The center and cyclicity problems for some analytic maps. (English) Zbl 1411.37050
Summary: The center variety and bifurcations of limit cycles from the center for maps \(f(x) = - \sum_{k = 0}^\infty a_k x^{k + 1}\) arising from \[ x + y + \sum_{j = 0}^n \alpha_{n - j, j} x^{n - j} y^j = 0 \] are considered. Motivated by a general result for \(n = 2 \ell + 1\) we investigate the center and cyclicity problem for \(n\) being even. We review results for \(n = 2\) and \(n = 4\) and perform the analysis for \(n = 6, 8, 10\). Finally, we state some conjectures for general \(n = 2 \ell\).
MSC:
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 93C55 | Discrete-time control/observation systems |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
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