Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

Freire, Emilio; Ponce, Enrique; Rodrigo, F.; Torres, Francisco

doi:10.1142/s0218127498001728

Cited by 255 publications

(213 citation statements)

References 7 publications

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“…Moreover in all cases that the origin is surrounded by a limit cycle, this is unique, stable if T C > 0 and unstable if T C < 0. Theorems 1 and 2 improve and extend cases studied in [9]. Here we give a shorter and clear proof using the techniques developed in [25].…”

Section: Theoremmentioning

confidence: 56%

Uniqueness and Non-uniqueness of Limit Cycles for Piecewise Linear Differential Systems with Three Zones and No Symmetry

Llibre

Ponce²,

Valls

2015

J Nonlinear Sci

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Abstract. Some techniques for proving the existence and uniqueness of limit cycles for smooth differential systems, are extended to continuous piecewiselinear differential systems. Then we obtain new results on the existence and uniqueness of limit cycles for systems with three linearity zones and without symmetry. We also reprove existing results of systems with two linearity zones giving shorter and clearer proofs.

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Section: Theoremmentioning

confidence: 56%

Uniqueness and Non-uniqueness of Limit Cycles for Piecewise Linear Differential Systems with Three Zones and No Symmetry

Llibre

Ponce²,

Valls

2015

J Nonlinear Sci

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“…We note that for these apparently simple systems, when they are continuous, some serious work is necessary for proving that they have at most one limit cycle, see [7] and [21]. This solved the conjecture of Lum and Chua [25] done in 1990 that such continuous differential systems can have at most one limit cycle.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 73%

Piecewise linear differential systems without equilibria produce limit cycles?

Llibre

Teixeira

2016

Nonlinear Dyn

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Abstract. In this article we study the planar piecewise differential systems formed by two linear differential systems separated by a straight line, such that both linear differential have no equilibria, neither real nor virtual.When the piecewise differential system is continuous, we show that the system has no limit cycles. But when the piecewise differential system is discontinuous, we show that it can have at most one limit cycle.

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“…In 1990, Lum and Chua conjectured that a continuous piecewise linear vector field in the plane with two zones has at most one limit cycle, see [14]. In 1998 this conjecture was proved by Freire, Ponce, Rodrigo and Torres in [8].…”

Section: Introductionmentioning

confidence: 99%

Center boundaries for planar piecewise-smooth differential equations with two zones

Buzzi

Pazim

Pérez-González

2017

Journal of Mathematical Analysis and Applications

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This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary. We prove that given a pair of systems that share a hyperbolic focus singularity p 0 , with the same orientation and opposite stability, and a ray Σ 0 with endpoint at the singularity p 0 , we can find a smooth manifold Ω such that Σ 0 ∪ {p 0 } ∪ Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached.

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Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

Cited by 255 publications

References 7 publications

Uniqueness and Non-uniqueness of Limit Cycles for Piecewise Linear Differential Systems with Three Zones and No Symmetry

Uniqueness and Non-uniqueness of Limit Cycles for Piecewise Linear Differential Systems with Three Zones and No Symmetry

Piecewise linear differential systems without equilibria produce limit cycles?

Center boundaries for planar piecewise-smooth differential equations with two zones

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