# zbMATH — the first resource for mathematics

Isochronic centers of a reversible cubic system. (Russian) Zbl 0989.34020
The authors study the reversible cubic system $\frac{dx}{dt} = -y(1+Dx+Px^2),\quad \frac{dy}{dt} = x - Ax^2-Cy^2-Kx^3-Mxy^2, \tag{1}$ where $$A$$, $$C$$, $$D$$, $$K$$, $$M$$ and $$P$$ are complex constants. They establish 12 classes of systems of type (1) for which a singular point $$O(0,0)$$ is an isochronic centre [see A. P. Vorob’ev, Dokl. Akad. Nauk BSSR 7, No. 3, 155-156 (1963; Zbl 0126.10403); K. S. Sibirskij, Algebraic invariants of differential equations and matrices, Kishinev (1976; Zbl 0334.34014); V. V. Amel’kin, N. A. Lukashevich and A. P. Sadovskij, Nonlinear oscillations in systems of second order, Minsk (1982; Zbl 0526.70024)].

##### MSC:
 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems, general theory
##### Keywords:
reversible cubic system; isochronic centre