Dynamics of a class of time-period strongly 2-cooperative system: integer-valued Lyapunov function and embedding property of limit sets

We construct an integer-valued Lyapunov function $\sigma(\cdot)$ for generalized negative cyclic feedback system; and prove that $\sigma(\cdot)$ on any $\omega$-limit set which generated by Poincar\'{e} mapping of bounded solution of such strongly $2$-cooperative system is constant. Therefore, the $\omega$-limit can be continuously embedded into a compact subset of a two-dimensional plane. Finally, a dissipative condition is given to ensure that all orbits of such system are bounded.