The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation–oscillations

The development of dissipative time scales, which are much faster than the rest, force the solution of a system of 1st order ODEs to land and then evolve on a slow invariant manifold. The slow evolution on this manifold can be approximated by a reduced system, which is constructed with asymptotic analysis and is free of the fast scales. In this work, the dynamics of the reduced system are compared to the fast and slow dynamics of the original system. The comparison is based on the van der Pol system, which is typical of those exhibiting relaxation–oscillation behavior and is characterized by successive dissipative and explosive phases. Special attention is given to the dissipative phase, during which the solution evolves on a slow invariant manifold and the fast/slow time scale gap tends to contract. It is shown that one cannot predict from the dynamics of the van der Pol system the approaching explosive phase and the associated degeneracy of the manifold. Such a prediction is possible only through the dynamics of the reduced system, which involves an explosive component that is absent in the original system. This explosive component accelerates as higher order effects are incorporated in the reduced system, resulting in a smaller gap between the fast time scale of the original system and the fast time scale of the reduced one. This feature sets a limit on the accuracy singular perturbation techniques can deliver, even in the presence of a reasonably large fast/slow time scale gap in the dynamics of the original system. Given a fast/slow system, predicting forthcoming changes in the dimensions of the manifold is very important when it is desired to identify the underlying major physical mechanisms or to control the process.

► The familiar asymptotic expansion y=y0+ε1y1+ε2y2+⋯y=y0+ε1y1+ε2y2+⋯ is shown to be inadequate.
► A more reliable expansion is provided by the expression y=y0+ε0y1+ε0ε1y2+⋯y=y0+ε0y1+ε0ε1y2+⋯.
► With this expression the degeneracy of slow invariant manifolds can be predicted.
► The level at which the fast and slow time scales communicate can also be identified.