Crossing limit cycles for discontinuous piecewise linear differential centers...
AbstractIn this paper we study the continuous and discontinuous planar piecewise differential systems formed by four linear centers separated by three parallel straight lines denoted by \(\Sigma...
View ArticlePolynomial Liénard systems with a nilpotent global center
AbstractA center for a differential system \(\dot{\textbf{x}}=f(\textbf{x})\) in \({\mathbb {R}}^2\) is a singular point p having a neighborhood U such that \(U\setminus \{p\}\) is filled with periodic...
View ArticleOn the Connection Between Global Centers and Global Injectivity in the Plane
AbstractIn this note we revisit a result of Sabatini relating global injectivity of polynomial maps to global centers in the plane. We deliver a generalization of this result for \(C^2\) maps defined...
View ArticleOn the Integrability of a Four-Prototype Rössler System
AbstractWe consider a four-prototype Rossler system introduced by Otto Rössler among others as prototypes of the simplest autonomous differential equations (in the sense of minimal dimension, minimal...
View ArticleOn the 16th Hilbert Problem for Discontinuous Piecewise Polynomial...
AbstractIn this paper we study the maximum number of limit cycles of the discontinuous piecewise differential systems with two zones separated by the straight line \(y=0\), in \(y\ge 0\) there is a...
View ArticleLimit cycles of a continuous piecewise differential system formed by a...
AbstractThe study of limit cycles of planar differential systems is one of the main and difficult problems for understanding their dynamics. Thus the objective of this paper is to study the limit...
View ArticleTopological Classification of Some SD Hamiltonian Systems
AbstractIn this paper we classify the phase portraits in the Poincaré disk of the Smooth and Discontinuous (SD) Hamiltonian system with the rational Hamiltonian function \(H(x,y)=y^2/2+P(x)/Q(x,y)\),...
View ArticleThe discontinuous matching of two globally asymptotically stable crossing...
AbstractA differential system in the plane \({\mathbb {R}}^2\) is globally asymptotically stable if it has an equilibrium point p and all the other orbits of the system tend to p in forward time. In...
View ArticleLimit Cycles of Continuous Piecewise Smooth Differential Systems
AbstractFrom the beginning of this century many articles have been published on the continuous and discontinuous piecewise differential systems specially in the plane. The big interest on these...
View ArticleThe limit cycles of a class of piecewise differential systems
AbstractIn this century many papers have been published on the piecewise differential systems in the plane. The increasing interest for this class of differential systems is motivated by their many...
View ArticleThe Markus–Yamabe Conjecture for Discontinuous Piecewise Linear Differential...
AbstractIn 1960 Markus and Yamabe made the conjecture that if a \(C^1\) differential system \(\dot{x} = F(x)\) in \(\mathbb {R}^n\) has a unique equilibrium point and DF(x) is Hurwitz for all \(x \in...
View ArticleThe 16th Hilbert Problem for Discontinuous Piecewise Linear Differential...
AbstractWe consider planar piecewise discontinuous differential systems formed by either linear centers or linear Hamiltonian saddles and separated by the algebraic curve \(y=x^n\) with \(n \ge 2\). We...
View ArticleLimit cycles of polynomial differential systems of degree 1 and 2 on the...
AbstractWe consider planar polynomial differential systems of degree 1 and 2 on the cylinder and we study their limit cycles. We prove that such linear differential systems have at most one limit cycle...
View ArticleOn the Poincaré–Bendixson Formula for Planar Piecewise Smooth Vector Fields
AbstractThe topological index, or simply the index, of an equilibrium point of a vector field is an integer which saves important information about the local phase portrait of the equilibrium point....
View ArticleInvariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three
AbstractWe discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem...
View ArticlePhase Portraits of a Class of Continuous Piecewise Linear Differential Systems
AbstractThe 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....
View ArticleLimit Cycles for Discontinuous Piecewise Differential Systems in $$\mathbb...
AbstractIn planar piecewise differential systems it is known that when the discontinuity curve is a straight line and both differential systems are linear centers, these piecewise differential systems...
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