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

Emilio Freire, Enrique Ponce, and Francisco Torres

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Abstract

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 Poincarç 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.

Article information

Source
Publ. Mat., Volume EXTRA (2014), 221-253.

Dates
First available in Project Euclid: 19 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.pm/1400505235

Mathematical Reviews number (MathSciNet)
MR3211836

Zentralblatt MATH identifier
1343.34077

Subjects
Primary: 34C05: Location of integral curves, singular points, limit cycles 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 37G15: Bifurcations of limit cycles and periodic orbits

Keywords
Discontinuous piecewise linear systems limit cycles bifurcations

Citation

Freire, Emilio; Ponce, Enrique; Torres, Francisco. The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. Publ. Mat. EXTRA (2014), 221--253. https://projecteuclid.org/euclid.pm/1400505235


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