•The Abel differential equation has been applied to modelize problems from ecology, control theory, electrical circuits, cosmology,...•The interest for understanding the dynamics of this equation is ...proved by the hundreds of papers dedicated to it in MathSciNet.•In this paper we characterize the phase portraits of the complex Abel polynomial differential equation z′=(z−a)(z−b)(z−c).•The real version of this equation depends on six parameters and consequently the classification of their phase portraits needs some work.
In this paper we characterize the phase portraits of the complex Abel polynomial differential equationsz˙=(z−a)(z−b)(z−c),with z∈C, a,b,c∈C. We give the complete description of their topological phase portraits in the Poincaré disc, i.e. in the compactification of R2 adding the circle S1 of the infinity.
Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we ...have two objectives.
First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, …
Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincaré compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincaré disc for the separable quadratic polynomial differential systems.
In this paper, we complete the classification on global topological structures of the three-dimensional cooperative Lotka-Volterra system with the identical intrinsic growth rate inside the Poincaré ...compactification of the positive octant of R3. Precisely, with the help of the replicator equations it is proved that this kind of system can have exactly 8 topologically different phase portraits. As a consequence, we obtain the necessary and sufficient conditions for the system to be bounded in the positive octant.
We describe the global dynamics in the Poincaré disc of the Higgins–Selkov modelx′=k0−k1xy2,y′=−k2y+k1xy2,where k0, k1, k2 are positive parameters, and of the Selkov ...modelx′=−x+ay+x2y,y′=b−ay−x2y,where a, b are positive parameters. We determine the regions of initial conditions with biological meaning.
Kukles Systems of Degree Three with Global Centers Dias, Fabio Scalco; Mello, Luis Fernando; Valls, Claudia
Boletim da Sociedade Brasileira de Matemática,
12/2023, Letnik:
54, Številka:
4
Journal Article
Recenzirano
A global center of a vector field in the plane is an equilibrium point such that the whole plane with the exception of the equilibrium point is filled with periodic orbits. We classify all Kukles ...systems of degree three that have global centers.
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