Branches of forced oscillations for periodically perturbed second-order autonomous ODEs on manifolds
The authors examine the structure of the set of forced oscillations of periodic perturbations of second-order autonomous ODEs of the type \[ x''_{\pi} = h(x,x') + \lambda f(t,x,x') \] where \(h: TM \rightarrow \mathbb{R}^{k}\) and \(f:\mathbb{R}\times TM \rightarrow \mathbb{R}^{k}\) are continuous vector fields, tangent to a smooth submanifold \(M\) of \(\mathbb{R}^{k}\), \(f\) is \(\omega\)-periodic in \(t\) and \(\lambda \geq 0.\) Here, \(TM\) denotes the tangent bundle of \(M.\) In particular, the authors provide sufficient conditions to prove the existence of a noncompact connected component of solutions \((\lambda,x) \), with \(x\) an \(\omega\)-periodic function, emanating from the set of equilibria for \(\lambda = 0.\) Moreover, this component is unbounded if \(M\) is complete. An application to a perturbed pendulum equation is given. It is very important, in the proofs, the use of the notion of degree of a second-order vector field.
parametrized second-order differential equations
scientific article; zbMATH DE number 1288031