A new method is presented to study bifurcation of no-return transition states (TSs) at potential saddles for systems of many degrees of freedom (dof). The method enables us to investigate analytically when and how the no-return TS bifurcates. Our method reveals a new aspect of bifurcation for systems of many dof, i.e., the action variables of the bath dof play a role of control parameters as long as they remain approximately conserved. As an illustrative example, we demonstrate our new method by using a three atomic exchange reaction. The bifurcation of no-return TSs gives rise to a short-lived intermediate state at the saddle, which results in the overestimation of the reaction rate. Hence, the understanding of the bifurcation of the no-return TS is crucial to capture the complexity in kinetics and dynamics of the reactions. The definability of no-return TSs in many-body chemical reactions is also addressed under the occurrence of bifurcation above the reaction threshold.