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Centers of planar polynomial systems. A review. (English) Zbl 1156.34315

The author surveys current research on polynomial systems \(dx/dt=\alpha x+\beta y+p(x,y)\), \(dy/dt=\gamma x+\delta y+q(x,y)\) with a center. Three cases are treated separately: (i) total degeneracy, \(\alpha+\beta=\gamma=\delta=0\); (ii) semidegeneracy, \(\alpha\delta-\beta\gamma=0, \alpha+\delta=0, \alpha^2+\beta^2+\gamma^2+\delta^2>0\); (iii) nondegeneracy, \(\alpha\delta-\beta\gamma\not=0\). By means of the coefficients of \(p\) and \(q\), identifications are discussed and integrable systems are considered. Geometrical classifications of centers are discussed in terms of the topology of their neighbourhood.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
37C27 Periodic orbits of vector fields and flows
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