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A new sufficient condition in order that the real Jacobian conjecture in R2 holds
https://ddd.uab.cat/record/239777
Let F = (f,g) : R2 → R2 be a polynomial map such that det(DF(x,y)) is nowhere zero and F(0,0) = (0,0). In this work we give a new sufficient condition for the injectivity of F. We also state a conjecture when det(DF(x,y)) = constant ≠ 0 and F(0,0) = (0,0) equivalent to the Jacobian conjecture. Giné, JaumeThu, 29 Apr 2021 11:48:26 GMThttps://ddd.uab.cat/record/2397772021Lower bounds for the local cyclicity for families of centers
https://ddd.uab.cat/record/236672
In this paper we are interested on how the local cyclicity of a family of centers depends on the parameters. This fact, was pointed out in [21], to prove that there exists a family of cubic centers, labeled by CD12 31 in [25], with more local cyclicity than expected. In this family there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some important gaps that we correct here. We take the advantage of better understanding of the bifurcation phenomenon in non generic cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorfic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields. Giné, JaumeFri, 12 Feb 2021 09:45:33 GMThttps://ddd.uab.cat/record/2366722021Vanishing set of inverse Jacobi multipliers and attractor/repeller sets
https://ddd.uab.cat/record/236646
In this paper, we study conditions under which the zero-set of the inverse Jacobi multiplier of a smooth vector field contains its attractor/repeller compact sets. The work generalizes previous results focusing on sink singularities, orbitally asymptotic limit cycles, and monodromic attractor graphics. Taking different flows on the torus and the sphere as canonical examples of attractor/repeller sets with different topologies, several examples are constructed illustrating the results presented. García, Isaac A.Fri, 12 Feb 2021 09:45:30 GMThttps://ddd.uab.cat/record/2366462021Formal Weierstrass nonintegrability criterion for some classes of polynomial differential systems in C²
https://ddd.uab.cat/record/228108
In this paper, we present a criterion for determining the formal Weierstrass nonintegrability of some polynomial differential systems in the plane C². The criterion uses solutions of the form y = f(x) of the differential system in the plane and their associated cofactors, where f(x) is a formal power series. In particular, the criterion provides the necessary conditions in order that some polynomial differential systems in C² would be formal Weierstrass integrable. Inside this class there exist non-Liouvillian integrable systems. Finally we extend the theory of formal Weierstrass integrability to Puiseux Weierstrass integrability. Giné, JaumeWed, 15 Jul 2020 12:00:35 GMThttps://ddd.uab.cat/record/2281082020On the mechanisms for producing linear type centers in polynomial differential systems
https://ddd.uab.cat/record/222633
In this paper we study the diﬀerent mechanisms that give rise to linear type centers for polynomial diﬀerential systems. The known mechanisms are the algebraic reversibility and the Liouville integrability. In this paper are discussed such mechanisms and established some open questions. The known mechanisms for the nilpotent and degenerate centers are also summarized. Giné, JaumeFri, 29 May 2020 18:25:45 GMThttps://ddd.uab.cat/record/2226332018Chiellini Hamiltonian Liénard differential systems
https://ddd.uab.cat/record/222609
We characterize the centers of the Chiellini Hamiltonian Liénard second-order differential equations x' = y, y' = −f(x)y−g(x) where g(x) = f(x)(k−α(1 + α)Rf(x)dx) with α,k ∈ R. Moreover we study the phase portraits in the Poincaré disk of these systems when f(x) is linear. Giné, JaumeTue, 26 May 2020 17:13:07 GMThttps://ddd.uab.cat/record/2226092019Strongly formal weierstrass non-integrability for polynomial differential systems in C2
https://ddd.uab.cat/record/221359
Recently a criterion has been given for determining the weakly formal Weierstrass non-integrability of polynomial differential systems in C2. Here we extend this criterion for determining the strongly formal Weierstrass non-integrability which includes the weakly formal Weierstrass non-integrability of polynomial differential systems in C2. The criterion is based on the solutions of the form y = f(x) with f(x) ∈ C[[x]] of the differential system whose integrability we are studying. The results are applied to a differential system that contains the famous force-free Duffing and the Duffing-Van der Pol oscillators. Giné, JaumeWed, 15 Apr 2020 15:23:10 GMThttps://ddd.uab.cat/record/2213592020Highest weak focus order for trigonometric Liénard equations
https://ddd.uab.cat/record/221357
Given a planar analytic differential equation with a critical point which is a weak focus of order k, it is well known that at most k limit cycles can bifurcate from it. Moreover, in case of analytic Liénard differential equations this order can be computed as one half of the multiplicity of an associated planar analytic map. By using this approach, we can give an upper bound of the maximum order of the weak focus of pure trigonometric Liénard equations only in terms of the degrees of the involved trigonometric polynomials. Our result extends to this trigonometric Liénard case a similar result known for polynomial Liénard equations. Gasull i Embid, ArmengolWed, 15 Apr 2020 15:23:09 GMThttps://ddd.uab.cat/record/2213572019The center problem for Z2-symmetric nilpotent vector fields
https://ddd.uab.cat/record/221353
We say that a polynomial differential system x˙=P(x,y), y˙=Q(x,y) having the origin as a singular point is Z-symmetric if P(−x,−y)=−P(x,y) and Q(−x,−y)=−Q(x,y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C first integral. However these two kinds of nilpotent centers are not characterized for different families of differential systems. Here we prove that the origin of any Z-symmetric system is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x,y)=y+⋯, where the dots denote terms of degree higher than two. Algaba, AntonioWed, 15 Apr 2020 15:23:09 GMThttps://ddd.uab.cat/record/2213532018Orbitally universal centers
https://ddd.uab.cat/record/221326
In this paper we define when a polynomial differential system is orbitally universal and we show the relevance of this notion in the classical center problem, i. e. in the problem of distinguishing between a focus and a center. Algaba, AntonioWed, 15 Apr 2020 15:23:06 GMThttps://ddd.uab.cat/record/2213262020Simultaneity of centres in Zq-equivariant systems
https://ddd.uab.cat/record/221302
We study the simultaneous existence of centres for two families of planar Zq-equivariant systems. First, we give a short review about Zq-equivariant systems. Next, we present the necessary and sufficient conditions for the simultaneous existence of centres for a Z2-equivariant cubic system and for a Z2- equivariant quintic system. Giné, JaumeWed, 15 Apr 2020 15:23:04 GMThttps://ddd.uab.cat/record/2213022018The 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/1694532016