On the loss of $L$-stability of the implicit Euler method for a linear problem

A number of important applied problems of chemical kinetics, biophysics, theory of electrical circuits are described by systems of stiff ordinary differential equations with given initial conditions. The one-step Runge–Kutta method is one of approaches for their numerical solution. The implicit Runge–Kutta methods are used for problems of small dimension. The so-called $ A $- and $ L $-stable methods are singled out among these algorithms. Usually, $ L $-stable methods much better cope with these problems. Namely, when we implement $ L $-stable methods we can choose the integration step much greater than in the implementation of $ A $-stable methods. The implicit Euler method is the simplest of these algorithms and it well proved itself.
In the article we consider an example of a linear autonomous system of ordinary differential equations depending on parameters. By choosing these parameters, as is wished stiff problem can be obtained. It is shown that for a particular choice of the parameters Euler implicit method will be ineffective. It is stable only under significant restrictions of the integration step. The construction of this example is based on some facts of the theory of the numerical solution of differential-algebraic equations of high index. The detailed computations are shown.