Time-reversal and the Bessel equation

The system of two damped/amplified oscillator equations is of widespread interest in the study of many physical problems and phenomena, from inflationary models of the Universe to thermal field theories, in condensed matter physics as well in high energy physics, and also in neuroscience. In this report we review the equivalence, in a suitable parametrization, between such a system of equations and the Bessel equations. In this connection, we discuss the breakdown of loop-antiloop symmetry, its relation with time-reversal symmetry and the mechanism of group contraction. Euclidean algebras such as e(2) and e(3) are also discussed in relation with Virasoro-like algebra.