Geometry of pursuit-evasion on second order rotation surfaces

Differential games with simple motion in a game space with non-unique short est geodesic line have phase portraits with complicated structure of singul ar surfaces. The successful solution of such games on a family of two-dimen sional cones [1] was based to the large extent on the parameter analysis of the problem. In the present paper the results of pursuit game on a cone of one sheet are extended to the game on the full two-sheet cone. The latter surface is included in one-parametric family of rotation surfaces (each of which is characterized by two more parameters), and the game on it is consi dered as the generating problem for the analysis of the games on the pertur bed surfaces. The paper continues the previous investigations of the author s. In [2], [3] some local results are obtained for the games on Riemannian manifolds. In [1], [2] the game problems are solved for a two-dimensional c one, in [4], [5] one sheet of a double-sheeted rotation hyperboloid is cons idered as game space and geometrical properties of trajectories are analyze d.