In this paper, we consider discrete time dissipativeness on quadratic difference forms and two-variable dipolynomial matrices. We show that if a system is dissipativeness for a supply rate on the dipolynomial ring, then there also exist storage functions and dissipation rates on the dipolynomial ring. Moreover, we clarify whether extremal storage functions for a given supply rates on the dipoly...

A single bunch of charged particles in a storage ring is a system of 10-10 coupled non-linear oscillators with radiative energy loss and compensated by the accelerating cavity. A stable collective mode in such system was induced and observed in the SPEAR storage ring using a turn-by-turn phase space monitor. New possibilities of using this mode as a tool to experimentally study non-linear dynam...