Singular point quantities and bifurcations of limit cycles for a class of quintic polynomial systems. (Chinese. English summary) Zbl 1105.34308
Summary: The bifurcations of limit cycles from a degenerate critical point and from infinity for a class of quintic polynomial systems are investigated. In the systems, the origin is a degenerate critical point and the equator contains no real critical point. Firstly, algebraic recursive formulas for computing singular point quantities of the origin and at infinity are derived, respectively. The first five singular point quantities at the origin and the first four singular point quantities at infinity for the system are derived in order to get conditions for a center and to investigate bifurcations of limit cycles. At last, the authors construct a quintic system which allows the appearance of five limit cycles in the neighhorhood of the origin and two limit cycles near infinity. As far as the authors know, this is the first time that the problem of limit cycles bifurcating from a degenerate singular point and from infinity is investigated under synchronous perturbed conditions.
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 |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
