Joan Torregrosa

Autonomous University of Barcelona, Cerdanyola del Vallès, Catalonia, Spain

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Publications (35)35.04 Total impact

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    ABSTRACT: This paper concerns the study of small-amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as an impassable grain. The canonical form of the unperturbed vector field is like a degenerate flow box. Near the singularity, the phase portrait consists of parallel fibers, all but one of which have no singular points, and at the singular fiber, there is one node. We demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic.
    Journal of Differential Equations 12/2014; 258(2). DOI:10.1016/j.jde.2014.09.024 · 1.57 Impact Factor
  • R. Prohens, J. Torregrosa
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    ABSTRACT: The second order Poincaré-Pontryagin-Melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers. This approach also allows us to give the shape and the period up to first order. We address these problems for some classes of Abel differential equations and quadratic isochronous vector fields in the plane. We prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order. In particular the configurations (2,0)(2,0) and (1,1)(1,1) are realizable when two centers are perturbed simultaneously. The required computations show that all the considered families share the same iterated rational trigonometric integrals.
    Physica D Nonlinear Phenomena 07/2014; 280. DOI:10.1016/j.physd.2014.05.002 · 1.83 Impact Factor
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    ABSTRACT: The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve.
    Journal of Differential Equations 11/2013; 255(9):2655-2671. DOI:10.1016/j.jde.2013.07.006 · 1.57 Impact Factor
  • M. Caubergh, J. Torregrosa
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    ABSTRACT: The results in this paper show that the cubic vector fields ẋ = -y + M(x, y) - y(x2 + y2), ẏ = x + N(x, y) + x( x2 + y2), where M, N are quadratic homogeneous polynomials, having simultaneously a center at the origin and at infinity, have at least 61 and at most 68 topologically different phase portraits. To this end, the reversible subfamily defined by M(x, y) = -γxy, N(x, y) = (γ - λ)x2 + α2λy2 with α, γ ∈ ℝ and λ ≠ 0, is studied in detail and it is shown to have at least 48 and at most 55 topologically different phase portraits. In particular, there are exactly five for γλ < 0 and at least 46 for γλ > 0. Furthermore, the global bifurcation diagram is analyzed.
    International Journal of Bifurcation and Chaos 09/2013; 23(09):50161-. DOI:10.1142/S0218127413501617 · 1.02 Impact Factor
  • Claudio Buzzi, Claudio Pessoa, Joan Torregrosa
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    ABSTRACT: This paper is mainly devoted to the study of the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line Σ and the singular point of the unperturbed system is in Σ. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirms that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For the latter systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.
    Discrete and Continuous Dynamical Systems 09/2013; 33(9). DOI:10.3934/dcds.2013.33.3915 · 0.92 Impact Factor
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    Set Perez-González, Joan Torregrosa
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    ABSTRACT: The period annuli of the planar vector field $x' = - y F(x,y), y' = x F(x,y),$ where the set $\{F(x,y)=0\}$ consists of $k$ different isolated points, is defined by $k+1$ concentric annuli. In this paper we perturb it with polynomials of degree $n$ and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of $k$ and $n.$ Additionally, we prove that the associated Abelian integral is piecewise rational and, when $k=1$, the provided upper bound is reached. Finally, the case $k=2$ is also treated.
    Bulletin des Sciences Mathématiques 08/2013; · 0.73 Impact Factor
  • R. Prohens, J. Torregrosa
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    ABSTRACT: In this work we are concerned with the problem of shape and period of isolated periodic solutions of perturbed analytic radial Hamiltonian vector fields in the plane. Françoise developed a method to obtain the first non vanishing Poincaré–Pontryagin–Melnikov function. We generalize this technique and we apply it to know, up to any order, the shape of the limit cycles bifurcating from the period annulus of the class of radial Hamiltonians. We write any solution, in polar coordinates, as a power series expansion in terms of the small parameter. This expansion is also used to give the period of the bifurcated periodic solutions. We present the concrete expression of the solutions up to third order of perturbation of Hamiltonians of the form H=H(r)H=H(r). Necessary and sufficient conditions that show if a solution is simple or double are also presented.
    Nonlinear Analysis 04/2013; 81:130–148. DOI:10.1016/j.na.2012.10.017 · 1.61 Impact Factor
  • J. Llibre, M. A. Teixeira, J. Torregrosa
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    ABSTRACT: In this paper, we provide a lower bound for the maximum number of limit cycles of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line. Here, we only consider nonsliding limit cycles. For those systems, the interior of any limit cycle only contains a unique equilibrium point or a unique sliding segment. Moreover, the linear differential systems that we consider in every half-plane can have either a focus (F), or a node (N), or a saddle (S), these equilibrium points can be real or virtual. Then, we can consider six kinds of planar discontinuous piecewise linear differential systems: FF, FN, FS, NN, NS, SS. We provide for each of these types of discontinuous differential systems examples with two limit cycles.
    International Journal of Bifurcation and Chaos 04/2013; 23(04):50066-. DOI:10.1142/S0218127413500661 · 1.02 Impact Factor
  • S. Pérez-González, J. Torregrosa
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    ABSTRACT: The period annuli of the planar vector field x′=−yF(x,y)x′=−yF(x,y), y′=xF(x,y)y′=xF(x,y), where the set {F(x,y)=0}{F(x,y)=0} consists of k different isolated points, is defined by k+1k+1 concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n. Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1k=1, the provided upper bound is reached. Finally, the case k=2k=2 is also treated.
    Bulletin des Sciences Mathématiques 01/2013; 138(1). DOI:10.1016/j.bulsci.2013.09.004 · 0.73 Impact Factor
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    Armengol Gasull, J. Tomas Lazaro, Joan Torregrosa
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    ABSTRACT: We analyze whether a given set of analytic functions is an Extended Chebyshev system. This family of functions appears studying the number of limit cycles bifurcating from some nonlinear vector field in the plane. Our approach is mainly based on the so called Derivation-Division algorithm. We prove that under some natural hypotheses our family is an Extended Chebyshev system and when some of them are not fulfilled then the set of functions is not necessarily an Extended Chebyshev system. One of these examples constitutes an Extended Chebyshev system with high accuracy.
    Journal of Mathematical Analysis and Applications 03/2012; 387(2). DOI:10.1016/j.jmaa.2011.09.019 · 1.12 Impact Factor
  • Armengol Gasull, Chengzhi Li, Joan Torregrosa
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    ABSTRACT: We prove that a family of functions defined through some definite integrals forms an extended complete Chebyshev system. The key point of our proof consists of reducing the study of certain Wronskians to the Gram determinants of a suitable set of new functions. Our result is then applied to give upper bounds for the number of isolated periodic solutions of some perturbed Abel equations.
    Journal of Differential Equations 01/2012; 252(2):1635–1641. DOI:10.1016/j.jde.2011.06.010 · 1.57 Impact Factor
  • Armengol Gasull, Chengzhi Li, Joan Torregrosa
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    ABSTRACT: Consider planar ordinary differential equations of the form ,, where C(x,y) is an algebraic curve. We are interested in knowing whether the existence of multiple factors for C is important or not when we study the maximum number of zeros of the Abelian integral M that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field. With this aim, we study in detail the case C(x,y)=(1−y)m, where m is a positive integer number and prove that m has essentially no impact on the number of zeros of M. This result improves the known studies on M. One of the key points of our approach is that we obtain a simple expression of M based on some successive reductions of the integrals appearing during the procedure.
    Nonlinear Analysis 01/2012; 75(1):278-285. DOI:10.1016/j.na.2011.08.032 · 1.61 Impact Factor
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    Hector Giacomini, Armengol Gasull, Joan Torregrosa
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    ABSTRACT: It is well-known that the existence of traveling wave solutions for reaction-diffusion partial differential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar differential equations. We introduce a method for finding explicit upper and lower bounds of these heteroclinic orbits. In particular, for the classical Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to locate these solutions analytically and with very high accuracy.
    Discrete and Continuous Dynamical Systems 12/2011; 33(8). DOI:10.3934/dcds.2013.33.3567 · 0.92 Impact Factor
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    ABSTRACT: In this paper we study the number of limit cycles bifurcating from isochronous surfaces of revolution contained in R3, when we consider polynomial perturbations of arbitrary degree. The method for studying these limit cycles is based on the averaging theory and on the properties of Chebyshev systems. We present a new result on averaging theory and generalizations of some classical Chebyshev systems which allow us to obtain the main results.
    Journal of Mathematical Analysis and Applications 09/2011; 381(1):414-426. DOI:10.1016/j.jmaa.2011.04.009 · 1.12 Impact Factor
  • M. Caubergh, J. Llibre, J. Torregrosa
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    ABSTRACT: We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax2 + bxy + cy2 + σ(dx - y)(x2 + y2), ẏ = x + δy + ex2 + fxy + gy2 + σ(x + dy) (x2 + y2), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincaré disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.
    International Journal of Bifurcation and Chaos 07/2011; 21(7):1831-1867. DOI:10.1142/S0218127411029501 · 1.02 Impact Factor
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    ABSTRACT: The authors consider the following system x ˙=((x 2 +y 2 -2)f(x,y,z)-z)x x 2 +y 2 +εP(x,y,z),y ˙=((x 2 +y 2 -2)f(x,y,z)-z)y x 2 +y 2 +εQ(x,y,z),z ˙=zf(x,y,z)+(x 2 +y 2 -2)+εR(x,y,z) defined in ℝ 3 ∖{(0,0,z)∣z∈ℝ}, where f(x,y,z)=1-(x 2 +y 2 -2) 2 -z 2 , ε is a small parameter, and P(x,y,z), Q(x,y,z), R(x,y,z) are polynomials. It is proved that there exist polynomial perturbations of degree d of the torus such that exactly ν limit cycles bifurcate for every ν∈{2,4,⋯,2(d+1)}.
    Advanced Nonlinear Studies 01/2011; 11(2). · 0.67 Impact Factor
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    Armengol Gasull, J. Tomas Lazaro, Joan Torregrosa
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    ABSTRACT: Consider the vector field $x'= -yG(x, y), y'=xG(x, y),$ where the set of critical points $\{G(x, y) = 0\}$ is formed by $K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree $n$ and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of $K$ and $n.$ Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for $K\le4$ we recover or improve some results obtained in several previous works.
    Nonlinear Analysis 12/2010; 75(13). DOI:10.1016/j.na.2012.04.033 · 1.61 Impact Factor
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    Armengol Gasull, Hector Giacomini, Joan Torregrosa
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    ABSTRACT: Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $\Phi(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${\Phi(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2})$. We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions.
    Nonlinearity 12/2009; 23(12). DOI:10.1088/0951-7715/23/12/001 · 1.20 Impact Factor
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    ABSTRACT: For every positive integer N ≥ 2 we consider the linear differential centre in 4 with eigenvalues ±i and ±Ni. We perturb this linear centre inside the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree N, i.e. where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree N. Then if the displacement function of order ϵ of the perturbed system is not identically zero, we study the maximal number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.
    Dynamical Systems 03/2009; 24(1):123-137. DOI:10.1080/14689360802534492 · 0.38 Impact Factor
  • A. Gasull, R. Prohens, J. Torregrosa
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    ABSTRACT: Consider the planar ordinary differential equation [(x)\dot]=-yF(x,y), [(y)\dot] =xF(x,y){\dot x=-yF(x,y), \dot y {=}xF(x,y)} , where the set {F(x,y)=0}{\{F(x,y)=0\}} consists of k non-zero points. In this paper we perturb this vector field with a general polynomial perturbation of degree n and study how many limit cycles bifurcate from the period annulus of the origin in terms of k and n. One of the key points of our approach is that the Abelian integral that controls the bifurcation can be explicitly obtained as an application of the integral representation formula of harmonic functions through the Poisson kernel.
    Journal of Dynamics and Differential Equations 11/2008; 20(4):945-960. DOI:10.1007/s10884-008-9112-7 · 1.00 Impact Factor