Non-local analysis of families of periodic solutions in autonomous systems (Q2761273)

The author considers the autonomous system \(\dot x = f(x,\lambda)\), \(x\in \mathbb{R}^n\), depending on parameter \(\lambda\), where \(f(x,\lambda)\in C^3(\mathbb{R}^n\times \mathbb{R},\mathbb{R}^n)\) and the system \(\dot x = f(x)\), \(x\in \mathbb{R}^n\), \(f(x)\in C^3(\mathbb{R}^n,\mathbb{R}^n)\). It is assumed that the latter system admits a single-valued integral \(H(x(t))\equiv\lambda = \text{const}\). For both systems, the author obtains an estimate on the minimal number of periodic solutions with arbitrarily large periods as well as an estimate on the minimal number of periodic solutions with a prescribed period. General results of the paper are applied to the investigation of an \(n\)-articulated pendulum.