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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.

37G15Bifurcations of limit cycles and periodic orbits
37G10Bifurcations of singular points
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
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