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14 p, 333.2 KB

Limit cycles of the discontinuous piecewise differential systems on the cylinder / Li, Jie (Sichuan University. Department of Mathematics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In order to understand the dynamics of the differential systems the limit cycles play a main role, but in general their study is not easy. These last years an increasing interest appeared for studying the limit cycles of some classes of discontinuous piecewise differential systems, due to the rich applications of this kind of differential systems. [...]
2022 - DOI (Digital Object Identifier)

10.11948/20220031
Journal of Applied Analysis and Computation, Vol. 12, Issue 3 (June 2022) , p. 952-963

2 documents

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On the limit cycles of the piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Salhi, Tayeb (University Mohamed El Bachir El Ibrahimi. Department of Mathematics (Algeria))
While the limit cycles of the discontinuous piecewise differential systems formed by two linear differential systems separated by one straight line have been studied intensively, and up to now there are examples of these systems with at most 3 limit cycles. [...]
2022 - DOI (Digital Object Identifier)

10.1016/j.chaos.2022.112256
Chaos, solitons and fractals, Vol. 160 (July 2022) , art. 112256

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10 p, 280.5 KB

The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Zhang, Xiang (Shanghai Jiao Tong University. School of Mathematical Sciences)
In 1960 Markus and Yamabe made the following conjecture: If a C1 differential system x˙ = F(x) in Rn has a unique equilibrium point and the Jacobian matrix of F(x) for all x ∈ Rn has all its eigenvalues with negative real part, then the equilibrium point is a global attractor. [...]
2021 - DOI (Digital Object Identifier)

10.1090/proc/15601
Proceedings of the American Mathematical Society, Vol. 149, Issue 12 (December 2021) , p. 5267-5274

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16 p, 666.6 KB

Limit cycles for two classes of control piecewise linear differential systems / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Oliveira, Regilene (Universidade de São Paulo. Instituto De Ciências Matemáticas e de Computação. Departamento de Matemática (Brazil)) ; Rodrigues, Camila A. B. (Universidade Federal de Santa Catarina. Departamento de Matemática (Brazil))
We study the bifurcation of limit cycles from the periodic orbits of 2n-dimensional linear centers x˙ = A0x when they are perturbed inside classes of continuous and discontinuous piecewise linear differential systems of control theory of the form x˙ = A0x+ ε(Ax+ ϕ(x) b), where ϕ is a continuous or discontinuous piecewise linear function, A0 is a 2n×2n matrix with only purely imaginary eigenvalues, ε is a small parameter, A is an arbitrary 2n×2n matrix, and b is an arbitrary vector of Rn.
2020 - DOI (Digital Object Identifier)

10.1007/s40863-020-00163-7
São Paulo Journal of Mathematical Sciences, Vol. 14, Issue 1 (June 2020) , p. 49-65

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17 p, 408.6 KB

Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Tang, Yilei (Shanghai Jiao Tong University. School of Mathematical Sciences)
We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line. [...]
2019 - DOI (Digital Object Identifier)

10.3934/dcdsb.2018236
Discrete and continuous dynamical systems. Series B, Vol. 24, Issue 4 (April 2019) , p. 1769-1784

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16 p, 360.0 KB

On the periodic solutions of the Milchelson continuous and discontinuous piecewise linear differential system / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Oliveira, Regilene (Universidade de Sâo Paulo (Brazil). Departamento de Matemática) ; Rodrigues, Camila A. B. (Universidade de São Paulo (Brasil). Departamento de Matemática)
Applying new results from the averaging theory for continuous and discontinuous differential systems, we study the periodic solutions of two distinct versions of the Michel- son differential system: a Michelson continuous piecewise linear differential system and a Michelson discontinuous piecewise linear differential system. [...]
2018 - DOI (Digital Object Identifier)

10.1007/s40314-016-0413-x
Computational & Applied Mathematics, Vol. 37, issue 2 (May 2018) , p. 1550-1561

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14 p, 356.4 KB

Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Zhang, Xiang (Shanghai Jiao Tong University. Department of Mathematics)
From the beginning of this century more than thirty papers have been published studying the limit cycles of the discontinuous piecewise linear differential systems with two pieces separated by a straight line, but it remains open the following question: what is the maximum number of limit cycles that this class of differential systems can have? Here we prove that when one of the linear differential systems has a center, real or virtual, then the discontinuous piecewise linear differential system has at most two limit cycles.
2018 - DOI (Digital Object Identifier)

10.1016/j.jmaa.2018.07.024
Journal of mathematical analysis and applications, Vol. 467, issue 1 (Nov. 2018) , p. 537-549

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16 p, 3.3 MB

The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems / Castillo, Juan (Universidad de Sonora (Mèxic). Departamento de Matemáticas) ; Verduzco, Fernando (Universidad de Sonora (Mèxic). Departamento de Matemáticas) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
The creation or destruction of a crossing limit cycle when a sliding segment changes its stability, is known as pseudo-Hopf bifurcation. In this paper, under generic conditions, we find an unfolding for such bifurcation, and we prove the existence and uniqueness of a crossing limit cycle for this family.
2017 - DOI (Digital Object Identifier)

10.1007/s11071-017-3766-9
Nonlinear dynamics, Vol. 90 (2017) , p. 1829-1840

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13 p, 346.6 KB

Piecewise linear differential systems without equilibria produce limit cycles? / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Teixeira, Marco Antonio (Universidade Estadual de Campinas (Brasil). Departamento de Matemática)
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.
2017 - DOI (Digital Object Identifier)

10.1007/s11071-016-3236-9
Nonlinear dynamics, Vol. 88 (2017) , p. 157-164

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