The author discusses the behavior of the approximate solution under a one-step method for a periodic solution of an autonomous differential equation \(\dot x(t)=f(x(t))\). Denote by \(\phi_ f\) the flow of f, i.e., \(\phi_ f(\tau,y)=x(\tau),\) where x is the solution with \(x(0)=y\). Assume that the periodic solution is hyperbolic, i.e., the eigenvalues \(\lambda\) of \(\partial_ x\phi_ f(T,x)\) other than l satisfy \(| \lambda | \neq 1\), where T is the period. Then the trajectory starting near the periodic one \(\Gamma^ 0\) either converges to \(\Gamma^ 0\) or gets far from it. The main result is the following: For \(k\geq 1\), \(f\in C^{k+2}({\mathfrak R}^ n)\) has a hyperbolic T-periodic trajectory \(\Gamma^ 0\). Denote \(\psi_ 0(x)=\phi_ f(\tau,x),\) for a fixed \(\tau\): \(0< \tau \leq T\). For \(\psi \in C^{k+1}({\mathfrak R}^ n)\), \(C^{k+1}\)-close enough to \(\psi_ 0\), the greatest \(\psi\)-invariant set in a neighborhood U is a \(C^ k\)-closed curve \(\Gamma\), which is \(O(\| \psi -\psi_ 0\|_{C^{h+1}})C^ k\)-close to \(\Gamma^ 0\). This is a slight improvement of the author's previous result [Appl. Math. Comput. (to appear)] which shows the \(C^ k\)-closedness is of \(O(h^ p)\) for the one-step method of order p with stepsize h.