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1.
58 p, 6.6 MB Phase Portraits of a Class of Continuous Piecewise Linear Differential Systems / Li, Jie (Southwest Petroleum University. School of Sciences) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
The phase portraits of the planar linear differential systems are very well known. This is not the case for the phase portraits of the planar continuous piecewise linear differential systems. In this paper we classify the phase portraits of the class of planar continuous piecewise linear differential systems of the form x˙=a|x|+by+c,y˙=α|x|+βy+γ, in the Poincaré disc when aβ- bα≠ 0, and prove the existence and uniqueness of limit cycles. [...]
2023 - 10.1007/s12591-023-00666-7
Differential Equations and Dynamical Systems, (November 2023)  
2.
Nonexistence and Uniqueness of Limit Cycles in a Class of Three-Dimensional Piecewise Linear Differential Systems / Chen, Ting (Guangdong University of Finance and Economics. School of Statistics and Mathematics) ; Huang, Lihong (Changsha University. College of Mathematics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
During the last twenty years there has been increasing interest in studying the piecewise differential systems, mainly due to their many applications in natural science and technology. Up to now the most studied differential systems are in dimension two, here we study them in dimension three. [...]
2023 - 10.1142/S021812742350075X
International journal of bifurcation and chaos in applied sciences and engineering, Vol. 33, Issue 6 (May 2023) , art. 2350075  
3.
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 - 10.1016/j.chaos.2022.112256
Chaos, solitons and fractals, Vol. 160 (July 2022) , art. 112256  
4.
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 - 10.1090/proc/15601
Proceedings of the American Mathematical Society, Vol. 149, Issue 12 (December 2021) , p. 5267-5274  
5.
18 p, 798.2 KB A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Zhang, X. (Shanghai Jiao Tong University. School of Mathematical Sciences and MOE-LSC (People's Republic of China))
In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in R and of the discontinuous piecewise differential systems in R formed by two linear differential systems separated by a straight line. [...]
2021 - 10.1016/j.exmath.2020.03.001
Expositiones Mathematicae, Vol. 39, Issue 1 (March 2021) , p. 48-61  
6.
15 p, 300.7 KB Piecewise linear differential systems with an algebraic line of separation / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Zhang, Xiang (Shanghai Jiao Tong University. Key Laboratory of Scientific and Engineering Computing (China))
We study the number of limit cycles of planar piecewise linear differential systems separated by a branch of an algebraic curve. We show that for each n ∈ N there exist piecewise linear differential systems separated by an algebraic curve of degree n having [n/2] hyperbolic limit cycles. [...]
2020
Electronic journal of differential equations, Vol. 2020, Issue 19 (2020) , p. 1-14
2 documents
7.
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 - 10.1007/s40863-020-00163-7
São Paulo Journal of Mathematical Sciences, Vol. 14, Issue 1 (June 2020) , p. 49-65  
8.
23 p, 764.3 KB Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields / Cardoso, Joäo L. (Universidade Federal de Goiás. Instituto de Matemática e Estatística (Brazil)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Novaes, Douglas D. (Universidade Estadual de Campinas. Departamento de Matematica (Brazil)) ; Tonon, Durval José (Universidade Federal de Goiás. Instituto de Matemática e Estatística (Brazil))
In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. [...]
2020 - 10.1080/14689367.2020.1722064
Dynamical Systems, vol. 35 3 (February 2020) p. 490-514  
9.
19 p, 420.8 KB Phase portraits of continuous piecewise linear Liénard differential systems with three zones / Li, Shimin (Guangdong University of Finance and Economics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Phase portraits are an invaluable tool in studying differential systems. Most of known results about global phase portraits are related to the smooth differential systems. This paper deals with a class of planar continuous piecewise linear Liénard differential systems with three zones separated by two vertical lines without symmetry. [...]
2019 - 10.1016/j.chaos.2018.12.037
Chaos, solitons and fractals, Vol. 120 (March 2019) , p. 149-157  
10.
22 p, 534.3 KB Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line / Li, Shimin (Guangdong University of Finance and Economics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. [...]
2019 - 10.1016/j.jde.2018.12.024
Journal of differential equations, Vol. 266, Issue 12 (June 2019) , p. 8094-8109  

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