The splitting of separatrices, the branching of solutions, and nonintegrability of many-dimensional systems. Application to the problem of the motion of a spherical pendulum with an oscillating suspension point

Some possibilities of the author's approach to the problem of nonintegrability of multidimensional systems related to the splitting of multidimensional separatrices and branching of solutions in the complex domain are discussed, using the example of the problem of the motion of a spherical pendulum with a suspension point performing small spatial periodic oscillations. Previous results are briefly reproduced and their generalizations are discussed, which are based on the calculation of a perturbation of the linear part of the Poincaré map at a hyperbolic point. We have succeeded in obtaining weaker conditions of nonintegrability, since this perturbation, generally speaking, violates the scalar nature of the restrictions of the linear part of the map to its two-dimensional expanding and contracting invariant subspaces. However, these conditions are expressed in terms of some repeated integrals because one must work in the second order of the perturbation theory. It is shown that in the case where the acceleration of the suspension point is represented by a function of complex time uni-valued over punctured vicinities of some isolated singularities, the nonintegrability conditions can be reduced to very simple ones in terms of certain local quantities associated with these singularities. The approach developed can be useful in problems where an unperturbed system possesses a symmetry leading to a degeneration, like the scalar nature of the restrictions of the linear part of the Poincaré map to its invariant subspaces.