Abstract
Using first integrals we introduce an alternative tool to the Poincaré maps in order to study the limit cycles of piecewise linear differential systems. It is also shown that an usual relaxation oscillation phenomenon occurrs in a two-parameter family of planar piecewise linear vector fields.
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L. A. Cherkas andL. I. Zhilevich,Some criteria for the absence of limit cycles and for the existence of a single limit cycle, Diff. Equation6 (1970), 891–897.
C. Christopher andJ. Llibre,Integrability via invariant algebraic curves for planar polynomial differential equations, Annals of Differential Equations5 (2000), 5–19.
G. Darboux,Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math. 2ème série2 (1878), 60–96; 123–144; 151–200.
F. Dumortier and R. Roussarie,Canard cycles and center manifolds, Mem. Amer. Math. Soc.577, 1996.
E. Freire, E. Ponce, F. Rodrigo, andF. Torres,Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg.11, (1988), 2073–2097.
M. W. Hirsch, C. C. Pugh and M. Shub,Invariant manifolds, Lecture Notes in Mathematics583, Springer-Verlag, 1977.
J. Llibre, E. Ponge and X. Zhang,Existence of piecewise linear differential systems with exactlyn limit cycles for n ∈ ℕ submitted to J. Differential Equations.
J. Llibre and X. Zhang,Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces, To appear in Rendiconti del circolo matematico di Palermo.
L. Perko,Differential equations and dynamical systems (thrid edition), Springer-Verlag, New York, 2001.
E. Ponce,On the Double-Zero Unfolding in Symmetric Piecewise Linear Systems, preprint, 2001.
R. Roussarie,Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Birkhäuser Verlag, 1998.
A. Teruel,Clasificación topológica de una familia de campos vectoriales lineales a trozos simetricos en el plano, Ph. D. Thesis, Universitat Autònoma de Barcelona, 2000.
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Llibre, J., Nuñez, E. & Teruel, A.E. Limit cycles for planar piecewise linear differential systems via first integrals. Qual. Th. Dyn. Syst. 3, 29–50 (2002). https://doi.org/10.1007/BF02969332
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DOI: https://doi.org/10.1007/BF02969332