Claudio Vidal

• University of Bío-Bío

We consider time-periodic Hamiltonians of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(t, {\textbf {Q}}, {\textbf {P}}, \epsilon )$$\end{document} where \...

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is...

The aim of this work is to prove analytically the existence of symmetric periodic solutions of the family of Hamiltonian systems with Hamiltonian function H(q1,q2,p1,p2)=12(q12+p12)+12(q22+p22)+aq14+bq12q22+cq24$$ H\left({q}_1,{q}_2,{p}_1,{p}_2\right)=\frac{1}{2}\left({q}_1^2+{p}_1^2\right)+\frac{1}{2}\left({q}_2^2+{p}_2^2\right)+a\kern0.1em {q}_1^...

We consider the motion of an infinitesimal mass under the Newtonian attraction of N point masses forming a ring plus a central body where a Manev potential ( $$-1/r + e/r^2$$ - 1 / r + e / r 2 , $$e \in {\mathbb {R}}$$ e ∈ R ), is applied to the central body. More precisely, the bodies are arranged in a planar ring configuration. This configuration...

Central configurations of the n-body problem play an important role in the study of celestial mechanics. In this paper, we study six-body planar central configurations having certain symmetries. Special attention is given to the existence results of such configurations. With analytic proofs, we show the existence of a new central configuration, whi...

In 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 Σ={(x,y)∈R2:x=-p,x=0,x=q,p,q>0}. We prove that when these piecewise differential systems are continuous they have no limit cycles. While for the discontinuous case we show...

We consider the planar double pendulum where its center of mass is attached in an elliptic orbit. We consider the case where the rods of the pendulum have variable length, varying according to the radius vector of the elliptic orbit. We make an Hamiltonian view of the problem, find four linearly stable equilibrium positions and construct the bounda...

We study convexity and symmetry of central configurations in the five body problem when three of the masses ara located at the vertices of an equilateral triangle, that we call Lagrange plus two central configurations. First, we prove that the two bodies out of the vertices of the triangle cannot be placed on certain lines. Next, we give a geometri...

In this paper, we study two classes of symmetric central configurations in the 5-vortex problem, namely the case with three vortices located at the vertices of an equilateral triangle and two vortices located symmetrically with respect to a perpendicular bisector of the triangle and the case with three vortices located at the vertices of an equilat...

The restricted elliptic isosceles three body problem (REI3BP) models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. The primaries of masses m1=m2 move along a degenerate Keplerian elliptic collision orbit (on a line) under their gravitational attraction, wherea...

We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or...

In the spatial Maxwell restricted \(N+1\)-body problem, the motion of an infinitesimal particle attracted by the gravitational field of (N) bodies is studied. These bodies are arranged in a planar ring configuration. This configuration consists of \(N-1\) primaries of equal mass m located at the vertices of a regular polygon that is rotating on its...

We consider the 2-body problem in the sphere \(\mathbb{S}^{2}\). This problem is modeled by a Hamiltonian system with \(4\) degrees of freedom and, following the approach given in [4], allows us to reduce the study to a system of \(2\) degrees of freedom. In this work we will use theoretical tools such as normal forms and some nonlinear stability r...

We provide canonical forms for the homogeneous polynomials of degree five. Then we characterize all the phase portraits in the Poincaré disk for all quartic homogeneous polynomial differential systems. More precisely, there are exactly 23 different topological phase portraits for the quartic homogeneous polynomial differential systems.

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solution...

In the framework of nonlinear stability of elliptic equilibria in Hamiltonian systems with \begin{document}$ n $\end{document} degrees of freedom we provide a criterion to obtain a type of formal stability, called Lie stability. Our result generalises previous approaches, as exponential stability in the sense of Nekhoroshev (excepting a few situati...

The elliptic isosceles restricted three body problem (REI3BP) models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. The primaries of masses $m_{1}=m_{2}$ move along a degenerate Keplerian elliptic collision orbit (on a line) under their gravitational attraction...

The dynamics of a test particle (a particle with zero vorticity) advected by the velocity field of 4 point-vortices forming a rhomboidal configuration with vorticities \(\Gamma _1= \Gamma _2= 1\) and \(\Gamma _3= \Gamma _4= \frac{b^2(3- b^2)}{3 b^2-1}\), is considered. We have two types of rhomboidal configuration: Rhombi type A if \(b \in [1,\sqrt...

We characterize the type stability of the five-vortex problem in the plane, where it is assumed that configuration is a rhombus vortex problem with a central vortex. More precisely, we suppose that the opposite vortices have the same vorticity Γ1=Γ2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usep...

We consider the dynamics of a passive tracer, advected by the presence of a latitudinal ring of identical point vortices. The corresponding instantaneous motion is modeled by a one degree of freedom Hamiltonian system. Such a dynamics presents a rich variety of behaviors with respect to the number of vortices, N, and the ring’s co-latitude, θo—or,...

In this work we examine trapezoidal central configurations in the planar four-vortex problem. More specifically, we consider the convex central configurations in which the convex quadrilateral has two parallel sides. With analytical arguments we classify all possible arrangements. Additionally, we prove the uniqueness of the trapezoidal central con...

The well-known problem of the nonlinear stability of L4 and L5 in the circular spatial restricted three-body problem is revisited. Some new results in the light of the concept of Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates for three specific values of the mass ratio that remained uncovered. Mor...

In this paper, a delayed with Holling type II functional response (Beddington‐DeAngelis) and Allee effect predator‐prey model is considered. The growth of the prey is affected by the parameter M, which defines the Allee effect. In addition, the delay τ also influences the logistic growth of the prey, which can be interpreted as the maturity time or...

In this work, we consider a generic 2-DOF autonomous Hamiltonian system, which linear part has all eigenvalues equal to zero. For these type of Hamiltonian, we provide, in an explicit and constructive way, the normal form up to the term of the fourth order. Additionally, we analyze the stability of the normalized Hamiltonian in several cases. Preci...

In this work we address the local behaviour at collision of the Kepler problem on surfaces of constant curvature. A full description of dynamics in the extended energy surface is given relying in the energy and mass values. Our analysis reveals remarkable differences with the classical Newtonian case. In particular, the collision surface does not c...

In this article we study the planar 4-body problem under homogeneous power-law potentials where the interaction between the bodies is given by r^{−a} , a>=4/3 (the Newtonian case corresponding to a = 3 and the vortex problem corresponding to a = 2). We study convex central configurations assuming two pairs of positive equal masses located at two ad...

LIMIT CYCLES OF A PERTURBATION OF A POLYNOMIAL HAMILTONIAN SYSTEMS OF DEGREE 4 SYMMETRIC WITH RESPECT TO THE ORIGIN - JAUME LLIBRE, PAULINA MARTÍNEZ, CLAUDIO VIDAL

A family of perturbed Hamiltonians in 1: −1:1 resonance depending on two real parameters is considered. We show the existence and stability of periodic solutions using reduction and averaging. In fact, there are at most thirteen families for every energy level h < 0 and at most twenty six families for every h > 0. The different types of periodic so...

We consider the Hamiltonian function defined by the cubic polynomial \(H= \frac{1}{2}(y_1^2+ y_2^2)+ V(x_1, x_2)\) where the potential \(V(x)= \delta V_2(x_1, x_2)+ V_3(x_1, x_2)\), with \(V_2(x_1, x_2)=\frac{1}{2}(x_1^2+x_2^2)\) and \( V_3(x_1, x_2)= \frac{1}{3} x_1^3+ f x_1 x_2^2+ g x_2^3\), with f and g are real parameters such that \(f \ne 0\)...

The aim of this paper is to analyze general dynamics features of a new Intraguild Predation (IGP) model where the top predator feeds only on the mesopredator and also affects its consumption rate. Important dynamical aspects of the model are described. Specifically, we prove that the trajectories of the associated system are bounded and defined for...

In this paper, we prove the instability of one equilibrium point in a Hamiltonian system with n-degrees of freedom under two assumptions: the first is the existence of multiple resonance of odd order s (without resonance of lower order) but with the possible existence of resonance of higher order; and the second is the existence of an invariant ray...

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and it...

We apply the averaging theory for proving the existence of twelve families of periodic orbits in a three-dimensional galactic Hamiltonian system.

The problem of attraction of an infinitesimal particle by the gravitational force induced by a massive body composed of a ring (in the form of a circle) of radius a and a punctual body at the center of the ring is considered. We study the singularity problem of the solutions associated with this problem, and we prove that all the singularities are...

We study the phase portraits on the Poincaré disc for all the linear type centers of polynomial Hamiltonian systems of degree 5 with Hamiltonian function H(x, y) = H1(x) + H2(y), where H1(x) = 1/2 x² + a3/3 x³ + a4/4 x⁴ + a5/5 x⁵ and H2(y) = 1/2 y² + b3/3 y³ + b4/4 y⁴ + b5/5 y⁵ as function of the six real parameters a3, a4, a5, b3, b4 and b5 with a...

We consider a restricted three body problem on surfaces of constant curvature. As in the classical Newtonian case the collision singularities occur when the position particle with infinitesimal mass coincides with the position of one of the primaries. We prove that the singularities due to collision can be locally (each one separately) and globally...

The existence and stability of periodic solutions for different types of perturbations associated to the Charged Restricted Circular Three Body Problem (shortly, CHRCTBP) is tackled using reduction and averaging theories as well as the technique of continuation of Poincaré for the study of symmetric periodic solutions. The determination of KAM 2-to...

In this paper we prove the instability of one equilibrium point in an autonomous Hamiltonian system with n-degrees of freedom under two assumptions: the first is the existence of a single resonance of order s (without resonance of lower order, but it could exist resonance of greater order); and the second is the existence of an invariant ray soluti...

The equations of motion of the Buckingham system are the ones of a two-body problem defined by the Hamiltonian $$ H= \frac{1}{2} \bigl(p_{x}^{2}+p_{y}^{2} \bigr) + A e^{-B \sqrt{x ^{2}+y^{2}}}-\frac{M}{ (x^{2}+y^{2} )^{3}}, $$ where \(A\), \(B\) and \(M\) are positive constants. The angular momentum \(p_{\theta }:= x p_{y}- y p_{x}\) and this Hamil...

We consider an integrable non-Hamiltonian system, which belongs to the quadratic Kukles differential systems. It has a center surrounded by a bounded period annulus. We study polynomial perturbations of such a Kukles system inside the Kukles family. We apply averaging theory to study the limit cycles that bifurcate from the period annulus and from...

We study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, per...

We consider a symmetric restricted three-body problem on surfaces Mκ² of constant Gaussian curvature κ≠0, which can be reduced to the cases κ=±1. This problem consists in the analysis of the dynamics of an infinitesimal mass particle attracted by two primaries of identical masses describing elliptic relative equilibria of the two body problem on Mκ...

We provide the phase portraits in the Poincare disk for all the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis given by the Hamiltonian function H(x, y) = 1/2 (x² + y²) + ax⁴y + bx²y³ + cy⁵ in function of its parameters.

In this paper we consider a symmetric restricted circular three-body problem on the surface S² of constant Gaussian curvature κ = 1. This problem consists in the description of the dynamics of an infinitesimal mass particle attracted by two primaries with identical masses, rotating with constant angular velocity in a fixed parallel of radius a ∈ (0...

In this paper we give new results for the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom. Our Main Theorem generalizes several results existing in the literature and mainly we give information in the critical cases (i.e., the condition of stabil...

This paper concerns with the study of the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with 1-degree of freedom in the degenerate case H = q⁴ + H5 + H6 +.... Our main results complement the study initiated by Markeev in [9].

We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential (−1/r+e/r2), e>0, is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stabilit...

Using the existence of integrable bi–almost‐periodic Green functions of linear homogeneous differential equations and the contraction fixed point, we are able to prove the existence of almost and pseudo–almost‐periodic mild solutions under quite general hypotheses for the differential equation with constant delay in a Banach space X, where τ>0 is...

The existence and stability of periodic solutions for an autonomous Hamiltonian system in 1:1:1 resonance depending on two real parameters α and β is established using reduction and averaging theories. The different types of periodic solutions as well as their bifurcation curves are characterized in terms of the parameters. The linear stability of...

The central configurations given by an equilateral triangle and a regular tetrahedron with equal masses at the vertices and a body at the barycenter have been widely studied in [9] and [14] due to the phenomena of bifurcation occurring when the central mass has a determined value . m*. We propose a variation of this problem setting the central mass...

A ratio dependent predator-prey model with stage structure for prey was investigated in [8]. There the authors mentioned that they were unable to show if such a model admits limit cycles when the unique equilibrium point E* at the positive octant is unstable. Here we characterize the existence of Hopf bifurcations for the systems. In particular we...

In this paper we study the global dynamics of the Hamiltonian systems , , where the Hamiltonian function H has the particular form , are polynomials, in particular H is the sum of the kinetic and a rational potential energies. Firstly, we provide the normal forms by a suitable μ-symplectic change of variables. Then, the global topological classific...

We investigate the existence of several families of symmetric periodic solutions as continuation of circular orbits of the Kepler problem for certain symmetric differentiable perturbations using an appropriate set of Poincaré-Delaunay coordinates which are essential in our approach. More precisely, we try separately two situations in an independent...

The aim of this paper is to prove the existence of a new symmetric family of periodic solutions of the generalized van der Waals Hamiltonian. In fact, we prove the existence of several families of first kind symmetric periodic solutions as continuation of circular orbits of the Kepler problem in the spatial case.

A ratio-dependent predator-prey model with stage structure for prey was investigated in [8]. There the authors mentioned that they were unable to show if such a model admits limit cycles when the unique equilibrium point E∗ at the positive octant is unstable. Here we characterize the existence of Hopf bifurcations for the systems. In particular we...

We classify and analyze the orbits of the Kepler problem on surfaces of constant curvature (both positive and negative, S² and ℍ², respectively) as functions of the angular momentum and the energy. Hill's regions are characterized, and the problem of time-collision is studied. We also regularize the problem in Cartesian and intrinsic coordinates, d...

The dynamics of a test particle (a particle with zero vorticity) advected by the velocity field of N point-vortices with vorticities Γj, j = 1, …N, is considered. Making an analogy with similar studies in celestial mechanics, we call such a study a “restricted N-vortex problem” or (N + 1)-vortex problem. In particular, we study and characterize the...

Recently some interest has appeared for the periodic FitzHugh–Nagumo differential systems. Here, we provide sufficient conditions for the existence of periodic solutions in such differential systems.

In this paper we describe a charged restricted circular three-body problem (CHRCTBP). Then, the particular equilibrium points, namely, the collinear equilibrium points , and and the isosceles triangle equilibrium and are characterized in an explicit way as functions of the parameters μ (parameter associated with the mass of the primaries) and β (pa...

In this paper we describe the flow of a predator-prey model where the predator feeds on two age classes of prey. Some basic features of the flow are proved under very mild hypothesis on the functional responses. To study the relationship between the ecological parameters and the behavioral ones, we also analyze a special case of the model where mod...

Analisys of Stability in Equilibrium Points of Difference Equations applied to Hamiltonian Systems

The elliptic isosceles restricted three-body problem (EIR3BP) with collision is defined as follows: Two point masses m 1 = m 2 move along a degenerate elliptic collision orbit under their gravitational attraction, then describe the motion of a third massless particle moving on a plane perpendicular to their line of motion and passing through the ce...

In the present paper,we consider the restricted planar (N+1)-body problem where N ≥ 3 bodies (called primaries) interacting with one another according to Newtonian law which are in the vertices of a regular N-gon with the origin at the center of masses of the coordinate system. We prove that the simultaneous binary collision between the infinitesim...

In the present paper, we consider the restricted planar (N+1)-body problem where N≥3 bodies (called primaries) interacting with one another according to Newtonian law which are in the vertices of a regular N-gon with the origin at the center of masses of the coordinate system. We prove that the simultaneous binary collision between the infinitesima...

In the present paper, we consider the restricted pla-nar (N + 1)-body problem where N ≥ 3 bodies (called primaries) interacting with one another according to Newtonian law which are in the vertices of a regular N-gon with the origin at the center of masses of the coordinate system. We prove that the simultaneous binary collision between the infinit...

This paper consider the trapezoidal collinear four--body problem as a model for binary-binary gravitational interaction in star clusters. It has two degrees of freedom, it also is a sub-problem of the trapezoidal four--body problem that has three degrees of freedom. We prove that all its singularities in the trapezoidal four-body problem are due to...

We introduce a circular restricted charged three-body problem on the plane. In this model, the gravitational and Coulomb forces, due to the primary bodies, act on a test particle; the net force exerted by some primary body on the test particle can be attractive, repulsive or null. The restricted problem is obtained by the general planar charged thr...

The aim of this paper is to characterize the exponential stability of linear systems of differential equations with slowly varying coefficients. Our approach is based on the freezing technique combined with new estimates for the norm of bounded linear operators. The main novelty of this work is that we use estimates for the absolute values of entri...

In this paper, we give necessary conditions for Lie-stability of the equilibrium solutions of autonomous and periodic Hamiltonian systems with n degrees of freedom which possess multiple resonances of arbitrary order. Necessary conditions for instability in the sense of Lyapunov in some cases of multiple resonances are provided. These conditions de...

Applying perturbation theory and symmetry conditions, we prove the existence of new families of periodic orbits for the 3-dimensional anisotropic Manev problem, which depends on three parameters. (C) 2015 AIP Publishing LLC.

In this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters. In particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the no...

Applying the averaging theory, we prove the existence of new families of periodic orbits for -dimensional type-galactic Hamiltonian systems.

In this work we study mechanical systems defined by polynomial potentials of degree on the plane, when the potential has a definite or semi-definite sign and the energy is non-negative. We give a global description of the flow for the non-negative potential case. Some partial results are obtained for the more complicated case of non-positive potent...

We started our studies with a planar Eulerian restricted four-body problem (ERFBP) where three masses move in circular orbits such that their configuration is always collinear. The fourth mass is small and does not influence the motion of the three primaries. In our model we assume that one of the primaries has mass 1 and is located at the origin a...

In the n-body problem, a collision singularity occurs when the position of two or more bodies coincide. By understanding the dynamics of collision motion in the regularized setting, a better understanding of the dynamics of near-collision motion is achieved. In this paper, we show that any double collision of the planar equilateral restricted four-...

We consider the Hamiltonian polynomial function H of degree fourth given by either H (x, y, px, py) = ( + ) + (x2 +y2) + V3(x;y) + V4(x,y), or H (x, y, px, py) = (- + ) + (-x2 +y2) + V3 (x, y) + V4 (x, y), where V3 (x, y) and V4 (x, y) are homogeneous polynomials of degree three and four, respectively. Our main objective is to prove the existence a...

This paper studies the Kummer–Schwarz differential equation ${2 \dot{x}\, \dddot{x}- 3 \ddot x^2=0}$ which is of special interest due to its relationship with the Schwarzian derivative. This differential equation is transformed into a first order differential system in ${\mathbb{R}^3}$ , and we provide a complete description of its global dynam...

We characterize the values of the parameters for which a zero--Hopf equilibrium point takes place at the singular points, namely, $O$ (the origin), $P_+$ and $P_-$ in the FitzHugh-Nagumo system. Thus we find two $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at the origin is a zero-Hopf equilibrium. For these...

Introduction: Oral anticoagulant therapy is an essential tool for prevention of thromboembolic events. The drugs most commonly used for this purpose are the vitamin K antagonists, which are monitored through the International Normalized Ratio (INR). Several factors affect the level of anticoagulation. Aim: To identify factors associated with an INR...

We consider the spatial isosceles Newtonian three-body problem, with one particle on a fixed plane, and the other two particles (with equal masses) located symmetrically with respect to this plane. Using variational methods, we find a one-parameter family of collision solutions for this system. All these solutions are periodic in a rotating frame.

We consider a system of five mass points r1, r2, r3, and r4 with masses m1 = m2 = m and m3=m4=m̃ moving about a single massive body r0 with mass m0 at its center which is assumed to be the origin of the coordinates system. We assume that the central body r0 makes a generalized force on the four mass points and that such a force is generated by a Ma...

The stability in the Lyapunov sense of an equilibrium position in a periodic Hamiltonian system with one degree of freedom is studied. It is assumed that the equilibrium is stable in the first approximation and that there exists an even resonance of order $k$ k , arbitrary. The critical case is considered, i.e., when the system of parameters is...

We study the central configurations (cc for short) for four masses arranged on a common circle (called co-circular cc) in two different situations, namely with no mass inside and later adding a fifth mass at the center of the circle. In the former, we focus the kite shape configurations by proving the existence of a one-parameter family of cc which...

We consider the Hamiltonian function defined by the cubic polynomial , where A, B, D ∈ ℜ are parameters and so H is an extension of the well known Hénon-Heiles problem. Our main contribution for D ≠ 0, A + B ≠ 0 and other technical restrictions are in three aspects: existence of periodic solutions, stability and instability of these periodic soluti...

The spatial restricted rhomboidal five-body problem, or shortly, SRRFBP, is a five body problem in which four positive masses, called the primaries, move two by two in coplanar circular motions with the center of mass fixed at the origin such that their configuration is always a rhombus, the fifth mass being negligible and not influencing the motio...

In this work we obtain necessary conditions for the instability in the Lyapunov sense of equilibrium points of autonomous and non-autonomous difference equations through the adaptation of differential methods and techniques due to Chetaev [3].

We present a geometric interpretation of the spectral stability of the triangular libration points in the charged three-body problem. We obtain that the spectral stability varies with the position of the center of mass of the three charges with respect to the circumcenter of the triangle configuration, which does not depend directly of the charges....

Using the notion of summable dichotomy in ordinary difference equations and the contraction and Schauder fixed point theorem, we obtain the existence of bounded and periodic solutions on ℤ under quite general hypotheses for non homogenous retarded difference equations.