Periodic perturbations with rotational symmetry of planar systems driven by a central force
The authors consider the planar system \[ \ddot{x}+g(|x|)x=\epsilon \nabla_x V(t,x), \;0<\epsilon \ll 1, \] where \(g:(0,+\infty)\rightarrow \mathbb{R}\) is a continuous differentiable function and \(V:\mathbb{R}\times(\mathbb{R}^2\setminus\{0\})\rightarrow \mathbb{R}\) is continuous, \(T\)-periodic in its first variable, and twice differentiable in its second variable. Under the assumption of the rotational symmetry in perturbing force i.e. \[ \nabla_xV(T,\mathcal{R}x)=\mathcal{R}\nabla_xV(t,x) \;\text{for every } t,x, \] where \(\mathcal{R}\) is the matrix of rotation, the existence of nearly circular periodic orbits is deduced. The result is the generalization, in the planar case only, of the 1989 work of \textit{A. Ambrosetti} and \textit{V. Coti Zelati} [Math. Z. 201, No. 2, 227--242 (1989; Zbl 0653.34032)] who assumed the perturbing potential to be even. It uses similar techniques and it is not extensible to higher dimensions as discussed by the authors.
scientific article; zbMATH DE number 6855895