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A sufficient condition in order that the Real Jacobian Conjecture in R^2 holds
https://ddd.uab.cat/record/169471
Let F=(f,g):\R^2\R^2 be a polynomial map such that DF(x) is different from zero for all x\R^2 and F(0,0) = (0,0). We prove that for the injectivity of F it is sufficient to assume that the higher homogeneous terms of the polynomials ff_x g g_x and f f_y g g_y do not have real linear factors in common. Braun, FranciscoMon, 23 Jan 2017 15:21:47 GMThttps://ddd.uab.cat/record/1694712016Reversible 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/1694692016Center problem for systems with two monomial nonlinearities
https://ddd.uab.cat/record/169455
We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees. Gasull i Embid, ArmengolMon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694552016Analytic 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/1694532016A 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/1507272014Polynomial and rational first integrals for planar homogeneous polynomial differential systems
https://ddd.uab.cat/record/150716
In this paper we find necessary and sufficient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic homogeneous polynomial differential systems. Giné, JaumeFri, 06 May 2016 08:59:51 GMThttps://ddd.uab.cat/record/1507162014Periodic solutions for nonlinear differential systems: The second order bifurcation function
https://ddd.uab.cat/record/150714
We are concerned here with the classical problem of Poincaré of persistence of periodic solutions under small perturbations. The main contribution of this work is to give the expression of the second order bifurcation function in more general hypotheses than the ones already existing in the literature. We illustrate our main result constructing a second order bifurcation function for the perturbed symmetric Euler top. Buica, AdrianaFri, 06 May 2016 08:59:51 GMThttps://ddd.uab.cat/record/1507142014Centers for a class of generalized quintic polynomial differential systems
https://ddd.uab.cat/record/150687
We classify the centers of the polynomial differential systems in R2 of degree d ≥ 5 odd that in complex notation writes as z˙ = iz + (zz¯)d−5/2 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5), where A, B, C, D, E, F ∈ C and either A = Re(D) = 0, or A = Im(D) = 0, or Re(A) = D = 0, or Im(A) = D = 0. Giné, JaumeFri, 06 May 2016 08:59:49 GMThttps://ddd.uab.cat/record/1506872014Universal centers and composition conditions
https://ddd.uab.cat/record/150654
In this paper we characterize the universal centers of the ordinary differential equations dρ/dθ =∑∞i=1 ai(θ)ρi+1, where ai(θ) are trigonometric polynomials, in terms of the composition conditions. These centers are closely related with the classical Poincaré center problem for planar analytic differential systems. Additionally we show that the notion of universal center is not invariant under changes of variables, and we also provide different families of universal centers. Finally we characterize all the universal centers for the quadratic polynomial differential systems. Giné, JaumeFri, 06 May 2016 08:54:16 GMThttps://ddd.uab.cat/record/1506542013Averaging theory at any order for computing periodic orbits
https://ddd.uab.cat/record/150627
We provide an explicit expression for the solutions of the perturbed first order differential equations. Using it we give an averaging theory at any order in ε for the following two kinds of analytic differential equations dx/dθ =Xk≥1εk Fk(θ, x), dx/dθ =Xk≥0εk Fk(θ, x). We apply these results for studying the limit cycles of planar polynomial differential systems after passing them to polar coordinates. Giné, JaumeFri, 06 May 2016 08:54:15 GMThttps://ddd.uab.cat/record/1506272013A note on Liouvillian first integrals and invariant algebraic curves
https://ddd.uab.cat/record/150613
In this paper we study the existence and non-existence of finite invariant algebraic curves for complex planar polynomial differential system having a Liouvillian first integral. Giné, JaumeFri, 06 May 2016 08:54:15 GMThttps://ddd.uab.cat/record/1506132013Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential sytems
https://ddd.uab.cat/record/150596
In this paper we find necessary and sufficient conditions in order that a planar quasi–homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasi–homogeneous polynomial differential system can be transformed into a differential system of the form u˙ = uf(v), ˙v = g(v) with f(v) and g(v) polynomials, and vice versa. Giné, JaumeFri, 06 May 2016 08:54:14 GMThttps://ddd.uab.cat/record/1505962013On the extensions of the Darboux theory of integrability
https://ddd.uab.cat/record/150592
Recently some extensions of the classical Darboux integrability theory to autonomous and non-autonomous polynomial vector fields have been done. The classical Darboux integrability theory and its recent extensions are based on the existence of algebraic invariant hypersurfaces. However the algebraicity of the invariant hypersurfaces is not necessary and the unique necessary condition is the algebraicity of the cofactors associated to them. In this note it is established a more general extension of the classical Darboux integrability theory. Giné, JaumeFri, 06 May 2016 08:54:14 GMThttps://ddd.uab.cat/record/1505922013On the center conditions for analytic monodromic degenerate singularities
https://ddd.uab.cat/record/150541
In this paper we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields can be applied independently that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allows to study monodromic degenerate singularities with characteristic directions. Giné, JaumeFri, 06 May 2016 08:49:23 GMThttps://ddd.uab.cat/record/1505412012A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter
https://ddd.uab.cat/record/150518
We deal with nonlinear T–periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation. In the end two applications are done. The key tool for proving the main result is the Lyapunov–Schmidt reduction method applied to the T–Poincaré–Andronov mapping. Buica, AdrianaFri, 06 May 2016 08:49:22 GMThttps://ddd.uab.cat/record/1505182012On Liouvillian integrability of the first-order polynomial ordinary differential equations
https://ddd.uab.cat/record/150496
Giné, JaumeFri, 06 May 2016 08:49:21 GMThttps://ddd.uab.cat/record/1504962012On the polynomial limit cycles of polynomial differential equations
https://ddd.uab.cat/record/150470
Giné, JaumeFri, 06 May 2016 08:30:51 GMThttps://ddd.uab.cat/record/1504702011Weierstrass integrability in Liénard differential systems
https://ddd.uab.cat/record/150438
In this work we study the Liénard differential systems that admit a Weierstrass first integral or a Weierstrass inverse integrating factor. Giné, JaumeFri, 06 May 2016 08:30:49 GMThttps://ddd.uab.cat/record/1504382011On the planar integrable differential systems
https://ddd.uab.cat/record/150422
Under very general assumptions we prove that the planar differential systems having a first integral are essentially the linear differential systems u˙ = u, ˙v = v. Additionally such systems always have a Lie symmetry. We improve these results for polynomial differential systems defined in R2 or C2. Giné, JaumeFri, 06 May 2016 08:30:48 GMThttps://ddd.uab.cat/record/1504222011Centers and isochronous centers for generalized quintic systems
https://ddd.uab.cat/record/145340
In this paper we classify the centers and the isochronous centers of certain polynomial differential systems in R2 of degree d ≥ 5 odd that incomplex notation can be written as z˙ = (λ + i)z + (zz¯)d−52 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5),where λ ∈ R and A, B, C, D, E, F ∈ C. Note that if d = 5 we obtain the full class of polynomial differential systems of the form a linear system with homogeneous polynomial nonlinearities of degree 5. Our study uses algorithms of computational algebra based on the Groebner basis theory and modular arithmetics for simplifying the computations. Giné, JaumeTue, 12 Jan 2016 16:38:10 GMThttps://ddd.uab.cat/record/1453402015Centers for the Kukles homogeneous systems with odd degree
https://ddd.uab.cat/record/145329
For the polynomial differential system x ̇ = −y, y ̇ = x Q n (x, y), where Q n (x, y) is a homogeneous polynomial of degree n there are the following two conjectures raised in 1999. (1) Is it true that the previous system for n 2 has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We prove both conjectures for all n odd. Giné, JaumeTue, 12 Jan 2016 16:38:09 GMThttps://ddd.uab.cat/record/1453292015Limit cycles bifurcating from planar polynomial quasi-homogeneous centers
https://ddd.uab.cat/record/145282
In this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center, which can be obtained using first order averaging method. This result improves the upper bounds given in [7]. Giné, JaumeTue, 12 Jan 2016 16:38:06 GMThttps://ddd.uab.cat/record/1452822015Integrability of a linear center perturbed by a fourth degree homogeneous polynomial
https://ddd.uab.cat/record/62394
In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones. Chavarriga Soriano, JavierFri, 17 Sep 2010 13:38:08 GMThttps://ddd.uab.cat/record/623941996Integrability of a linear center perturbed by a fifth degree homogeneous polynomial
https://ddd.uab.cat/record/13061
In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones. Chavarriga Soriano, JavierFri, 24 Nov 2006 18:10:24 GMThttps://ddd.uab.cat/record/130611997The null divergence factor
https://ddd.uab.cat/record/12917
Let (P,Q) be a C1 vectorfield defined in a open subset U ⊂ R2. We call a null divergence factor a C1 solution V (x, y) of the equation P ∂V ∂x + Q∂V ∂y = ∂P ∂x + ∂Q ∂y V . In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paperw e show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method to generate integrable systems. Chavarriga Soriano, JavierWed, 22 Nov 2006 16:01:53 GMThttps://ddd.uab.cat/record/129171997