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

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