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A criticality result for polycycles in a family of quadratic reversible centers
https://ddd.uab.cat/record/199353
We consider the family of dehomogenized Loud's centers Xµ_=y(x-1)∂ₓ + (x + Dx² + Fy²)_y, where µ=(D,F)єR², and we study the number of critical periodic orbits that emerge or dissapear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family {Xµ,µ є R²} distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set ΓB of codimension 1 in R². In the present paper we succeed in proving that a subset of ΓB has criticality equal to one. Rojas, DavidMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993532018Analytic tools to bound the criticality at the outer boundary of the period annulus
https://ddd.uab.cat/record/199343
In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most n 0 critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family ẍ= xp − xq , p, q ∈ R with p > q. Mañosas Capellades, FrancescMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993432018On the Chebyshev property of certain Abelian integrals near a polycycle
https://ddd.uab.cat/record/199323
F. Dumortier and R. Roussarie formulated in (Discrete Contin. Dyn. Syst. 2 (2009) 723–781] a conjecture concerning the Chebyshev property of a collection I₀,I₁,. . . ,In of Abelian integrals arising from singular perturbation problems occurring in planar slow-fast systems. The aim of this note is to show the validity of this conjecture near the polycycle at the boundary of the family of ovals defining the Abelian integrals. As a corollary of this local result we get that the linear span ⟨I₀,I₁,. . . ,In⟩ is Chebyshev with accuracy k = k(n). Marín Pérez, DavidMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993232018An effective algorithm to compute Mandelbrot sets in parameter planes
https://ddd.uab.cat/record/182522
In 2000 McMullen proved that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algo- rithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps. Garijo, AntoniTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825222017Study of the period function of a two-parameter family of centers
https://ddd.uab.cat/record/182501
In this paper we study the period function of ẍ = (1 x) p − (1 x) q , with p, q ∈ R and p > q. We prove three independent results. The first one establishes some regions in the parameter space where the corresponding center has a monotonous period function. This result extends the previous ones by Miyamoto and Yagasaki for the case q = 1. The second one deals with the bifurcation of critical periodic orbits from the center. The third one is addressed to the critical periodic orbits that bifurcate from the period annulus of each one of the three isochronous centers in the family when perturbed by means of a one-parameter deformation. These three results, together with the ones that we obtained previously on the issue, leads us to propose a conjectural bifurcation diagram for the global behaviour of the period function of the family. Mañosas Capellades, FrancescTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1825012017The criticality of centers of potential systems at the outer boundary
https://ddd.uab.cat/record/169462
The number of critical periodic orbits that bifurcate from the outer boundary of a potential center is studied. We call this number the criticality at the outer boundary. Our main results provide sufficient conditions in order to ensure that this number is exactly 0 and 1. We apply them to study the bifurcation diagram of the period function of X = −y∂ x ((x 1) p − (x 1) q )∂ y with q < p. This family was previously studied for q = 1 by Y. Miyamoto and K. Yagasaki. Mañosas Capellades, FrancescMon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694622016Analytical tools to study the criticality at the outer boundary of potential centers
https://ddd.uab.cat/record/167879
Rojas Pérez, DavidMon, 19 Dec 2016 00:27:26 GMThttps://ddd.uab.cat/record/167879urn:ISBN:9788449064760Universitat Autònoma de Barcelona,2016Unfolding of saddle-nodes and their Dulac time
https://ddd.uab.cat/record/163673
In this paper we study unfoldings of saddle-nodes and their Dulac time. By unfolding a saddle-node, saddles and nodes appear. In the first result (Theorem A) we give a uniform asymptotic expansion of the trajectories arriving at the node. Uniformity is with respect to all parameters including the unfolding parameter bringing the node to a saddle-node and a parameter belonging to a space of functions. In the second part, we apply this first result for proving a regularity result (Theorem B) on the Dulac time (time of Dulac map) of an unfolding of a saddle-node. This result is a building block in the study of bifurcations of critical periods in a neighborhood of a polycycle. Finally, we apply Theorem A and Theorem B to the study of critical periods of the Loud family of quadratic centers and we prove that no bifurcation occurs for certain values of the parameters (Theorem C). Mardesic, PavaoMon, 05 Sep 2016 20:35:14 GMThttps://ddd.uab.cat/record/1636732016Algebraic and analytical tools for the study of the period function
https://ddd.uab.cat/record/150683
In this paper we consider analytic planar differential systems having a first integral of the form H(x, y) = A(x) + B(x)y + C(x)y2 and an integrating factor κ(x) not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results. Garijo, AntoniFri, 06 May 2016 08:59:49 GMThttps://ddd.uab.cat/record/1506832014Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals
https://ddd.uab.cat/record/150616
We develop techniques for the verification of the Chebyshev property of Abelian integrals. These techniques are a combination of theoretical results, analysis of asymptotic behavior of Wronskians, and rigorous computations based on interval arithmetic. We apply this approach to tackle a conjecture formulated by Dumortier and Roussarie in [Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009), 723–781], which we are able to prove for q < 2. Figueras, Jordi-LluísFri, 06 May 2016 08:54:15 GMThttps://ddd.uab.cat/record/1506162013The period function of generalized Loud's centers
https://ddd.uab.cat/record/150602
In this paper a three parameter family of planar differential systems with homogeneous nonlinearities of arbitrary odd degree is studied. This family is an extension to higher degree of the Loud's systems. The origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function. We study the bifurcation diagram of this function focusing our attention on the bifurcations occurring at the polycycle that bounds the period annulus of the center. Moreover we determine some regions in the parameter space for which the corresponding period function is monotonous or it has at least one critical period, giving also its character (maximum or minimum). Finally we propose a complete conjectural bifurcation diagram of the period function of these generalized Loud's centers. Marín Pérez, DavidFri, 06 May 2016 08:54:14 GMThttps://ddd.uab.cat/record/1506022013Study of the period function of a biparametric family of centers
https://ddd.uab.cat/record/150584
Rojas Pérez, DavidFri, 06 May 2016 08:54:14 GMThttps://ddd.uab.cat/record/1505842013Bifurcation of local critical periods in the generalized Loud's system
https://ddd.uab.cat/record/150533
We study the bifurcation of local critical periods in the differential system (x˙ = −y + Bxn−1y,y˙ = x + Dxn + F xn−2y2, where B, D, F ∈ R and n > 3 is a fixed natural number. Here by "local" we mean in a neighbourhood of the center at the origin. For n even we show that at most two local critical periods bifurcate from a weak center of finite order or from the linear isochrone, and at most one local critical period from a nonlinear isochrone. For n odd we prove that at most one local critical period bifurcates from the weak centers of finite or infinite order. In addition, we show that the upper bound is sharp in all the cases. For n = 2 this was proved by Chicone and Jacobs in [Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486] and our proof strongly relies on their general results about the issue. Villadelprat, JordiFri, 06 May 2016 08:49:22 GMThttps://ddd.uab.cat/record/1505332012A Chebyshev criterion for Abelian integrals
https://ddd.uab.cat/record/150466
We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced. Grau, MaiteFri, 06 May 2016 08:30:51 GMThttps://ddd.uab.cat/record/1504662011Bounding the number of zeros of certain Abelian integrals
https://ddd.uab.cat/record/150430
In this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n Abelian integrals to have at most n + k − 1 zeros counted with multiplicities. This condition involves the functions in the integrand of the Abelian integrals and it can be checked, in many cases, in a purely algebraic way. Mañosas Capellades, FrancescFri, 06 May 2016 08:30:49 GMThttps://ddd.uab.cat/record/1504302011On the wave length of smooth periodic traveling waves of the Camassa-Holm equation
https://ddd.uab.cat/record/145296
This paper is concerned with the wave length of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height a (or ''peak-to-peak amplitude''). Our main result establishes monotonicity properties of the map a (a), i. e. , the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated to the equation, which distinguish between the two possible qualitative behaviours of (a), namely monotonicity and unimodality. The key point is to relate (a) to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems. Geyer, AnnaTue, 12 Jan 2016 16:38:07 GMThttps://ddd.uab.cat/record/1452962015The period function of the generalized Lotka-Volterra centers
https://ddd.uab.cat/record/44168
"Vegeu el resum a l'inici del document del fitxer adjunt". Villadelprat Yagüe, JordiMon, 13 Jul 2009 13:09:21 GMThttps://ddd.uab.cat/record/44168Centre de Recerca Matemàtica2005Càlcul numèric
https://ddd.uab.cat/record/28565
Alsedà, LluísMon, 15 Sep 2008 10:23:40 GMThttps://ddd.uab.cat/record/285652002-03