Periodic solutions of difference Runge-Kutta schemes (Q811096)

An \(\omega\)-periodic solution \(x=\phi (t)\) of a system of differential equations \(x'(t)=g(t,x)\) is studied where \(g\in C^ 2\), g is \(\omega\)- periodic with respect to the first argument, and the system \(z'=A(t)z,\) \(A(t)=\frac{\partial}{\partial x}g(t,\phi (t))\) has no non-trivial \(\omega\)-periodic solution. This system is investigated by means of an explicit Runge-Kutta method of the second order. It is shown that there exists an only \(\omega\)-periodic solution of this method in a neighbourhood of \(x=\phi (t).\)