On the periodic solutions of a perturbed double pendulum
We provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations.
math.DS↗
arXiv subjects
Jaume Llibre
Publications and source records attributed to Jaume Llibre.
Periodic orbits for a class of galactic potentials
In this work, applying general results from averaging theory, we find periodic orbits for a class of Hamiltonian systems $H$ whose potential models the motion of elliptic galaxies.
math.CA↗
Analytic integrability of the Bianchi Class A cosmological models with 0<= k<1
There are many works studying the integrability of the Bianchi class A cosmologies with k=1. Here we characterize the analytic integrability of the Bianchi class A cosmological models when 0<= k<1.
math-ph↗
A Note on the First Integrals of Vector Fields with Integrating Factors and Normalizers
We prove a sufficient condition for the existence of explicit first integrals for vector fields which admit an integrating factor. This theorem recovers and extends previous results in the literature on the integrability of vector fields which are volume preserving and possess nontrivial normalizers. Our approach is geometric and coordinate-free and hence it works on any smooth orientable manifold.
math.DS↗
A restricted version of the Hilbert's 16th problem for quadratic vector fields
The restricted version of the Hilbert 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector fields $"σ$--distant from centers and $\ka $-distant from singular quadratic vector fields" provided that the limit cycles are $"δ$--distant from singular points and infinity".
math.DS↗
On the centers of the weight-homogeneous polynomial vector fields on the plane
We classify all the centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4.
math.DS↗
Phase portraits for quadratic homogeneous polynomial vector fields on S^2
Let X be a homogeneous polynomial vector field of degree 2 on S^2. We show that if X has at least a non--hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on S^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16^{th} Hilbert's problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on S^2 of degree n.
math.DS↗
On Mitropol'skii Yu.A.'s Theorem on Periodic Solutions of Systems of Nonlinear Differential Equations with Non-Differentiable Right-Hand-Sides
The smooth second Bogolyubov's theorem is generalized for Lipschitz systems.
math.CA↗
Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator
In this paper we study the existence, uniqueness and asymptotic stability of the periodic solutions for a Lipschitz system with a small right hand side. Classical hypotheses in the periodic case of second Bogolyubov's theorem imply our ones. By means of the results established we construct the curves of dependence of the amplitude of asymptotically stable $2π$--periodic solutions of the nonsmooth van der Pol oscillator on the detuning parameter and the amplitude of the perturbation. After, we compare the resonance curves obtained, with the resonance curves of the classical van der Pol oscillator which were first constructed by Andronov and Witt.
math.CA↗
Lines of Principal Curvature on Canal Surfaces in R^3
In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Riccati equations, it is established that canal surfaces have at most two isolated periodic principal lines. Examples of canal surfaces with two simple and one double periodic principal lines are given.
math.DG↗
The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems
In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré--Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincaré--Liapunov method.
math.DS↗
Periods, Lefschetz numbers and entropy for a class of maps on a bouquet of circles
We consider some smooth maps on a bouquet of circles. For these maps we can compute the number of fixed points, the existence of periodic points and an exact formula for topological entropy. We use Lefschetz fixed point theory and actions of our maps on both the fundamental group and the first homology group.
math.DS↗
Periods for holomorphic maps via Lefschetz numbers
We characterise the set of fixed points of a class of holomorphic maps on complex manifolds with a prescribed homology. Our main tool is the Lefschetz number and the action of maps on the first homology group.
math.DS↗
Planar cubic polynomial differential systems with the maximum number of invariant straight lines
We classify all cubic systems possessing the maximum number of invariant straight lines (real or complex) taking into account their multiplicities. We prove that there are exactly 23 topological different classes of such systems. For every class we provide the configuration of its invariant straight lines in the Poincare disc. Moreover, every class is characterized by a set of affine invariant conditions.
math.DS↗
Multiplicity of Invariant Algebraic Curves and Darboux Integrability
We define four different kinds of multiplicity of an invariant algebraic curve for a given polynomial vector field and investigate their relationships. After taking a closer look at the singularities and at the line of infinity, we improve the Darboux theory of integrability using these new notions of multiplicity.
math.DS↗