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The cubic polynomial differential systems with two circles as algebraic limit cycles
https://ddd.uab.cat/record/199332
In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles. Giné, JaumeMon, 12 Nov 2018 12:11:49 GMThttps://ddd.uab.cat/record/1993322018Centers for the Kukles homogeneous systems with even degree
https://ddd.uab.cat/record/182729
Giné, JaumeThu, 07 Dec 2017 14:53:38 GMThttps://ddd.uab.cat/record/1827292017Centers for generalized quintic polynominal differential systems
https://ddd.uab.cat/record/182538
Giné, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825382017Center Problem for trigonometric Liénard systems
https://ddd.uab.cat/record/182524
We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Liénard systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems and the characterization of some subfields of the quotient field of the ring of trigonometric polynomials. Our results are also applied to some particular subfamilies of planar trigonometric Liénard systems. The results obtained are reminiscent of the ones for planar polynomial Liénard systems but the proofs are different. Gasull i Embid, ArmengolTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825242017On the Integrability of Liénard systems with a strong saddle
https://ddd.uab.cat/record/182516
We study the local analytic integrability for real Li\'enard systems, x=y-F(x), y= x, with F(0)=0 but F'(0)0, which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the [p:-q] resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the [p:-q] resonant saddle into a strong saddle. Giné, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825162017Integrability conditions of a resonant saddle in generalized Liénard-like complex polynomial differential systems
https://ddd.uab.cat/record/182512
We consider a complex differential system with a resonant saddle at the origin. We compute the resonant saddle quantities and using Gröbner bases we find the integrability conditions for such systems up to a certain degree. We also establish a conjecture about the integrability conditions for such systems when they have arbitrary degree. Giné, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825122017Integrability of Liénard systems with a weak saddle
https://ddd.uab.cat/record/182506
We characterize the local analytic integrability of weak saddles for complex Liénard systems. Gasull i Embid, ArmengolTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1825062017Centers of weight-homogeneous polynomial vector fields on the plane
https://ddd.uab.cat/record/182495
We characterize all centers of a planar weight-homogeneous polynomial vector fields. Moreover we classify all centers of a planar weight-homogeneous polynomial vector fields of degrees 6 and 7. Giné, JaumeTue, 28 Nov 2017 07:46:28 GMThttps://ddd.uab.cat/record/1824952017Analytic reducibility of nondegenerate centers: Cherkas systems
https://ddd.uab.cat/record/169493
In this paper we study the center problem for polynomial differential systems and we prove that any center of an analytic differential system is analytically reducible. We also study the centers for the Cherkas polynomial differential systems x˙ = y, y˙ = P0(x) + P1(x)y + P2(x)y2, where Pi(x) are polynomials of degree n, P0(0) = 0 and P′0(0) < 0. Computing the focal values we find the center conditions for such systems for degree 3, and using modular arithmetics for degree 4. Finally we do a conjecture about the center conditions for Cherkas polynomial differential systems of degree n. Giné, JaumeMon, 23 Jan 2017 15:21:48 GMThttps://ddd.uab.cat/record/1694932016Averaging methods of arbitrary order, periodic solutions and integrability
https://ddd.uab.cat/record/169481
In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods. Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability. For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number N such that if its first N averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for n = 2. Furthermore, in a punctured neighborhood of the origin, the system is C^ completely integrable for n > 2 provided that each periodic orbit has a trivial holonomy. Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities. Giné, JaumeMon, 23 Jan 2017 15:21:47 GMThttps://ddd.uab.cat/record/1694812016A 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/1505182012