Integrability and dynamics of a simplified class B laser system

A simplified class B laser system is a family of differential polynomial systems of degree two depending on the parameters a and b. Its rich dynamics has already been observed in 1980s, see Arecchi et al. [Opt. Commun. 51, 308-314 (1984)] and Politi et al. [Phys. Rev. A 33, 4055 (1986)], and still nowadays, it attracts the interest of the researchers. In this paper, we characterize its dynamics near infinity for all values of the parameters. When a = 0, the partial integrability was already proved by Oppo and Politi [Z. Phys. B Con. Mat. 59, 111-115 (1985)]. Here, we prove that for a = 0, it is completely integrable with two independent first integrals given by Liouvillian functions, and we present a complete study of its dynamics. When a ≠ 0, we study its dynamics in the Poincaré ball B 3, i.e., the interior of this ball is identified with R 3 and its boundary the two-dimensional sphere S 2 is identified with the infinity of R 3.