Foreword

Foreword to the i-Math DocCourse

Computational Methods in Dynamical Systems and Applications

by

Carles Simó

Some items to take into account when studying a model

Formulation of the model, how faithful it is, check against experimental data (not only numerical). Iterate on formulation and checks.

Detect symmetries, groups acting on the system, scaling, changes of variables.

Basic invariant objects: fixed points, periodic solutions, invariant tori, dependence on parameters, continuation, bifurcations, unfolding. Also topological constraints.

Centre/stable/unstable manifolds, homo/heteroclinic connections, chaotic dynamics. Prediction vs ``partial prediction''. Attention to regularity and formal aspects, bounds on errors. Perturbative vs ``beyond the perturbative'' domains.

Dynamical/statistical indicators. Meaning and interpretation. Invariant measures.

Numerics and symbolics to check/suggest analytic studies. The ``experimental'' part of maths. Precision, error propagation, complexity, efficiency. Graphic tools. Interpretation of results.

The aim of the courses is to provide you with the background and methodology to study the previous items for both discrete and continuous systems, conservative or dissipative, mainly in the finite-dimensional case, despite some of them, with suitable modifications, can be successfully used for infinite-dimensional models.

Going back near 120 years, when people realized that most of the problems in dynamics cannot be solved explicitly but some particular solutions, like the periodic solutions, can be extremely valuable to give a hint on the dynamics, we can recall a well-known
sentence of Poincaré:

ce qui nous rends ces solutions périodiques si précieuses, c'est qu'elles sont, pour ainsi dire, la seule brèche par où nous puissions essayer de pénétrer dans une place jusqu'ici réputée inabordable

(from the end of paragraph 36 of the first volume of his Méthodes nouvelles de la Mécanique Céleste, 1892), that is:

what makes these periodic solutions so precious is that they are, so to say, the only breach through which we can try to penetrate into a region reputed, up to now, to be unapproachable

Now, already in the XXI century, the words of Poincaré still apply to give valuable information. But the sentence has to be updated. We can say, concerning conservative systems:

The stability properties of these orbits play a key role. In the linearly stable case we can expect to have nearby KAM tori. In
the unstable case what we have are the invariant manifolds, stable/unstable, of the centre manifold of the periodic orbit. They also play a key role. The splitting properties of these manifolds, the existence of homo/heteroclinic connections between different invariant objects, gives the way to organize globally the orbits in the phase space. Poincar\'e itself was aware of the role of splitting and homoclinic phenomena.

Furthermore, the same ideas can be applied to conservative and dissipative systems, continuous or discrete, finite or infinite dimensional. And it is fully relevant to consider all the elements of the ``skeleton'' of the dynamics (fixed points, periodic and quasi-periodic solutions, invariant manifolds and their relative position as depending on parameters. It is not just the phase space E what is relevant, but the product of phase and parameter spaces E x P. This gives the framework to analyze bifurcations and unfold degenerate cases.