Simple-non-autonomous differential equations with many limit cycles. (English) Zbl 1152.34326
Summary: Consider the family of differential equations on the cylinder, \(\frac{dx}{dt}= a(t)+b(t)|x|\), where \(x,t\in\mathbb R\), and \(a, b\) are real, 1-periodic and smooth functions. The solutions satisfying \(x(0)=x(1)\) are called periodic orbits of the equation. The periodic orbits that are isolated in the set of all the periodic orbits are usually called limit cycles. We give a proof, which is self contained, that there is no upper bound for the number of limit cycles of the above type of equations.
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
