Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems
Jaume Giné Maite Grau Jaume Llibre
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 $\dot{u} \, = \, u f(v)$, $\dot{v} \, = \, g(v)$ with $f(v)$ and $g(v)$ polynomials, and vice versa.
keywords: polynomial first integral rational first integral. Quasi--homogeneous polynomial differential equations integrability problem
Essential perturbations of polynomial vector fields with a period annulus
Adriana Buică Jaume Giné Maite Grau
Chicone--Jacobs and Iliev found the essential perturbations of quadratic systems when considering the problem of finding the cyclicity of a period annulus. Given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which the expressions of its Poincaré--Liapunov constants are known, we give the structure of its $k$-th Melnikov function. This allows to find the essential perturbations in concrete cases. We study here in detail the essential perturbations for all the centers of the differential systems \begin{eqnarray} \dot{x} = -y + P_{\rm d}(x,y), \quad \dot{y} = x + Q_{d}(x,y), \end{eqnarray} where $P_d$ and $Q_d$ are homogeneous polynomials of degree $d$, for $ d=2$ and $ d=3$.
keywords: bifurcation. non-degenerated center Melnikov functions essential perturbation Cyclicity
On the stability of periodic orbits for differential systems in $\mathbb{R}^n$
Armengol Gasull Héctor Giacomini Maite Grau
We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of $n-1$ codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of the periodic orbits in several examples, including a periodic solution found by Steklov studying the rigid body dynamics.
keywords: rigid body dynamics Periodic orbit characteristic multipliers Steklov periodic orbit. invariant curve Mathieu's equation

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