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Periodic solutions of linear, Riccati, and Abel dynamic equations
https://ddd.uab.cat/record/204401
We study the number of periodic solutions of linear, Riccati and Abel dynamic equations in the time scales setting. In this way, we recover known results for corresponding differential equations and obtain new results for associated difference equations. In particular, we prove that there is no upper bound for the number of isolated periodic solutions of Abel difference equations. One of the main tools introduced to get our results is a suitable Melnikov function. This is the first time that Melnikov functions are used for dynamic equations on time scales. Bohner, MartinThu, 16 May 2019 13:36:19 GMThttps://ddd.uab.cat/record/2044012019On central configurations of the κn-body problem
https://ddd.uab.cat/record/204399
We consider planar central configurations of the Newtonian κn-body problem consisting in κ groups of regular n-gons of equal masses, called (κ,n)-crown. We derive the equations of central configurations for a general (κ,n)-crown. When κ=2 we prove the existence of a twisted (2,n)-crown for any value of the mass ratio. Moreover, for n=3,4 and any value of the mass ratio, we give the exact number of twisted (2,n)-crowns, and describe their location. Finally, we conjecture that for any value of the mass ratio there exist exactly three (2,n)-crowns for n≥5. Barrabés, EstherThu, 16 May 2019 13:36:19 GMThttps://ddd.uab.cat/record/2043992019Singularities of inner functions associated with hyperbolic maps
https://ddd.uab.cat/record/204390
Evdoridou, VasilikiThu, 16 May 2019 13:36:18 GMThttps://ddd.uab.cat/record/2043902019Zero entropy for some birational maps of C²
https://ddd.uab.cat/record/204387
In this study, we consider a special case of the family of birational maps f:C² → C² , which were dynamically classified by [13]. We identify the zero entropy subfamilies of f and explicitly give the associated invariant fibrations. In particular, we highlight all of the integrable and periodic mappings. Cima, AnnaThu, 16 May 2019 13:36:18 GMThttps://ddd.uab.cat/record/2043872019Rational maps with Fatou components of arbitrarily large connectivity
https://ddd.uab.cat/record/199372
We study the family of singular perturbations of Blaschke products B_a,(z)=z^3-a1- ^2. We analyse how the connectivity of the Fatou components varies as we move continuously the parameter . We prove that all possible escaping configurations of the critical point c_-(a,) take place within the parameter space. In particular, we prove that there are maps B_a, which have Fatou components of arbitrarily large finite connectivity within their dynamical planes. Canela Sánchez, JordiMon, 12 Nov 2018 12:11:51 GMThttps://ddd.uab.cat/record/1993722018Simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli
https://ddd.uab.cat/record/199360
We study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations. The cubic planar system (x',y')= (-y((x-1)² + y²),x((x-1)² + y²) has simultaneously a center at the origin and at infinity. We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli first separately and second simultaneously. This problem is an generalization of PerTor2014 to the piecewise systems class. When the polynomial perturbation has degree n, we prove that the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of zeros. When the perturbation is cubic, the same degree than the unperturbed vector field, the maximum number of limit cycles, up to first order perturbation, from the inner and outer annuli is 9 and 8, respectively. But, when the simultaneous bifurcation problem is considered, 12 limit cycles exist. These limit cycles appear in three type of configurations: (9,3), (6,6) and (4,8). In the non-piecewise scenario only 5 limit cycles were found. Da Cruz, Leonardo Pereira CostaMon, 12 Nov 2018 12:11:50 GMThttps://ddd.uab.cat/record/1993602018Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center
https://ddd.uab.cat/record/199322
From the beginning of this century more than thirty papers have been published studying the limit cycles of the discontinuous piecewise linear differential systems with two pieces separated by a straight line, but it remains open the following question: what is the maximum number of limit cycles that this class of differential systems can have? Here we prove that when one of the linear differential systems has a center, real or virtual, then the discontinuous piecewise linear differential system has at most two limit cycles. Llibre, JaumeMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993222018On extended chebyshev systems with positive accuracy
https://ddd.uab.cat/record/182730
A classical necessary condition for an ordered set of n+1 functions F to be an ECT-system in a closed interval is that all the Wronskians do not vanish. With this condition all the elements of Span(F) have at most n zeros taking into account the multiplicity. Here the problem of bounding the number of zeros of Span(F) is considered as well as the effectiveness of the upper bound when some Wronskians vanish. For this case we also study the possible configurations of zeros that can be realized by elements of Span(F). An application to count the number of isolated periodic orbits for a family of nonsmooth systems is performed. Novaes, Douglas D.Thu, 07 Dec 2017 15:24:47 GMThttps://ddd.uab.cat/record/1827302017Bifurcation of 2-periodic orbits from non-hyperbolic fixed points
https://ddd.uab.cat/record/182539
We introduce the concept of 2-cyclicity for families of one-dimensional maps with a non-hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus. This new concept is useful in order to study the number of 2-periodic orbits that can bifurcate from the fixed point. As an application we study the 2-cyclicity of some natural families of polynomial maps. Cimà, AnnaTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825392018Center boundaries for planar piecewise-smooth differential equations with two zones
https://ddd.uab.cat/record/182517
This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary. We prove that given a pair of systems that share a hyperbolic focus singularity p 0 , with the same orientation and opposite stability, and a ray Σ 0 with endpoint at the singularity p 0 , we can find a smooth manifold Ω such that Σ 0 ∪ p 0 ∪ Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached. Buzzi, Claudio A.Tue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825172017Limit cycles of discontinuous piecewise polynomial vector fields
https://ddd.uab.cat/record/182510
When the first average function is non-zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the center x= -y((x^2 y^2)/2)^m and y= x((x^2 y^2)/2)^m with m 1, when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces. The positive integers m, n and k are arbitrary. The main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields. de Carvalho, TiagoTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1825102017Study 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/1825012017Global behaviour of the period function for some degenerate centers
https://ddd.uab.cat/record/182499
We study the global behaviour of the period function on the period annulus of degenerate centers for two families of planar polynomial vector fields. These families are the quasi-homogeneous vector fields and the vector fields given by the sum of two quasi-homogeneous Hamiltonian ones. In the first case we prove that the period function is globally decreasing, extending previous results that deal either with the Hamiltonian quasi-homogeneous case or with the general homogeneous situation. In the second family, and after adding some more additional hypotheses, we show that the period function of the origin is either decreasing or has at most one critical period and that both possibilities may happen. This result also extends some previous results that deal with the situation where both vector fields are homogenous and the origin is a non-degenerate center. Álvarez, Maria JesúsTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1824992017On the dynamics of a model with coexistence of three attractors: a point, a periodic orbit and a strange attractor
https://ddd.uab.cat/record/182494
For a dynamical system described by a set of autonomous differential equations, an attractor can be either a point, or a periodic orbit, or even a strange attractor. Recently a new chaotic system with only one parameter has been presented where besides a point attractor and a chaotic attractor, it also has a coexisting attractor limit cycle which makes evident the complexity of such a system. We study using analytic tools the dynamics of such system. We describe its global dynamics at infinity, and show that it has no Darboux first integrals. Additionally, we characterize its Hopf bifurcations. Llibre, JaumeTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1824942017A convex representation of totally balanced games
https://ddd.uab.cat/record/182464
We analyze the least increment function, a convex function of n variables associated to an n-person cooperative game. Another convex representation of cooperative games, the indirect function, has previously been studied. At every point the least increment function is greater than or equal to the indirect function, and both functions coincide in the case of convex games, but an example shows that they do not necessarily coincide if the game is totally balanced but not convex. We prove that the least increment function of a game contains all the information of the game if and only if the game is totally balanced. We also give necessary and sufficient conditions for a function to be the least increment function of a game as well as an expression for the core of a game in terms of its least increment function. Bilbao, Jesús MarioMon, 27 Nov 2017 12:03:28 GMThttps://ddd.uab.cat/record/1824642012Reversible nilpotent centers with cubic homogeneous nonlinearities
https://ddd.uab.cat/record/169469
We provide 13 non-topological equivalent classes of global phase portraits in the Poincaré disk of reversible cubic homogeneous systems with a nilpotent center at origin, which complete the classification of the phase portraits of the nilpotent centers with cubic homogeneous nonlinearities. Dukarić, MašaMon, 23 Jan 2017 15:21:47 GMThttps://ddd.uab.cat/record/1694692016Analytic nilpotent centers as limits of nondegenerate centers revisited
https://ddd.uab.cat/record/169453
We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré-Liapunov method to detect centers with purely imaginary eigenvalues can be used to detect nilpotent centers. García, Isaac A.Mon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694532016Existence and uniqueness of limit cycles for generalized -Laplacian Liénard equations
https://ddd.uab.cat/record/169444
The Liénard equation x′′ + f(x)x′ + g(x) = 0 appears as a model in many problems of science and engineering. Since the first half of the 20th century, many papers have appeared providing existence and uniqueness conditions for limit cycles of Li´enard equations. In this paper we extend some of these results for the case of the generalized ϕ-Laplacian Liénard equation, (ϕ(x′))′ + f(x)ψ(x′) + g(x) = 0. This generalization appears when derivations of the equation different from the classical one are considered. In particular, the relativistic van der Pol equation, [x′/p1 − (x′/c)2]′+ µ(x2 − 1)x′ + x = 0, has a unique periodic orbit when µ 6= 0. Pérez-González, SetMon, 23 Jan 2017 15:21:45 GMThttps://ddd.uab.cat/record/1694442016Integrability and algebraic entropy of k-periodic non-autonomous Lyness recurrences
https://ddd.uab.cat/record/150735
This work deals with non-autonomous Lyness type recurrences of the form xn+2 = an + xn+1xn, where {an}n is a k-periodic sequence of positive numbers with minimal period k. We treat such non-autonomous recurrences via the autonomous dynamical system generated by the birational mapping Fak ◦ Fak−1 ◦ · · · ◦ Fa1 where Fa is defined by Fa(x, y) = (y,a+yx). For the cases k ∈ {1, 2, 3, 6} the corresponding mappings have a rational first integral. By calculating the dynamical degree we show that for k = 4 and for k = 5 generically the dynamical system in no longer rationally integrable. We also prove that the only values of k for which the corresponding dynamical system is rationally integrable for all the values of the involved parameters, are k ∈ {1, 2, 3, 6}. Cima, AnnaFri, 06 May 2016 08:59:52 GMThttps://ddd.uab.cat/record/1507352014Limit cycles for discontinuous quadratic differential systems with two zones
https://ddd.uab.cat/record/150734
In this paper we study the maximum number of limit cycles given by the averaging theory of first order for discontinuous differential systems, which can bifurcate from the periodic orbits of the quadratic isochronous centers ˙x = −y + x2, ˙y = x + xy and ˙x = −y + x2 − y2, y˙ = x + 2xy when they are perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity y = 0. Comparing the obtained results for the discontinuous with the results for the continuous quadratic polynomial differential systems, this work shows that the discontinuous systems have at least 3 more limit cycles surrounding the origin than the continuous ones. Llibre, JaumeFri, 06 May 2016 08:59:52 GMThttps://ddd.uab.cat/record/1507342014Bifurcation diagram and stability for a one-parameter family of planar vector fields
https://ddd.uab.cat/record/150733
We consider the 1-parameter family of planar quintic systems, ˙x = y3−x3, y˙ = −x + my5, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter m is in (0. 36, 0. 6). In this paper, using the Bendixon-Dulac theorem, we give a new unified proof of all the previous results, we shrink this to (0. 547, 0. 6), and we prove the hyperbolicity of the limit cycle. We also consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally we answer an open question about the change of stability of the origin for an extension of the above systems. García Saldaña, Johanna DeniseFri, 06 May 2016 08:59:52 GMThttps://ddd.uab.cat/record/1507332014A method for characterizing nilpotent centers
https://ddd.uab.cat/record/150727
To characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest, first for the problem of distinguishing between a focus and a center, and after for studying the bifurcation of limit cycles from it or from its period annulus. We give an effective algorithm in the search of necessary conditions for detecting nilpotent centers based in recent developments. Moreover we survey the last results on this problem and illustrate our approach by means of examples. Giné, JaumeFri, 06 May 2016 08:59:52 GMThttps://ddd.uab.cat/record/1507272014Approximating Mills ratio
https://ddd.uab.cat/record/150707
Consider the Mills ratio f(x) =1 − Φ(x)/φ(x), x ≥ 0, where φ is the density function of the standard Gaussian law and Φ its cumulative distribution. We introduce a general procedure to approximate f on the whole [0, ∞) which allows to prove interesting properties where f is involved. As applications we present a new proof that 1/f is strictly convex, and we give new sharp bounds of f involving rational functions, functions with square roots or exponential terms. Also Chernoff type bounds for the Gaussian Q–function are studied. Gasull i Embid, ArmengolFri, 06 May 2016 08:59:51 GMThttps://ddd.uab.cat/record/1507072014Spatial bi-stacked central configurations formed by two dual regular polyhedra
https://ddd.uab.cat/record/150695
In this paper we prove the existence of two new families of spatial stacked central configurations, one consisting of eight equal masses on the vertices of a cube and six equal masses on the vertices of a regular octahedron, and the other one consisting of twenty masses at the vertices of a regular dodecahedron and twelve masses at the vertices of a regular icosahedron. The masses on the two different polyhedra are in general different. We note that the cube and the octahedron, the dodecahedron and the icosahedron are dual regular polyhedra. The tetrahedron is itself dual. There are also spatial stacked central configurations formed by two tetrahedra, one and its dual. Corbera Subirana, MontserratFri, 06 May 2016 08:59:50 GMThttps://ddd.uab.cat/record/1506952014A simple solution of some composition conjectures for Abel equations
https://ddd.uab.cat/record/150650
Trigonometric Abel differential equations appear when one studies the number of limit cycles and the center-focus problem for certain families of planar polynomial systems. Inside trigonometric Abel equations there is a class of centers, the composition centers, that have been widely studied during these last years. We fully characterize this type of centers. They are given by the couples of trigonometric polynomials for which all the generalized moments vanish and also coincide with the strongly persistent centers. This result solves the so called Composition Conjecture for trigonometric Abel differential equations. We also prove the equivalent version of this result for Abel equations with polynomial coefficients. Cima, AnnaFri, 06 May 2016 08:54:16 GMThttps://ddd.uab.cat/record/1506502013