Dipòsit Digital de Documents de la UAB
https://ddd.uab.cat
Dipòsit Digital de Documents de la UAB latest documentscaWed, 21 Aug 2019 18:24:26 GMTInvenio 1.1.6ddd.bib@uab.cat36022125https://ddd.uab.cat/img/uab/ddd_logo.gifDipòsit Digital de Documents de la UAB
https://ddd.uab.cat
Search Search this site:p
https://ddd.uab.cat/search
Limit cycles for discontinuous planar piecewise linear differential systems separated by an algebraic curve
https://ddd.uab.cat/record/204393
We study how to change the maximum number of limit cycles of the discontinuous piecewise linear differential systems with only two pieces in function of the degree of the discontinuity of the algebraic curve between the two linear differential systems. These discontinuous differential systems appear frequently in applied sciences. Llibre, JaumeThu, 16 May 2019 13:36:18 GMThttps://ddd.uab.cat/record/2043932019The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems
https://ddd.uab.cat/record/204391
We provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane. Llibre, JaumeThu, 16 May 2019 13:36:18 GMThttps://ddd.uab.cat/record/2043912019Limit 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/1993222018The number of polynomial solutions of polynomial Riccati equations
https://ddd.uab.cat/record/169484
Consider real or complex polynomial Riccati differential equations a(x) y=b_0(x) b_1(x)y b_2(x)y^2 with all the involved functions being polynomials of degree at most . We prove that the maximum number of polynomial solutions is 1 (resp. 2) when 1 (resp. =0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2 (resp. 3) when 2 (resp. =1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain. Gasull i Embid, ArmengolMon, 23 Jan 2017 15:21:47 GMThttps://ddd.uab.cat/record/1694842016Averaging 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/1694812016Limit cycles of linear vectors on manifolds
https://ddd.uab.cat/record/169478
It is well known that linear vector fields on the manifold R^n cannot have limit cycles, but this is not the case for linear vector fields on other manifolds. We study the periodic orbits of linear vector fields on different manifolds, and motivate and present an open problem on the number of limit cycles of linear vector fields on a class of C^1 connected manifold. Llibre, JaumeMon, 23 Jan 2017 15:21:47 GMThttps://ddd.uab.cat/record/1694782016Liouvillian integrability versus Darboux polynomials
https://ddd.uab.cat/record/169448
In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system x ̇ = y, y ̇ = − f (x)y − g(x) with deg f > deg g is not Liouvillian integrable. Llibre, JaumeMon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694482016Darboux integrability and Algebraic limit cycles for a class of polynomial differential Systems
https://ddd.uab.cat/record/150749
This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems ˙x = λx − y + Pn+1(x, y) + xF2n(x, y), ˙y = x + λy + Qn+1(x, y)+yF2n(x, y), where Pi(x, y), Qi(x, y) and Fi(x, y) are homogeneous polynomials of degree i. Inside this class we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 we present necessary and sufficient conditions for being Darboux integrable. Cao, JinlongFri, 06 May 2016 08:59:53 GMThttps://ddd.uab.cat/record/1507492014On the analytic integrability of the 5-dimensional Lorenz system for the gravitiy-wave activity
https://ddd.uab.cat/record/150747
The 5-dimensional Lorenz system for the coupled Rosby and gravity waves has exactly two independent analytic first integrals. Llibre, JaumeFri, 06 May 2016 08:59:53 GMThttps://ddd.uab.cat/record/1507472014Local Darboux first integrals of analytic differential systems
https://ddd.uab.cat/record/150739
In this paper we discuss local and formal Darboux first integrals of analytic differential systems, using the theory of Poincaré Dulac normal forms. We study the effect of local Darboux integrability on analytic normalization. Moreover we determine local restrictions on classical Darboux integrability of polynomial systems. Llibre, JaumeFri, 06 May 2016 08:59:52 GMThttps://ddd.uab.cat/record/1507392014On the limit cycles of the polynomial differential systems with a linear node and homogeneous nonlinearities
https://ddd.uab.cat/record/150722
We consider the class of polynomial differential equations ˙x = λx + Pn(x, y), y˙ = µy + Qn(x, y) in R2 where Pn(x, y) and Qn(x, y) are homogeneous polynomials of degree n > 1 and λ 6= µ, i. e. the class of polynomial differential systems with a linear node with different eigenvalues and homogeneous nonlinearities. For this class of polynomial differential equations we study the existence and non–existence of limit cycles surrounding the node localized at the origin of coordinates. Llibre, JaumeFri, 06 May 2016 08:59:51 GMThttps://ddd.uab.cat/record/1507222014On the Darboux integrability of the polynomial differential systems
https://ddd.uab.cat/record/150526
This is a survey on recent results on the Darboux integrability of polynomial vector fields in Rn or Cn with n ≥ 2. We also provide an open question and some applications based on the existence of such first integrals. Llibre, JaumeFri, 06 May 2016 08:49:22 GMThttps://ddd.uab.cat/record/1505262012Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds
https://ddd.uab.cat/record/150474
Liu, FeiFri, 06 May 2016 08:30:51 GMThttps://ddd.uab.cat/record/1504742011On the Hopf-zero bifurcation of the Michelson system
https://ddd.uab.cat/record/150468
Applying a new result for studying the periodic orbits of a differential system via the averaging theory, we provide the first analytic proof on the existence of a Hopf–zero bifurcation for the Michelson system x2 x˙ = y, y˙ = z, z˙ = c2 − y − , 2 at c = 0. Moreover our method estimates the shape of this periodic orbit in function of c > 0 sufficiently small. Llibre, JaumeFri, 06 May 2016 08:30:51 GMThttps://ddd.uab.cat/record/1504682011Generalized rational first integrals of analytic differential systems
https://ddd.uab.cat/record/150420
In this paper we mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances. The main results extend some of the previous related ones, for instance the classical Poincaré's one [16], the Furta's one [8], part of Chen et al's ones [4], and the Shi's one [18]. The key point in the proof of our main results is that functionally independence of generalized rational functions implies the functionally independence of their lowest order rational homogeneous terms. Cong, WangFri, 06 May 2016 08:30:48 GMThttps://ddd.uab.cat/record/1504202011On polynomial integrability of the Euler equations on so(4)
https://ddd.uab.cat/record/145370
In this paper we prove that the Euler equations on the Lie algebra so(4) with a diagonal quadratic Hamiltonian either satisfy the Manakov condition, or have at most four functionally independent polynomial first integrals. Also we provide necessary conditions in order that the Euler equations can have a fourth functionally independent polynomial first integral via the Kowalevsky exponents. Llibre, JaumeTue, 12 Jan 2016 16:38:12 GMThttps://ddd.uab.cat/record/1453702015The Completely Integrable Differential Systems are Essentially Linear Differential Systems
https://ddd.uab.cat/record/145335
Let ˙x = f(x) be a C k autonomous differential system with k ∈ N ∪ {∞, ω} defined in an open subset Ω of R n. Assume that the system ˙x = f(x) is C r completely integrable, i. e. there exist n−1 functionally independent first integrals of class C r with 2 ≤ r ≤ k. If the divergence of system ˙x = f(x) is non–identically zero, then any Jacobian multiplier is functionally independent of the n − 1 first integrals. Moreover the system ˙x = f(x) is C r−1 orbitally equivalent to the linear differential system ˙y = y in a full Lebesgue measure subset of Ω. For Darboux and polynomial integrable polynomial differential systems we characterize their type of Jacobian multipliers. Llibre, JaumeTue, 12 Jan 2016 16:38:10 GMThttps://ddd.uab.cat/record/1453352015Vector fields with homogeneous nonlinearities and many limit cycles
https://ddd.uab.cat/record/145332
Consider planar real polynomial differential equations of the form x'=Lx X_n(x), where x=(x,y) R^2, L is a 2×2 matrix and X_n is a homogeneous vector field of degree n > 1. Most known results about these equations, valid for infinitely many n, deal with the case where the origin is a focus or a node and give either non-existence of limit cycles or upper bounds of one or two limit cycles surrounding the origin. In this paper we improve some of these results and moreover we show that for n 3 odd there are equations of this form having at least (n 1)/2 limit cycles surrounding the origin. Our results include cases where the origin is a focus, a node, a saddle or a nilpotent singularity. We also discuss a mechanism for the bifurcation of limit cycles from infinity. Gasull i Embid, ArmengolTue, 12 Jan 2016 16:38:10 GMThttps://ddd.uab.cat/record/1453322015On the limit cycles of linear differential systems with homogeneous nonlinearities
https://ddd.uab.cat/record/145301
We consider the class of polynomial differential systems of the form x'= x - y P_n (x, y), y'=x y Q_n (x, y), where P_n and Q_n are homogeneous polynomials of degree n. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence. Llibre, JaumeTue, 12 Jan 2016 16:38:08 GMThttps://ddd.uab.cat/record/1453012015Rational first integrals in the Darboux theory of integrability in Cn
https://ddd.uab.cat/record/44096
"Vegeu el resum a l'inici del document del fitxer adjunt". Llibre, JaumeMon, 13 Jul 2009 13:09:07 GMThttps://ddd.uab.cat/record/44096Centre de Recerca Matemàtica2008Hopf bifurcation in higher dimensional differential systems via the averaging method
https://ddd.uab.cat/record/44095
"vegeu el resum a l'inici del document del fitxer adjunt". Llibre, JaumeMon, 13 Jul 2009 13:09:07 GMThttps://ddd.uab.cat/record/44095Centre de Recerca Matemàtica2008Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds
https://ddd.uab.cat/record/44094
"Vegeu el resum a l'inici del document del fitxer adjunt". Liu, FeiMon, 13 Jul 2009 13:09:07 GMThttps://ddd.uab.cat/record/44094Centre de Recerca Matemàtica2008