Integrability of a linear center perturbed by a fourth degree homogeneous polynomial

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

Integrability of a linear center perturbed by a fifth degree homogeneous polynomial

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

Integrability of a linear center perturbed by a fifth degree homogeneous polynomial

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

Integrability of a linear center perturbed by a fourth degree homogeneous polynomial

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

The null divergence factor

Let $(P, Q)$ be a $C^{1}$ vector field defined in a open subset $U \subset R^{2}$. We call a null divergence factor a $C^{1}$ solution $V(x, y)$ of the equation $P \frac{\partial V}{\partial x} + Q \frac{\partial V}{\partial y} = \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}\right) \, V$. 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 paper we 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

Isochronicity conditions for some planar polynomial systems II

We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $$\dot x=-y+A(x,y),\;\dot y=x+B(x,y),$$ where $A,\;B\in \mathbb{R}[x,y]$, which can be reduced to the Li\'enard type equation. When $deg(A)\leq 4$ and $deg(B) \leq 4$, using the so-called C-algorithm we found $36$ new families of isochronous centers. When the Urabe function $h=0$ we provide an explicit general formula for linearization. This paper is a direct continuation of \cite{BoussaadaChouikhaStrelcyn2010} but can be read independantly

The null divergence factor

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

Microfiltracion apical de pasta preparadas en base a hidroxido de calcio con diferentes vehiculos, para apexificacion. Estudio in vitro.

61 p.El objetivo de este estudio In Vitro fue determinar la microfiltración apical que sufre el Hidróxido de calcio aplicado con diferentes vehículos. Para esto se recolectaron 70 dientes uniradiculares (comprobado radiográficamente), luego se les cortaron las coronas y 1mm del 1/3 apical, luego fueron instrumentados con fresas Peeso nº II, irrigando con suero. Luego de esto se pincelaron todas las muestras con esmalte de uñas desde 2mm del ápice hacia coronal. Las muestras fueron divididas en 7 grupos de 10 ejemplares cada uno, conformando 7 grupos con los siguientes rellenos: 10 muestra rellenas con vidrio ionómero( control-),10 muestra no rellenas (control +),10 muestras rellenas con hidróxido de calcio sólo,10 muestras rellenas con pasta de hidróxido de calcio más suero,10 muestra rellenas con pasta de hidróxido de calcio más propilenglicol,10 muestra rellenas con pasta de hidróxido de calcio más agua destilada y 10 muestra rellenas con pasta de hidróxido de calcio más anestesia. Luego se sellaron en coronal con Chemfil, para posteriormente ser suspendidas en laminas de plumavit rotuladas por grupo, en placas petri que contenían azul de metileno y estaban dentro de un baño termoregulado a 37ºC , 100% de humedad por 24 hrs. Luego se hicieron 2 cortes longitudinales a cada raíz, se separaron las mitades y se cuantificó la microfiltración con una lupa del articulador Panadent calibrada en décimas de mm. Los datos fueron analizados estadísticamente usando el test ANOVA , test de Tukey y Duncan con un nivel de significancia de p≤0.05, encontrándose diferencias significativas en la micofiltración apical de las pastas en base a hidróxido de calcio con vehículos acuosos y viscosos. Se encontró una diferencia altamente significativa entre las pasta en que se uso como vehículo agua destilada, anestesia y suero, que presentaron filtraciones promedio 11,73;11,99; 10,25 mm. respectivamente, y el grupo que utilizó un vehículo viscoso que fue el propilenglicol cuyo promedio fue 6,67mm

Number and Amplitude of Limit Cycles emerging from {\it Topologically Equivalent} Perturbed Centers

We consider three examples of weekly perturbed centers which do not have {\it geometrical equivalence}: a linear center, a degenerate center and a non-hamiltonian center. In each case the number and amplitude of the limit cycles emerging from the period annulus are calculated following the same strategy: we reduce of all of them to locally equivalent perturbed integrable systems of the form: $dH(x,y)+\epsilon(f(x,y)dy-g(x,y)dx)=0$, with $H(x,y)={1/2}(x^2+y^2)$. This reduction allows us to find the Melnikov function, $M(h)=\int_{H=h}fdy-gdx$, associated to each particular problem. We obtain the information on the bifurcation curves of the limit cycles by solving explicitly the equation $M(h)=0$ in each case.Comment: 17 pages, 0 figure