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A Formal Series Approach to the Center Manifold Theorem

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Abstract

This paper considers near-equilibrium systems of ordinary differential equations with explicit separation of the slow and stable manifolds. Formal B-series like those previously used to analyze highly oscillatory systems or to construct modified equations are employed here to construct expansions of the change of variables, the center invariant manifold and the reduced model. The new approach may be seen as a process of reduction to a normal form, with the surprising advantage, as compared to the standard view conveyed by the celebrated center manifold theorem, that it is possible to recover the complete solution at any time through an explicitly computable way.

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Notes

  1. Several articles otherwise consider the situation where the matrix \(\Lambda \) exhibits a periodic dependency in time [6, 18].

  2. The equation satisfied by h follows by substituting \(z=\varepsilon h(x)\) into \(\dot{z} = - \Lambda z + \varepsilon g(x,z)\).

  3. Up to our knowledge, this is the first time that an explicit and generic representation of \(x_0^\varepsilon \) is provided.

  4. In order to manipulate finite series, we have considered here polynomial functions. Though appropriate truncations of the series employed in this paper, it is expected that the result persists with exponentially small error terms in \(\varepsilon \) under less restrictive assumptions.

  5. The main reason for considering this change of variable is that (2.1) is compatible for the composition of B-series, in contrast to (1.1).

  6. Note that for all \(\lambda \in \mathbb {R}\), \(f_k(x,\lambda z) = \lambda ^k f_k(x,z)\) and similarly for \(g_k\).

  7. See [2] for the first definition of the star product and [14] for the introduction of B-series.

  8. The rigorous definition of the star product requires a few notations (see [8]): an ordered subtree of \(w \in \mathfrak {T}\) is a subset s of the set of all |w| colored-indexed vertices of w, which is connected (by edges of the tree w) and contains the root of w (if \(s \ne \emptyset \)). The collection of all ordered subtrees of w is denoted by \(\mathcal{S}(w)\). Associated with an ordered subtree s are, on the one hand, the collection \(w \setminus s\) of rooted trees that remain when the vertices of the subtree s together with its adjacent edges are removed from the tree w and, on the other hand, the rooted tree \(w''_s\) given by the vertices of s with root and edges from tree w. With these notations at hand, the coefficient \(c_w\) may be computed through the formula

    $$\begin{aligned} (a \star b)_w = \sum _{s \in \mathcal{S}(w)} b(w''_s) \prod _{w' \in w \setminus s} a(w'). \end{aligned}$$

  9. This equation is documented in almost every book on the center manifold theorem, see, for instance, [5].

  10. It is actually obtained as the result of a very high-precision simulation.

  11. In the same way that [10] has paved the way for [4, 9].

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Authors and Affiliations

  1. IRMAR, University of Rennes 1, Rennes, France

    F. Castella & J. Sauzeau

  2. INRIA Rennes, ENS Rennes and IRMAR, University of Rennes 1, Rennes, France

    Ph. Chartier

Authors
  1. F. Castella
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  2. Ph. Chartier
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  3. J. Sauzeau
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Correspondence to Ph. Chartier.

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Communicated by Hans Munthe-kaas.

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Castella, F., Chartier, P. & Sauzeau, J. A Formal Series Approach to the Center Manifold Theorem. Found Comput Math 18, 1397–1434 (2018). https://doi.org/10.1007/s10208-017-9371-y

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