et al.  Piecewise smooth dynamical systems , 2008, Scholarpedia.

[18]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[19]  Armengol Gasull,et al.  Degenerate hopf bifurcations in discontinuous planar systems , 2001 .

[20]  C. Budd,et al.  Review of ”Piecewise-Smooth Dynamical Systems: Theory and Applications by M. di Bernardo, C. Budd, A. Champneys and P. 2008” , 2020 .