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The limit cycles of a second-order system of differential equations: the method of small forms. (English. Russian original) Zbl 1192.34043

Russ. Math. 53, No. 8, 60-68 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 8, 73-82 (2009).
The paper deals with the problem of the existence of limit cycles for a system in the form
\[ \dot{x}=-y+ \sum_{i+j=k_0}^{n} (a_{ij}+ \mu_{ij})x^i y^j,\quad \dot{y}=x+ \sum_{i+j=k_0}^{n} (b_{ij}+ \lambda_{ij})x^i y^j, \]
where \(a_{ij},\) \(b_{ij}\) are real constants, \(\mu_{ij},\) \(\lambda_{ij}\) are parameters, \(k_0\geq 2.\) For this purpose, the method of small forms, based on the representation of a solution as a sum of forms with respect to the initial value and the parameter, is proposed. The author constructs a polynomial, whose positive roots allow to obtain the number of limit cycles in a sufficiently small neighborhood of the equilibrium point.

MSC:

34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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References:

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