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Periodic perturbations with rotational symmetry of planar systems driven by a central force. (English) Zbl 1392.34045

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 A. Ambrosetti and 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.

MSC:

34C25 Periodic solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations

Citations:

Zbl 0653.34032
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References:

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