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Bifurcation of limit cycles from a centre in $\Bbb R^{4}$ in resonance $1:N$. (English) Zbl 1167.37027
The authors consider the linear differential centre $\dot{x}=Ax$ in $\mathbb{R}^{4}$ with eigenvalues $\pm i$ and $\pm Ni$ for every positive integer $N\geq2.$ They perturb this linear centre inside the class of all polynomial differential systems of the linear plus a homogeneous nonlinearity of degree $N$ form, i.e., $\dot{x}=Ax+\varepsilon F(x),$ where every component of $F(x)$ is a linear polynomial plus a homogeneous of degree $N.$ Then, if the displacement function of order $\varepsilon$ of the perturbed system is not identically zero, the authors study the maximum number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.

##### MSC:
 37G15 Bifurcations of limit cycles and periodic orbits 37G10 Bifurcations of singular points 37J40 Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
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