An initial bound state and an autoionizing state of a complex atom (or ion) which are resonantly coupled by a radiation field can both be characterized in a Floquet approach by complex quasienergies. For certain values of the field's intensity and frequency, the two quasienergies become equal, giving rise to laser-induced degenerate states (LIDS). This phenomenon is demonstrated by applying the ab initio, fully nonperturbative $R$-matrix Floquet theory to Ar, ${\mathrm{H}}^{\ensuremath{-}}$, and He. The general case is considered by constructing models containing discrete states and continua. Results for one- and for two-photon resonances are presented and suggestions for experimental observation are given.