One-diemnsional motion of inelastic balls. I: Reduction to discrete time

One-dimensional motion along the interval between walls is considered for two balls. The balls and walls are assumed to be absolutely rigid but not absolutely elastic. The next law is supposed for inelastic collision: the relative velocity of walls is multiplied by constant less than unity in absolute value. Therefore energy is lost at collision. The simplest mechanism is introduced for the influx of energy: during free motion the balls acelerate in propotion to their velocities. An a priori estimate is established for solutions. For the degenerate cases (at some values of collision constants) attracting invariant manifolds are found which consist of two-point cycles.