In the dymanics of numerical methods for nonlinear differential equations there is a growing interest in specifying procedures that avoid the introduction of spurious solutions by time discretization [see \textit{A. Iserles} and the authors, SIAM J. Numer. Anal. 28, No. 6, 1723-1751 (1991)]. In the present paper and for the equation: \(du/dt=G(u)\), \(u(t)\in R^ m\), \(u(0)=0\), a sequence \(U_ n\in R^ m\) is constructed satisfying: (1) \(U_{n+1}-U_ n=\Delta t[(1-\theta)G(U_ n)+\theta G(U_{n+1})]\), \(\theta\in[0,1]\). If \(G(u)\) is splitted in linear and nonlinear components, \(G(u)=Au+H(u)\), the semiimplicit procedure: (2) \(U_{n+1}- U_ n=\Delta t[AU_{n+1}+H(U_ n)]\) is also considered. The authors look for spurious asymptotic states; their main theorem is: If \(\theta=1/2\) the method (1) cannot have period 2 solutions in \(n\). The method (2) and the method (1) for \(\theta\neq 1/2\) can have period \(2\) solutions in \(n\). These period 2 solutions are spurious. An application to some semilinear parabolic equations is given.