Periodic solutions of nonlinear periodic differential systems with a small parameter
Adriana Buică Jean–Pierre Françoise Jaume Llibre
Communications on Pure & Applied Analysis 2007, 6(1): 103-111 doi: 10.3934/cpaa.2007.6.103
We deal with nonlinear periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide sufficient conditions in order that some of the periodic orbits of this invariant manifold persist after the perturbation. These conditions are not difficult to check, as we show in some applications. The key tool for proving the main result is the Lyapunov--Schmidt reduction method applied to the Poincaré--Andronov mapping.
keywords: Periodic solution Lyapunov--Schmidt reduction. averaging method
Essential perturbations of polynomial vector fields with a period annulus
Adriana Buică Jaume Giné Maite Grau
Communications on Pure & Applied Analysis 2015, 14(3): 1073-1095 doi: 10.3934/cpaa.2015.14.1073
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

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