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Limit cycles and integrability in a class of systems with high-order nilpotent critical points. (English) Zbl 1309.34043

The authors consider the family of planar analytic differential equations \[ x'=y+f(x^n,y), \]
\[ y'=-x^{2n-1}+x^{n-1}g(x^n,y) \] with \(n\geq2\) and \(f\) and \(g\) starting at least with second order terms. They have a nilpotent critical point at the origin. They observe that when \(n\) is even the origin has to be a reversible center and that when \(n\) is odd the natural change of variables \(u=x^n\), \(v=y\) together with a time-reparametrization, write them as \[ u'=nv+nf(u,v), \]
\[ v'=-u+g(u,v). \] Since the origin of these new systems is a non-degenerate weak-focus, any of the classical methods for computing Lyapunov constants can be applied to decide whether the origin is a center or a focus. The concrete example with \(f\) and \(g\) homogeneous of degree 3 is developed in the paper.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37C27 Periodic orbits of vector fields and flows
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