One-dimensional motion of inelastic balls. II: 2-tied quasicycles (Q1920815)

The paper is a continuation of the investigation considering one-dimensional motion of two balls along the interval between two walls [the authors, ibid., 34, No. 6, 1017-1026 (1993; Zbl 0861.70015); translation from Sib. Mat. Zh. 34, No. 6, 23-33 (1993)]. The balls and walls are supposed to be absolutely rigid but inelastic. During free motion, the balls accelerate in proportion to their velocity. In the first part of this, paper, a reduction of the problem to discrete time (the Poincaré-section map) was brought to the simplest form by using the projective transformation \(S\) (or the semigroup \(\{S^n\})\) in the projective plane. The authors gave a priori upper estimate for velocity and considered the degenerate (trajectory tends to a cycle as time \(t\to \infty)\) case. In the second part the authors investigate the non-degenerate case of motion. They consider special cases for some values of collision constants near ``degenerate'' values. For these slightly ``non-degenerate'' constants the attracting invariant manifold consisting of cycles (and decomposed in the degenerate case) exhibits more complicated invariant sets attracting all trajectories. The authors obtain a measure which is invariant under the Poincaré map and prove the ``mixing'' property, i.e. establish the chaotic behavior of the system in the classical sense.