arXiv subjects
Publications and source records attributed to Armengol Gasull.
We give lower bounds in terms of~$n,$ for the number of limit cycles of polynomial vector fields of degree~$n,$ having any prescribed invariant algebraic curve. By applying them when the ovals of this curve are also algebraic limit cycles we obtain a new recurrent property for the Hilbert numbers. Finally, we apply our results to two important families of models: Kolmogorov systems and a general family of systems appearing in Game Theory.
We study the $T$-periodic solutions of the real Riccati differential equation $x' = x^2 + \gamma(t),$ where $x=x(t)$ and $\gamma$ is a $T$-periodic function. Our goal is to define a real-valued discriminant $\Delta_{\gamma}$ that determines whether the equation admits two, one, or no $T$-periodic solutions, in analogy with the classical discriminant of the quadratic algebraic equations. This problem is closely related to a question posed by Coppel concerning the characterization of bifurcation curves for planar quadratic differential systems. Although our result does not cover all periodic Riccati differential equations, many of them can be transformed into this particular form.
We study two natural families of methods of order $n\ge 2$ that are useful for solving numerically one variable equations $f(x)=0.$ The first family consists on the methods that depend on $x,f(x)$ and its successive derivatives up to $f^{(n-1)}(x)$ and the second family comprises methods that depend on $x,g(x)$ until $g^{\circ n}(x),$ where $g^{\circ m}(x)=g(g^{\circ (m-1)}(x))$ and $g(x)=f(x)+x$. The first family includes the well-known Newton, Chebyshev, and Halley methods, while the second one contains the Steffensen method. Although the results for the first type of methods are well known and classical, we provide new, simple, detailed, and self-contained proofs.
We consider the family of piecewise linear maps $F(x,y)=\left(|x| - y + a, x - |y| + b\right),$ where $(a,b)\in \mathbb{R}^2$. In previous work, we identified that certain maps of this class possess one-dimensional invariant sets, planar graphs, that capture the global dynamics of the system. Within these graphs, chaotic dynamics emerge for certain parameter values, leading to an intermediate dynamical regime between regular behavior and full-plane chaos. In the present study, we revisit this family and analyze in detail the topological entropy as a function of a bifurcation parameter, finding that transitions from positive to zero entropy are continuous for certain parameter values and discontinuous for others. We also provide a methodology for determining arbitrarily sharp rational bounds for the bifurcation values at which this transition occurs. Finally, motivated by the limitations of numerical simulations in detecting the complex dynamics within these graphs, we prove that for some parameter values, there exists a full-measure set in these graphs where orbits converge to at most three omega-limit sets, which, when the parameter values are rational, correspond to periodic orbits.
Let $\dot{z}=f(z)$ be a holomorphic differential equation with center at $p$. In this paper we are concerned about studying the piecewise perturbation systems $\dot{z}=f(z)+\epsilon R^\pm(z,\overline{z}),$ where $R^\pm(z,\overline{z})$ are complex polynomials defined for $\pm\operatorname{Im}(z)> 0.$ We provide an integral expression, similar to an Abelian integral, for the period annulus of $p.$ The zeros of this integral control the bifurcating limit cycles from the periodic orbits of this annular region. This expression is given in terms of the conformal conjugation between $\dot{z}=f(z)$ and its linearization $\dot{z}=f'(p)z$ at $p$. We use this result to control the simultaneous bifurcation of limit cycles of the two annular periods of $\dot{z}={\rm i} (z^2-1)/2$, after both complex and holomorphic piecewise polynomial perturbations. In particular, as far as we know, we provide the first proof of the existence of non nested limit cycles for piecewise holomorphic systems.
We prove that the period function of the center at the origin of the $\mathbb{Z}_k$-equivariant differential equation $\dot{z}=iz+a(z\overline{z})^nz^{k+1}, a\ne0,$ is monotonous decreasing for all $n$ and $k$ positive integers, solving a conjecture about them. We show this result as corollary of proving that the period function of the center at the origin of a sub-family of the reversible quadratic centers is monotonous decreasing as well.
This paper studies the number of centers and limit cycles of the family of planar quartic polynomial vector fields that has the invariant algebraic curve $(4x^2-1)(4y^2-1)=0.$ The main interest for this type of vector fields comes from their appearance in some mathematical models in Game Theory composed by two players. In particular, we find examples with five nested limit cycles surrounding the same singularity, as well as examples with four limit cycles formed by two disjoint nests, each one of them with two limit cycles. We also prove a Berlinski\u \i's type result for this family of vector fields.
We consider the family of piecewise linear maps $$F_{a,b}(x,y)=\left(|x| - y + a, x - |y| + b\right),$$ where $(a,b)\in \mathbb{R}^2$. This family belongs to a wider one that has deserved some interest in the recent years as it provides a framework for generalized Lozi-type maps. Among our results, we prove that for $a\ge 0$ all the orbits are eventually periodic and moreover that there are at most three different periodic behaviors formed by at most seven points. For $a<0$ we prove that for each $b\in\mathbb{R}$ there exists a compact graph $\Gamma,$ which is invariant under the map $F$, such that for each $(x,y)\in \mathbb{R}^2$ there exists $n\in\mathbb{N}$ (that may depend on $x$) such that $F_{a,b}^n(x,y)\in \Gamma.$ We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all $(a,b)\in\mathbb{R}^2$ obtaining, among other results, a full characterization of when $F_{a,b}|_{\Gamma}$ has positive or zero entropy.
Let $X$ be a planar smooth vector field with a polycycle $\Gamma^n$ with $n$ sides and all its corners, that are at most $n$ singularities, being hyperbolic saddles. In this paper we study the cyclicity of $\Gamma^n$ in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least $k,$ for any $k\leqslant n.$ Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when $X$ is polynomial there is a polynomial perturbation (in general with degree much higher that the one of $X$) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world.
Let $\mathcal{H}(n)$ be the maximum number of limit cycles that a planar polynomial vector field of degree $n$ can have. In this paper we prove that $\mathcal{H}(n)$ is realizable by structurally stable vector fields with only hyperbolic limit cycles and that it is a strictly increasing function whenever it is finite.
We study the number of limit cycles that a planar polynomial vector field can have as a function of its number $m$ of monomials. We prove that the number of limit cycles increases at least quadratically with $m$ and we provide good lower bounds for $m\leqslant10$.
In this paper we are concerned with determining lower bounds of the number of limit cycles for piecewise polynomial holomorphic systems with a straight line of discontinuity. We approach this problem with different points of view: study of the number of zeros of the first and second order averaging functions, or with the control of the limit cycles appearing from a monodromic equilibrium point via a degenerated Andronov-Hoph type bifurcation, adding at the very end the sliding effects. We also use the Poincaré-Miranda theorem for obtaining an explicit piecewise linear holomorphic systems with 3 limit cycles, result that improves the known examples in the literature that had a single limit cycle.
We study the dynamics of the piecewise planar rotations $F_{\lambda}(z)=\lambda (z-H(z)), $ with $z\in\C$, $H(z)=1$ if $\mathrm{Im}(z)\ge0,$ $H(z)=-1$ if $\mathrm{Im}(z)<0,$ and $\lambda=\mathrm{e}^{i \alpha} \in\C$, being $\alpha$ a rational multiple of $\pi$. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of $F_\lambda$, with a period $\ell,$ that depends on the connected component. Furthermore, $F_\lambda^\ell $ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides.
We study three questions related to Machin's type formulas. The first one gives all two terms Machin formulas where both arctangent functions are evaluated $2$-integers, that is values of the form $b/2^a$ for some integers $a$ and~$b$. These formulas are computationally useful because multiplication or division by a power of two is a very fast operation for most computers. The second one presents a method for finding infinitely many formulas with $N$ terms. In the particular case $N=2$ the method is quite useful. It recovers most known formulas, gives some new ones, and allows to prove in an easy way that there are two terms Machin formulas with Lehmer measure as small as desired. Finally, we correct an oversight from previous result and give all Machin's type formulas with two terms involving arctangents of powers of the golden section.
For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Lienard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.
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.
We describe the global dynamics of some pointwise periodic piecewise linear maps in the plane that exhibit interesting dynamic features. For each of these maps we find a first integral. For these integrals the set of values are discrete, thus quantized. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of open tiles of certain regular or uniform tessellations. The action of the maps on each invariant set of tiles is described geometrically.
It is well known that the existence of traveling wave solutions (TWS) for many partial differential equations (PDE) is a consequence of the fact that an associated planar ordinary differential equation (ODE) has certain types of solutions defined for all time. In this paper we address the problem of persistence of TWS of a given PDE under small perturbations. Our main results deal with the situation where the associated ODE has a center and, as a consequence, the original PDE has a continuum of periodic traveling wave solutions. We prove that the TWS that persist are controlled by the zeroes of some Abelian integrals. We apply our results to several famous PDE, like the Ostrovsky, Klein-Gordon, sine-Gordon, Korteweg-de Vries, Rosenau-Hyman, Camassa-Holm, and Boussinesq equations.