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Averaging analysis of a perturbated quadratic center. (English) Zbl 0992.34024
The authors consider planar polynomial dynamical systems. Using the averaging theory for studying limit cycle bifurcations, they prove that if the quadratic system with a center at the origin $dx/dt=-y(1+x),$ $dy/dt=x(1+x)$ will be perturbed by polynomials of degree $n$ to the polynomial systems $dx/dt=-y(1+x)+\varepsilon p(x,y),$ $dy/dt=x(1+x)+\varepsilon q(x,y)$, then for sufficiently small $\varepsilon$ it is possible to obtain at most $n$ hyperbolic limit cycles surrounding the origin.

34C07Theory of limit cycles of polynomial and analytic vector fields
34C29Averaging method
34C05Location of integral curves, singular points, limit cycles (ODE)
34C08Connections of ODE with real algebraic geometry
37G15Bifurcations of limit cycles and periodic orbits
34C23Bifurcation (ODE)
Full Text: DOI
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[5] Li, C.; Llibre, J.; Zhang, Z.: Weak focus, limit cycles and bifurcations for bounded quadratic systems. J. differential equations 116, 193-222 (1995) · Zbl 0823.34043
[6] Pontrjagin, L. S.: Über autoschwingungssysteme, die den hamiltonschen nahe liegen. Physikalische zeitschrift der sowjetunion 6, No. 1--2, 25-28 (1934) · Zbl 61.1478.02
[7] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, 2nd Edition, Springer, Berlı\acute{}n, 1996. · Zbl 0854.34002