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The shape of limit cycles that bifurcate from non-Hamiltonian centers. (English) Zbl 0999.34028

Consider planar systems of the form \[ dx/dt = P \equiv -y+f_0(x,y) + \varepsilon f_1(x,y),\;dy/dt = Q \equiv x+g_0 (x,y) + \varepsilon g_1(x,y),\tag \(*\) \] where \(\varepsilon\) is a small parameter, under the assumptions: (i) \(x=y=0\) is a center, (ii) \(f_0,f_1,g_0,g_1\) are analytic in a neighborhood of the origin, (iii) \(f_0,g_0\) have no linear terms. The authors exploit the fact, that if \(V\) is a differentiable solution to the partial differential equation \(PV_x + QV_y = (P_x+Q_y)V\), then any limit cycle to \((*)\) is contained in the set \(V^{-1} (0)\), to derive a method to determine the number, position and shape of limit cycles to \((*)\) for small \(\varepsilon\). The method is illustrated by an example.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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