This work presents a mathematical description of light beam behavior inside a double phase-conjugated resonator when an optical device capable of generating Duffing-like map dynamics is introduced in the cavity. It will be shown that the internal optical device may control the light spatial dynamics inside a ring resonator with double phase-conjugated mirrors. The resonator is constructed using one perfect plane mirror intercalated with two ideal phase-conjugated mirrors. The ABCD matrix formalism used to find the general iterative expressions describes the optical beam state after n round trips, showing paraxiality conservation along the entire process. The characteristics of the internal optical device are discussed and its transformation matrix is presented.

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