In this paper, we analyze stability, bifurcations, and limit cycles for the cubic self-assembling micelle system with chemical sinks using algebraic methods and provide a complete classification of the stability and types of steady states in the hyperbolic case. Hopf bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation are also analyzed. Exact algebraic conditions on the four parameters of the system are derived to describe the stability and types of steady states and the kinds of bifurcations. It is shown that three limit cycles can be constructed from a Hopf bifurcation point by small perturbation