Ten limit cycles around a center-type singular point in a 3-d quadratic system with quadratic perturbation. (English) Zbl 1336.34051
The authors study local limit cycle bifurcations in three dimensional autonomous systems of ordinary differential equations with quadratic nonlinearities. It is shown that there are such systems with at least 10 small-amplitude limit cycles around a singular point. To construct the example, the authors compute the focus values of the system, apply the normal forms method and use resultants to analyze the zero set of the focus values.
Reviewer: Valery Romanovski (Maribor)
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 |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C45 | Invariant manifolds for ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
Keywords:
3-d quadratic system; focus; center; limit cycle; Hopf bifurcation; center manifold; normal form; focus value computationReferences:
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