Oscillatory Solutions in the Planar Restricted Three-Body Problem.
Kneading theory of Lorenz maps
Darbouxian integrability for polynomial vector fields on the 2-dimensional sphere.
Restricted three-body problems and the non-regularization of the elliptic collision restricted isosceles three-body problem.
Chordal cubic systems.
We classify the phase portraits of the cubic systems in the plane such that they do not have finite critical points, and the critical points on the equator of the Poincaré sphere are isolated and have linear part non-identically zero.
Homogeneous polynomial vector fields of degree 2 on the 2-dimensional sphere.
Let X be a homogeneous polynomial vector field of degree 2 on S having finitely many invariant circles. Then, we prove that each invariant circle is a great circle of S, and at most there are two invariant circles. We characterize the global phase portrait of these vector fields. Moreover, we show that if X has at least an invariant circle then it does not have limit cycles.
Hoptf bifurcation from infinity for planar control systems.
Symmetric piecewise linear bi-dimensional systems are very common in control engineering. They constitute a class of non-differentiable vector fields for which classical Hopf bifurcation theorems are not applicable. For such systems, sufficient and necessary conditions for bifurcation of a limit cycle from the periodic orbit at infinity are given.
On the number of fixed points for a continuous map of a finite connected graph.
The full periodicity kernel of the trefoil
We consider the following topological spaces: , , , , , et . Set . An map is a continuous self-map of having the branching point fixed. We denote by the set of periods of all periodic points of . The set is the of if it satisfies the following two conditions: (1) If is an map and , then . (2) If is a set such that for every map , implies , then . In this paper we compute the full periodicity kernel of and .
Results and open questions on some invariants measuring the dynamical complexity of a map
Let f be a continuous map on a compact connected Riemannian manifold M. There are several ways to measure the dynamical complexity of f and we discuss some of them. This survey contains some results and open questions about relationships between the topological entropy of f, the volume growth of f, the rate of growth of periodic points of f, some invariants related to exterior powers of the derivative of f, and several invariants measuring the topological complexity of f: the degree (for the case...
Periods of Morse-Smale diffeomorphisms of 𝕊²
The aim of this paper is to describe the set of periods of a Morse-Smale diffeomorphism of the two-dimensional sphere according to its homotopy class. The main tool for proving this is the Lefschetz fixed point theory.
Quadratic systems with a unique finite rest point.
We study phase portraits of quadratic systems with a unique finite singularity. We prove that there are 111 different phase portraits without limit cycles and that 13 of them are realizable with exactly one limit cycle. In order to finish completely our study two problems remain open: the realization of one topologically possible phase portrait, and to determine the exact number of limit cycles for a subclass of these systems.
Quadratic vector fields with a weak focus of third order.
We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle.
self-maps on closed manifolds with finitely many periodic points all of them hyperbolic
Let be a connected closed manifold and a self-map on . We say that is almost quasi-unipotent if every eigenvalue of the map (the induced map on the -th homology group of ) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of as eigenvalue of all the maps with odd is equal to the sum of the multiplicities of as eigenvalue of all the maps with even. We prove that if is having finitely many periodic points all of them hyperbolic,...
Polynomial systems: a lower bound for the weakened 16th Hilbert problem.
In this paper we provide the greatest lower bound about the number of (non-infinitesimal) limit cycles surrounding a unique singular point for a planar polynomial differential system of arbitrary degree.
A classification of braid types for periodic orbits of diffeomorphisms of surfaces of genus one with topological entropy zero.
We classify the braid types that can occur for finite unions of periodic orbits of diffeomorphisms of surfaces of genus one with zero topological entropy.
Abelian integrals of quadratic Hamiltonian vector fields with an invariant straight line.
We prove that the lowest upper bound for the number of isolated zeros of the Abelian integrals associated to quadratic Hamiltonian vector fields having a center and an invariant straight line after quadratic perturbations is one.
Dynamics of a Lotka-Volterra map
Given the plane triangle with vertices (0,0), (0,4) and (4,0) and the transformation F: (x,y) ↦ (x(4-x-y),xy) introduced by A. N. Sharkovskiĭ, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior...
Periods for holomorphic maps via Lefschetz numbers.
When singular points determine quadratic systems.
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