The splitting of separatrices, the branching of solutions and non-integrability in the problem of the motion of a spherical pendulum with an oscillating suspension point

Abstract

The effectiveness of the results obtained previously in [Dovbysh SA. Transversal intersection of separatrices and non-existence of an analytical integral in multidimensional systems. In: Ambrosetti A, Dell Antonio GF, editors. Variational and Local Methods in the Study of Hamiltonian Systems. Singapore, etc: World Scientific; 1995. p. 156–65; Dovbysh SA. Transversal intersection of separatrices, the structure of a set of quasi-random motions and the non-existence of an analytic integral in multidimensional systems. Uspekhi Mat Nauk 1996; 51(4): 153–54; Dovbysh SA. Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I. Basic result: Separatrices of hyperbolic periodic points. Collect Math 1999; 50(2): 119–97; Dovbysh SA. Branching of the solutions in the complex domain from the point of view of symbolic dynamics and the non-integrability of multidimensional systems. Dokl Ross Akad Nauk 1998; 361(3): 303–6] on the non-integrability of multidimensional systems is illustrated using the example of the problem of the motion of a spherical pendulum with a suspension point performing small periodic oscillations. With this aim, the splitting of the separatrices of the unstable equilibrium position and the branching of the solutions are investigated. It is shown that the separatrices are split for any law of motion of the suspension point, and a simple criterion of the presence of their transversal intersection is obtained. The validity of the non-integrability result, based on a combination of the conditions related to the splitting of multidimensional separatrices and to the branching of the solutions, is also pointed out.