Geometry of pursuit-evasion on second order rotation surfaces

Differential games with simple motion in a game space with non-unique shortest geodestic line have phase portraits with complicated structure of singular surfaces. The successful solution of such gams on a family of two-dimensional 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 or more parameters), and the game on it is considered as the generating problem for the analysis of the games on the perturbed surfaces. The paper continues the previous investigations of the authors. 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 cone, in [4], [5] one sheet of a double-sheeted rotation hyperboloid is considered as game space and geometrical properties of trajectories are analyzed.