Periodic solutions and dynamics of a multimolecular reaction system

In this paper, we consider a general multimolecular reaction system, which appears in biochemistry as a theoretical problem of concentration kinetics and in mathematics as a special polynomial vector field of high degree. We shall investigate its global dynamics and discuss existence and nonexistence of periodic solutions. Although the case of trimolecular reactions and some other special cases were studied extensively, it remains difficult to discuss the general case, that there is involved a lot complicated computation for polynomials of any given degree. In this paper, special techniques are used in computation of Lyapunov numbers for Hopf bifurcation, construction of Dulac auxiliary functions for nonexistence of periodic solution, and determination of qualitative properties of degenerate equilibria.