Numerical integration of differential equations in the presence of first integrals: observer method

We introduce a simple and powerful procedure-the observer method-in order to obtain a reliable method of numerical integration over an arbitrary long interval of time for systems of ordinary differential equations having first integrals. This aim is achieved by a modification of the original system such that the level manifold of the first integrals becomes a local attractor. We provide a theoretical justification of this procedure. We report many tests and examples dealing with a large spectrum of systems with different dynamical behaviour. The comparison with standard and symplectic methods of integration is also provided.