Periodic Solutions of a Class of Second-Order Differential Equation

We study the periodic solutions of the second-order differential equations of the form where the functions, , and are periodic of period in the variable t.

PHASE PORTRAITS OF THE LESLIE-GOWER SYSTEM

In this paper we characterize the phase portraits of the Leslie-Gower model for competition among species.We give the complete description of their phase portraits in the Poincarédisc(i.e.,in the compactification of R^(2) adding the circle S1 of the infinity)modulo topological equivalence.It is well-known that the equilibrium point of the Leslie-Gower model in the interior of the positive quadrant is a global attractor in this open quadrant,and in this paper we characterize where the orbits attracted by this equilibrium born.

Integrability of Hamiltonian Systems with Two Degrees of Freedom and Homogenous Potential of Degree Zero

We provide necessary conditions in order that the Hamiltonian systems with Hamiltonian ,?and one of the following potentials ?are integrable in the Liouville sense.

Simultaneous Periodic Orbits Bifurcating from Two Zero-Hopf Equilibria in a Tritrophic Food Chain Model

We are interested in the coexistence of three species forming a tritrophic food chain model. Considering a linear grow for the lowest trophic species or prey, and a type III Holling functional response for the middle and highest trophic species (first and second predator respectively). We prove that this model exhibits two small amplitud periodic solutions bifurcating simultaneously each one from one of the two zero-Hopf equilibrium points that the model has adequate values of its parameters. As far as we know, this is the first time that the phenomena appear in the literature related with food chain models.

具有全局中心的平面多项式Hamilton系统

本文给出平面多项式Hamilton系统具有全局中心的充要条件,这刻画出整个实平面有叶层结构,即它被该系统对应的多项式Hamilton函数的水平集(即实代数曲线)分层,而每一层对应的实代数曲线都是卵形线(即紧闭分枝).进一步地,本文证明,对于任意正整数n和任意一种类型中心(即线性型、幂零型和退化型),都存在2n+1次平面多项式Hamilton系统使其具有该类型的全局中心,而2n次平面多项式Hamilton系统没有全局中心.作为特殊情形,本文也给出平面三次多项式Hamilton系统具有退化型中心的充要条件.

一类pitchfork分支的全局相图

本文研究二次微分系统dt/dr=y^2?y?x,dy/dt=x^2?μx?y,其中μ∈R.该系统是刻画pitchfork分支的重要例子.本文给出该系统在Poincare圆盘上的全局相图。

Liouvillian and Analytic Integrability of the Quadratic Vector Fields Having an Invariant Ellipse

We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2＋y2-1＋y（ax＋by＋c）,y=x（ax＋by＋c）,and the ellipse becomes x2＋y2=1.We prove that（i） this quadratic system is analytic integrable if and only if a=0;（ii） if x2＋y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2＋y2=1 is not a limit cycle;and（iii） if x2＋y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.

NEW FAMILIES OF CENTERS AND LIMIT CYCLES FOR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH HOMOGENEOUS NONLINEARITIES

We consider the class of polynomial differential equations x = -y+Pn(x,y), y = x + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. Inside this class we identify a new subclass of systems having a center at the origin. We show that this subclass contains at least two subfamilies of isochro-nous centers. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centers when we perturb them inside the class of all polynomial differential systems of the above form. In particular, we present a function whose simple zeros correspond to the limit cycles vvhich bifurcate from the periodic orbits of Hamiltonian systems.

On the Darboux Integrability of the Hindmarsh–Rose Burster

We study the Hindmarsh-Rose burster which can be described by the differential system x^·=y-x^3＋bx^2＋I-z,y^·=1-5x^2-y,z^·=μ（s（x-x0）-z）where b, I, μ, s, x0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist.

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 ＋ y^2 ＋ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.

Weierstrass Integrability of Complex Differential Equations

We characterize the complex differential equations of the form dy/dx=a_(n)(x)y^)n_+a_(n-1)(x)y^(n-1)+…+a_(1)(x)y+a_(0)(x) where a_(j)(x) are meromorphic functions in the variable x for j = 0,..., n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.

Periodic Orbits for Some Systems of Delay Differential Equations

We provide sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay. We extend Kaplan-Yorke＇s method for finding periodic orbits from a delay differential equation with several delays to a system of delay differential equations with a unique delay.