The paper presents an analysis of the Rosenbrock integration method applied to a stiff system of the form \[ (1)\qquad \dot x = f(t,x,y,\varepsilon ) + \varepsilon ^{ - 1} A(t)y,\quad \dot y = g(t,x,y,\varepsilon ) + \varepsilon ^{ - 1} \mu (t)By. \] This equation possesses the following desirable model properties. (a) It permits the simultaneous occurrence of smooth and transient solution components. (b) It contains a small parameter admitting a transition to arbitrarily high stiffness. (c) The Jacobian matrix has a time-dependent eigensystem. (d) It contains nonlinear terms. Provided certain assumptions have been satisfied, a characteristic of (1) is that for given initial vectors $x(0) = x_0 ,y(0) = y_0 $\[ (2)\qquad \| {x(t,\varepsilon )} \| = O(1),\quad \| {y(t,\varepsilon )} \| = O(\varepsilon ),\quad \varepsilon \to 0,\quad t \in (0,T],\quad T{\text{finite}}.\] Our analysis will be directed towards obtaining criteria which guarantee a similar behavior for finite sequences of Rosenbrock approximatio...