Numerical treatment of non-integrable dynamical systems

A systematic and detailed discussion of the gravitational spring-pendulum problem is given for the first time. A procedure is developed for the numerical treatment of non-integrable dynamical systems which possess certain properties in common with the gravitational problem. The technique is important because, in contrast to previous studies, it discloses completely the structure of two-dimensional periodic motion by examining the stability of the one-dimensional periodic motion. Through the parameters of this stability, points have been predicted from which the one-dimensional motion bifurcates into two-dimensional motion. Consequently, families of two-dimensional periodic solutions emanated from these points are studied. These families constitute the generators of the mesh of all the families of periodic solutions of the problem.