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Jaume Llibre

Publications and source records attributed to Jaume Llibre.

At least 19 recordsLinked to original sources
arXiv2026

Singular barriers and quartic integrability breaking in the TTW system

We study the competition between resonance-induced integrability breaking and centrifugal confinement in a symmetric quartic deformation of the classical $k=1$ Tremblay--Turbiner--Winternitz (TTW) system, $H=\frac{1}{2}\left(P_X^2+P_Y^2\right)+X^2+Y^2 +\frac{\gamma}{X^2}+\frac{\gamma}{Y^2}+\kappa X^2Y^2.$ The inverse-square terms are the centrifugal remnants of an $SO(2)\times SO(2)$ reduction of a four-dimensional oscillator, while the $X^2Y^2$ interaction couples the reduced radial modes and connects the Smorodinsky--Winternitz and Contopoulos limits. For $\gamma>0$, the axes are impenetrable and split configuration space into invariant sectors. For sufficiently small nonzero $\kappa$, resonant averaging constructs a phase-locked periodic orbit on each energy surface $H=h>4\sqrt{\gamma}$ whose fixed-energy reduced Poincar\'e map has no unit characteristic multiplier. Poincar\'e's criterion therefore excludes a second independent $C^1$ first integral in any invariant neighbourhood of this orbit. At finite coupling, Poincar\'e sections and finite-time Lyapunov maps show the breakup of invariant curves and the growth of chaotic layers. Comparison with the barrier-free limit separates two effects: the quartic interaction breaks integrability, whereas the singular barriers reduce phase-space connectivity and chaotic transport without restoring it.

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arXiv2026

Limit cycles in piecewise smooth systems with circular switching manifold

We study limit cycles in piecewise complex systems with switching manifold $\mathbb{S}^1$. Using M\"obius transformations we establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and algebraic structure. For piecewise polynomial holomorphic systems we obtain lower bounds on the number of limit cycles via second-order averaging and, for low degrees, via Lyapunov quantities. For piecewise antiholomorphic systems we prove upper bounds: at most $3$ limit cycles in the linear case and $10$ in the quadratic case. We also prove a rigidity theorem: when both components admit classical holomorphic normal forms at the origin no crossing limit cycles exist. Finally, we construct explicit algebraic limit cycles in the circular context, providing, as far as we know the first such examples in the literature.

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arXiv2025

Global Dynamics Of Quadratic And Cubic Planar Quasi-homogeneous Differential Systems

In this paper we obtain the global dynamics and phase portraits of quadratic and cubic quasi-homogeneous but non-homogeneous systems. We first prove that all planar quadratic and cubic quasi-homogeneous but non-homogeneous polynomial systems can be reduced to three homogeneous ones. Then for the homogeneous systems, we employ blow-up method, normal sector method, Poincar\'e compactification and other techniques to discuss their dynamics. Finally we characterize the global phase portraits of quadratic and cubic quasi-homogeneous but non-homogeneous polynomial systems.

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arXiv2025

The dynamics of the Ehrhard-M\"uller system with invariant algebraic surfaces

In this paper we study the global dynamics of the Ehrhard-M\"uller differential system \[ \dot{x} = s(y - x), \quad \dot{y} = rx - xz - y + c, \quad \dot{z} = xy - z, \] where $s$, $r$ and $c$ are real parameters, and $x$, $y$, and $z$ are real variables. We classify the invariant algebraic surfaces of degree $2$ of this differential system. After we describe the phase portraits in the Poincar\'e ball of this differential system having one of this invariant algebraic surfaces. The Poincar\'e ball is the closed unit ball in $\mathbb{R}^3$ whose interior has been identified with $\mathbb{R}^3$, and his boundary, the $2$-dimensional sphere $\mathbb{S}^2$, has been identified with the infinity of $\mathbb{R}^3$. Note that in the space $\mathbb{R}^3$ we can go to infinity in as many as directions as points has the sphere $\mathbb{S}^2$. A polynomial differential system as the Ehrhard-M\"uller system can be extended analytically to the Poincar\'e ball, in this way we can study its dynamics in a neigborhood of infinity. Providing these phase portraits in the Poincar\'e ball we are describing the dynamics of all orbits of the Ehrhard-M\"uller system having an invariant algebraic surface of degree $2$.

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arXiv2025

Planar Kolmogorov systems with infinitely many singular points at infinity

We classify the global dynamics of the five-parameter family of planar Kolmogorov systems \begin{equation*} \begin{split} \dot{y}&=y \left( b_0+ b_1 y z + b_2 y + b_3 z\right), \dot{z}&=z\left( c_0 + b_1 y z + b_2 y + b_3 z\right), \end{split} \end{equation*} which is obtained from the Lotka-Volterra systems of dimension three. These systems have infinitely many singular points at inifnity. We give the topological classification of their phase portraits in the Poincar\'e disc, so we can describe the dynamics of these systems near infinity. We prove that these systems have 13 topologically distinct global phase portraits.

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arXiv2025

Global phase portraits of a predator-prey system

We classify the global dynamics of a family of Kolmogorov systems depending on three parameters which has ecological meaning as it modelizes a predator-prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincar\'e disc, so we provide all the possible distinct dynamics of these systems.

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arXiv2025

Phase portraits of a family of Kolmogorov systems depending on six parameters

Consider a general $3$-dimensional Lotka-Volterra system with a rational first integral of degree two of the form $H=x^i y^j z^k$. The restriction of this Lotka-Volterra system to each surface $H(x,y,z)=h$ varying $h\in \mathbb{R}$ provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form $x^\ell y^m e^{st}$ they reduce to the Kolmogorov systems \begin{equation*} \begin{split} \dot{x}&=x \left( a_0- \mu (c_1 x + c_2 z^2 + c_3 z)\right),\\ \dot{z}&=z\left( c_0+ c_1 x + c_2 z^2 + c_3 z\right). \end{split} \end{equation*} In this paper we classify the phase portraits in the Poincar\'e disc of all these Kolmogorov systems which depend on six parameters.

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arXiv2024

Limit cycles and chaos in planar hybrid systems

In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic dynamics.

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arXiv2023

Global centers of a class of cubic polynomial differential systems

A difficult classical problem in the qualitative theory of differential systems in the plane $\mathbb{R}^2$ is the center-focus problem, i.e. to distinguish between a focus and a center. Another difficult problem is to distinguish inside a family of centers the ones which are global. A global center is a center $p$ such that $\mathbb{R}^2\setminus\{p\}$ is filled with periodic orbits. In this paper we classify the global centers of the family of real polynomial differential systems of degree $3$ that in complex notation write $$ i\dot{w}=w-A_3\overline{w}^2-A_4w^3-A_5w^2\overline{w}-A_6w\overline{w}^2, $$ where $w=x+iy$ and $A_k\in\mathbb{C}$ for $k=3,4,5,6$.

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arXiv2023

Existence of a cylinder foliated by periodic orbits in the generalized Chazy differential equation

The generalized Chazy differential equation corresponds to the following two-parameter family of differential equations \begin{equation*}\label{gcdeq} \dddot x+|x|^q \ddot x+\dfrac{k |x|^q}{x}\dot x^2=0, \end{equation*} which has its regularity varying with $q$ , a positive integer. Indeed, for $q=1$ it is discontinuous on the straight line $x=0$, whereas for $q$ a positive even integer it is polynomial, and for $q>1$ a positive odd integer it is continuous but not differentiable on the straight line $x=0$. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for $q=2$ and $k=3$. In this paper, we prove analytically the existence of such periodic solutions. Our strategy allows to establish sufficient conditions ensuring that the generalized Chazy differential equation, for $k=q+1$ and any positive integer $q$ , has actually an invariant topological cylinder foliated by periodic solutions in the $(x,\dot x,\ddot x)$-space. In order to set forth the bases of our approach, we start by considering $q=1,2,3$, which are representatives of the different classes of regularity. For an arbitrary positive integer $q$, an algorithm is provided for checking the sufficient conditions for the existence of such an invariant cylinder, which we conjecture that always exists. The algorithm was successfully applied up to $q=100$.

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arXiv2022

Limit cycles of linear vector fields on $(\mathbb{S}^2)^m \times \mathbb{R}^n$

It is well known that linear vector fields defined in $\mathbb{R}^n$ can not have limit cycles, but this is not the case for linear vector fields defined in other manifolds. We study the existence of limit cycles bifurcating from a continuum of periodic orbits of linear vector fields on manifolds of the form $(\mathbb{S}^2)^m \times \mathbb{R}^n$ when such vector fields are perturbed inside the class of all linear vector fields. The study is done using the averaging theory. We also present an open problem concerning the maximum number of limit cycles of linear vector fields on $(\mathbb{S}^2)^m \times \mathbb{R}^n$.

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arXiv2022

Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centers

These last years an increasing interest appeared for studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. One of the difficulties for understanding the dynamics of these systems is the study their limit cycles. In this paper we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line, and formed by combinations of linear centers (consequently isochronous) and cubic isochronous centers with homogeneous nonlinearities. For these classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles.

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arXiv2022

On the equilateral pentagonal central configurations

An equilateral pentagon is a polygon in the plane with five sides of equal length. In this paper we classify the central configurations of the $5$-body problem having the five bodies at the vertices of an equilateral pentagon with an axis of symmetry. We prove that there are two unique classes of such equilateral pentagons providing central configurations, one concave equilateral pentagon and one convex equilateral pentagon, the regular one. A key point of our proof is the use of rational parameterizations to transform the corresponding equations, which involve square roots, into polynomial equations.

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arXiv2021

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for $y<0$ and a virtual one for $y>0$, and such that the real center is a global center. Then, working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one additional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.

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arXiv2021

Dynamics of a generalized Rayleigh system

Consider the first order differential system given by \begin{equation*} \begin{array}{l} \dot{x}= y, \qquad \dot{y}= -x+a(1-y^{2n})y, \end{array} \end{equation*} where $a$ is a real parameter and the dots denote derivatives with respect to the time $t$. Such system is known as the generalized Rayleigh system and it appears, for instance, in the modeling of diabetic chemical processes through a constant area duct, where the effect of adding or rejecting heat is considered. In this paper we characterize the global dynamics of this generalized Rayleigh system. In particular we prove the existence of a unique limit cycle when the parameter $a\ne 0$.

math.CA↗
arXiv2021

Periods of Morse--Smale diffeomorphisms on $\mathbb{S}^n$, $\mathbb{S}^m \times \mathbb{S}^n$, $\mathbb{C}P^n}$ and $\mathbb{H}P^n}$

We study the set of periods of the Morse--Smale diffeomorphisms on the $n$-dimensional sphere $\mathbb{S}^n$, on products of two spheres of arbitrary dimension $\mathbb{S}^m \times \mathbb{S}^n$ with $m \neq n$, on the $n$-dimensional complex projective space $\mathbb{C}\text{\textbf{P}}^n}$ and on the $n$-dimensional quaternion projective space $\mathbb{H}\text{\textbf{P}}^n}$. We classify the minimal sets of Lefschetz periods for such Morse--Smale diffeomorphisms. This characterization is done using the induced maps on the homology. The main tool used is the Lefschetz zeta function.

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