Complex methods for bounds on the number of periodic solutions with an application to a neural model. (English) Zbl 1493.34124
The author is interested in determining an upper bound on the number of periodic solutions of some first order scalar differential equation
\[
x' = f(t,x),
\]
where \(f:{\mathbb R}\times{\mathbb R}\to{\mathbb R}\) is a smooth function, periodic in \(t\). After a friendly exposition of a method of Ilyashenko involving complex analysis, he applies it to the model of a single neuron or pool of neurons, thus obtaining an upper bound on the number of periodic solutions. Some open problems are also proposed.
Reviewer: Alessandro Fonda (Trieste)
MSC:
| 34C25 | Periodic solutions to ordinary differential equations |
| 30E99 | Miscellaneous topics of analysis in the complex plane |
| 92C20 | Neural biology |
| 37C60 | Nonautonomous smooth dynamical systems |
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