Some unexpected results on the Brillouin singular equation: fold bifurcation of periodic solutions
In the paper the following singular parametric equation \[ \ddot x + \beta (1-\cos t)x =\frac1 x \eqno{(1)} \] is investigated. Such an equation describes the dynamics of an electron beam under the influence of the Brillouin flow, and is called Brillouin electron beam focusing equation. The authors discuss the existence of a classical \(2 \pi\) periodic positive solution of (1) in terms of the values of the parameter \(\beta\). They prove the existence of a threshold value \(\beta^*\) such that the branch of the solutions, parametrized by \(\beta\), undergoes a fold bifurcation at \(\beta=\beta^*\). As a consequence, they are able to state also a multiplicity result near bifurcation. The clever argument is based on numerical techniques combined with fixed point theorems. First, the authors find numerically an approximate \(2 \pi\) periodic solution, and then by a fixed point argument, they prove that a `real' \(2 \pi\) periodic solution exists near the `approximate' one. Finally they show how this technique can be generalized to the equation \[ \ddot x + \beta (1-\cos t)x =\frac1{x^p} \] for \(p=2,3\).
scientific article; zbMATH DE number 6875584
scientific article