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Chordal cubic systems.

Marc CarbonellJaume Llibre — 1989

Publicacions Matemàtiques

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.

Jaume LlibreClaudio Pessoa — 2006

Extracta Mathematicae

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.

Jaume LlibreEnrique Ponce — 1997

Publicacions Matemàtiques

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.

The full periodicity kernel of the trefoil

Carme LeseduarteJaume Llibre — 1996

Annales de l'institut Fourier

We consider the following topological spaces: O = { z : | z + i | = 1 } , O 3 = O { z : z 4 [ 0 , 1 ] , Im z 0 } , O 4 = O { z : z 4 [ 0 , 1 ] } , 1 = O : | z - i | = 1 } { z : z [ 0 , 1 ] } , 2 = 1 { z : z 2 [ 0 , 1 ] } , et T = { z : z = cos ( 3 θ ) e i θ , 0 θ 2 π } . Set E { O 3 , O 4 , 1 , 2 , T } . An E map f is a continuous self-map of E having the branching point fixed. We denote by Per ( f ) the set of periods of all periodic points of f . The set K is the of E if it satisfies the following two conditions: (1) If f is an E map and K Per ( f ) , then Per ( f ) = . (2) If S is a set such that for every E map f , S Per ( f ) implies Per ( f ) = , then K S . In this paper we compute the full periodicity kernel of O 3 , O 4 , 1 , 2 and T .

Results and open questions on some invariants measuring the dynamical complexity of a map

Jaume LlibreRadu Saghin — 2009

Fundamenta Mathematicae

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...

Quadratic systems with a unique finite rest point.

Bartomeu CollArmengol GasullJaume Llibre — 1988

Publicacions Matemàtiques

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.

Joan C. ArtésJaume Llibre — 1997

Publicacions Matemàtiques

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.

C 1 self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

Jaume LlibreVíctor F. Sirvent — 2016

Mathematica Bohemica

Let X be a connected closed manifold and f a self-map on X . We say that f is almost quasi-unipotent if every eigenvalue λ of the map f * k (the induced map on the k -th homology group of X ) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of λ as eigenvalue of all the maps f * k with k odd is equal to the sum of the multiplicities of λ as eigenvalue of all the maps f * k with k even. We prove that if f is C 1 having finitely many periodic points all of them hyperbolic,...

Dynamics of a Lotka-Volterra map

Francisco BalibreaJuan Luis García GuiraoMarek LampartJaume Llibre — 2006

Fundamenta Mathematicae

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...

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