Study of the passage from a dissipative to a conservative state in two-dimensional nonlinear systems of ordinary differential equations

We consider an approximate model of a space pendulum described by a system of two equations. We construct a two-dimensional diagram in the parameter space showing the passage from a dissipative to a conservative state in this system. We show that the Poincaré map of this system can be represented as the square of some other map; this fact is used to explain some specific features of the structure of the bifurcation diagram of the original map.