Article
Mathematics, Applied
Maria Elisa Anacleto, Jaume Llibre, Claudia Valls, Claudio Vidal
Summary: This study demonstrates that discontinuous planar piecewise differential systems formed by linear centers can have at most three limit cycles, and presents examples with zero, one, two, or three limit cycles.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics, Interdisciplinary Applications
Jaume Llibre, Claudia Valls
Summary: This paper studies the maximum number of crossing limit cycles in a class of discontinuous piecewise differential systems composed of linear Hamiltonian saddles or linear centers, separated by a conic intersecting the conic in two points. The extended 16th Hilbert problem is solved for this type of systems.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Interdisciplinary Applications
Bilal Ghermoul, Jaume Llibre, Tayeb Salhi
Summary: First, we studied planar continuous piecewise differential systems separated by the straight line x = 0, which consist of a linear isochronous center in x > 0 and an isochronous quadratic center in x < 0. We proved that these systems cannot have crossing periodic orbits, and therefore do not have crossing limit cycles. Additionally, we examined another type of piecewise differential systems with crossing periodic orbits and limit cycles, where x > 0 has a general quadratic isochronous center and x < 0 has an arbitrary quadratic isochronous center. We found that the maximum number of crossing limit cycles for these systems is one, and provided examples of systems with one crossing limit cycle. Consequently, we have successfully solved the extension of the 16th Hilbert problem.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
Article
Mathematics, Applied
Joyce A. Casimiro, Jaume Llibre
Summary: Many articles have been published on continuous and discontinuous piecewise differential systems in the plane since the beginning of this century. The increasing number of applications for modeling natural phenomena has led to great interest in these systems. One major challenge in understanding the dynamics of planar differential systems is controlling their limit cycles. Most papers studying continuous piecewise differential systems have used straight lines as the separating boundary. However, this work considers continuous piecewise differential systems separated by a circle and consisting of one linear and one quadratic differential center. The goal is to determine the maximum number of limit cycles such a continuous piecewise differential system can exhibit.
RESULTS IN MATHEMATICS
(2023)
Article
Multidisciplinary Sciences
Jaume Llibre, Claudia Valls
Summary: This study focuses on the maximum number of limit cycles in planar piecewise differential systems formed by linear Hamiltonian saddles. The findings demonstrate that the number of limit cycles varies depending on the continuity and separation of the systems, with different scenarios leading to different outcomes.
Article
Mathematics, Interdisciplinary Applications
Bilal Ghermoul, Jaume Llibre, Tayeb Salhi
Summary: This study investigates the crossing periodic orbits and limit cycles in a planar piecewise-continuous differential systems separated by the straight-line x = 0. In the region x > 0, there is a general quadratic isochronous center x = -y + x(2), y = x(1 + y) after an affine transformation. In the region x < 0, there exists an arbitrary quadratic isochronous center, except for the quadratic isochronous center x = -y + x(2) - y(2), y = x(1 + 2y), which has been studied in [Ghermoul et al., 2021]. For these piecewise-continuous differential systems, the upper bound for crossing limit cycles is 2, and specific examples with one crossing limit cycle exist.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
Article
Mathematics, Applied
Jaume Llibre, Marco Antonio Teixeira
Summary: Many papers have been published in the past 20 years on piecewise differential systems in the plane, which have a wide range of applications in modeling natural phenomena. This paper focuses on studying the periodic orbits of a class of piecewise differential systems where the line of separation between the two systems is continuous in one part and discontinuous in the other. The results show that these continuous-discontinuous systems cannot have limit cycles but can have a continuum of periodic orbits.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Engineering, Mechanical
Meriem Barkat, Rebiha Benterki, Jaume Llibre
Summary: This paper investigates the limit cycles of a class of piecewise differential systems and proves that a maximum of 5 crossing limit cycles can exist in this family of systems with a linear center and a polynomial first integral of degree 2n.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Jaume Llibre, Tayeb Salhi
Summary: This paper studies the limit cycle quantity of the discontinuous piecewise differential systems formed by a linear differential system and a quadratic polynomial differential system separated by one straight line. Using up to seventh order averaging theory, it is proved that this system can have 8 limit cycles.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
Zhengkang Li, Xingbo Liu
Summary: This paper examines impact limit cycles in planar piecewise linear hybrid systems composed of center type vector fields and reset maps on impact surfaces. By utilizing Poincare map and first integral, the study provides an estimation of the maximum number of two-zone and three-zone impact limit cycles. It is shown that when the hybrid systems are separated by a single straight line, they can have at most one two-zone impact limit cycle, whereas when separated by two parallel straight lines, these systems can have at most two three-zone impact limit cycles.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Joao Medrado, Bruno Rodrigues de Freitas
Summary: In this study, a class of three-dimensional piecewise linear differential systems with two zones separated by a plane is considered. It is shown that such systems can exhibit unique limit cycles, one-parameter family of periodic orbits, scrolls, and invariant cylinders.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Interdisciplinary Applications
Johana Jimenez, Jaume Llibre, Claudia Valls
Summary: We study two families of piecewise linear Hamiltonian systems without equilibria in R-2 and determine the maximum number and positions of the crossing limit cycles for each family. This solves the extended 16th Hilbert problem for each family.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
Article
Engineering, Mechanical
Man Jia, Youfeng Su, Hebai Chen
Summary: This paper investigates the global dynamics of a continuous planar piecewise linear differential system with three zones and provides the global phase portraits and complete bifurcation diagram in the Poincare disc. It also demonstrates its application in a second-order memristor oscillator and validates the theoretical results through numerical phase portraits.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Zhengkang Li, Xingbo Liu
Summary: This paper studies the limit cycles in discontinuous piecewise linear planar systems without equilibria. It is proved that such systems can have at most two limit cycles, and these limit cycles must intersect the nonregular separation line in two or four points. The stability of various limit cycles is also proven based on Poincare map. Concrete examples are provided to illustrate the main results.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
Article
Mathematics, Applied
Zhengkang Li, Xingbo Liu
Summary: In this paper, the existence of limit cycles for discontinuous planar piecewise linear systems with three zones and two parallel straight lines is investigated. The maximum number of limit cycles for systems with different types of focus-center boundaries is presented, showing that these systems can have at most three limit cycles, with two of four intersection points type and one of two intersection points type.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Tao Li, Jaume Llibre
Summary: This paper studies the maximum number of limit cycles of discontinuous piecewise differential systems separated by a straight line. It provides upper bounds for the maximum number of limit cycles for these systems and shows that these bounds can be reached for certain parameter values.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Jaume Llibre, Claudia Valls
Summary: This paper investigates the Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems. The findings show that the conjecture holds true in R-2, but fails in R-n (n>=3). Additionally, for discontinuous systems, the conjecture is also false in R-R (n>=2).
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Jean -Marc Ginoux, Jaume Llibre
Summary: In this paper, a new approach to studying families of periodic orbits of a Hamiltonian system using the averaging theory is presented. The study involves computing a new family of periodic orbits of the extended Van der Pol oscillator, which has two degrees of freedom.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Engineering, Mechanical
Meriem Barkat, Rebiha Benterki, Jaume Llibre
Summary: This paper investigates the limit cycles of a class of piecewise differential systems and proves that a maximum of 5 crossing limit cycles can exist in this family of systems with a linear center and a polynomial first integral of degree 2n.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
Jaume Llibre, Claudia Valls
Summary: This article investigates the characteristics and phase portraits of generalized van der Pol systems. The conditions for the origin to be a center are characterized, and the global phase portraits are provided when f(y) = a1y + a2y2.
BULLETIN DES SCIENCES MATHEMATIQUES
(2023)
Article
Mathematics, Applied
Jaume Llibre
Summary: The article discusses the Painleve-Ince differential equation and provides its phase portrait in the Poincare disc.
RESULTS IN MATHEMATICS
(2023)
Article
Mathematics
Jie Li, Jaume Llibre
Summary: This paper classifies the phase portraits in the Poincare disc for the Lienard equation $x\ddot{x} +f(x)\dot{x} + g(x) = 0$, with four different cases: $f(x) = 0$ and $g(x)$ is a quadratic or cubic polynomial; $f(x)$ is a quadratic or cubic polynomial and $g(x) = 0$.
BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA
(2023)
Article
Mathematics, Applied
Francisco Braun, Jaume Llibre
Summary: In this article, a result of Sabatini regarding the relationship between global injectivity of polynomial maps and global centers in the plane is revisited. A generalization of this result for C-2 maps defined on connected sets is presented, taking into account the shape of the image without using Hadamard's invertibility theorem.
DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Jaume Llibre, Claudia Valls
Summary: In this study, the authors investigate the chaotic behavior of a four-prototype Rossler system, which is considered as a simple autonomous differential equation with chaotic behavior. The integrability of the system is analyzed from different perspectives, and it is shown that the system is neither Darboux integrable nor C-1-integrable.
MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY
(2023)
Article
Astronomy & Astrophysics
Elbaz I. I. Abouelmagd, Juan Luis Garcia Guirao, Jaume Llibre
Summary: This paper investigates the existence of periodic orbits under quantum corrections for perturbed third-body motion. Two approaches are used to analyze the first (second) types of periodic orbits. Conditions are provided to demonstrate that circular (elliptical) periodic orbits in the rotating Kepler problem (RKP) can persist in the perturbed motion of the third body under quantum corrections, where a massive primary body exerts excessive gravitational force over a smaller primary body. The primaries move in circular (elliptical) orbits around each other, with the assumption of a sufficiently small mass ratio. The existence of these two orbit types is proven using Poincare's terminology for quantized perturbed motion.
Article
Multidisciplinary Sciences
Rebiha Benterki, Jaume Llibre
Summary: This paper investigates the phase portraits of five classes of homogeneous Hamiltonian polynomial differential systems in the Poincare disc. These phase portraits are symmetric with respect to the origin of coordinates and exhibit 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincare disc.
Article
Mathematics, Applied
Laurent Cairo, Jaume Llibre
Summary: The quadratic polynomial differential systems in a plane are the simplest nonlinear differential systems and have been extensively studied due to their nonlinearity and wide range of applications. These systems can be classified into ten classes, and this article provides all topologically different phase portraits in the Poincare disc for two of these classes.
Article
Mathematics, Interdisciplinary Applications
J. Llibre, C. Pantazi
Summary: Any singular irreducible cubic curve can be written as y(2) = x(3), y(2) = x(2)(x + 1), or y(2) = x(2)(x - 1) after an affine transformation. We classify the phase portraits of quadratic polynomial differential systems with the invariant cubic y(2) = x(2)(x + 1). There are 63 different topological phase portraits for such systems and we analyze the bifurcations between them. The phase portraits have no limit cycles and range from having a center to multiple polycycles. The maximum number of separartices and canonical regions vary among the phase portraits.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2023)
Article
Mathematics, Applied
Jaume Llibre
Summary: In this study, we consider planar continuous piecewise differential systems separated by a parabola. We prove that the systems formed by a linear center and a quadratic center have at most one limit cycle, and we find systems in this class exhibiting one limit cycle.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2023)
Article
Mathematics, Applied
Jaume Llibre
Summary: Due to their applications, the study of continuous or discontinuous piecewise differential systems has gained significant interest. Limit cycles are crucial in analyzing planar differential systems. Previous studies primarily focused on systems separated by a straight line, but this research considers planar continuous piecewise differential systems separated by a parabola. The study proves that systems with a linear center and a quadratic center separated by a parabola can have at most one limit cycle, and there are systems in this class that exhibit one limit cycle. Thus, this research solves the extension of the 16th Hilbert problem for this class of differential systems.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2023)
