Higher-Order Averaging, Formal Series and Numerical Integration II: The Quasi-Periodic Case

Abstract

The paper considers non-autonomous oscillatory systems of ordinary differential equations with d≥1 non-resonant constant frequencies ω ₁,…,ω _d . Formal series like those used nowadays to analyze the properties of numerical integrators are employed to construct higher-order averaged systems and the required changes of variables. With the new approach, the averaged system and the change of variables consist of vector-valued functions that may be written down immediately and scalar coefficients that are universal in the sense that they do not depend on the specific system being averaged and may therefore be computed once and for all given ω ₁,…,ω _d . The new method may be applied to obtain a variety of averaged systems. In particular, we study the quasi-stroboscopic averaged system characterized by the property that the true oscillatory solution and the averaged solution coincide at the initial time. We show that quasi-stroboscopic averaging is a geometric procedure, because it is independent of the particular choice of co-ordinates used to write the given system. As a consequence, quasi-stroboscopic averaging of a canonical Hamiltonian (respectively, of a divergence-free) system results in a canonical (respectively, in a divergence-free) averaged system. We also study the averaging of a family of near-integrable systems where our approach may be used to construct explicitly d formal first integrals for both the given system and its quasi-stroboscopic averaged version. As an application we construct three first integrals of a system that arises as a nonlinear perturbation of five coupled harmonic oscillators with one slow frequency and four resonant fast frequencies.

Appendix: Multiplication of Coefficients

In this appendix we give details of the product ∗ in $\mathcal {G}$. We hope that this will increase the readability of the paper; for a more complete treatment the reader is referred to [15], Chap. III.

Given $\delta\in \mathbb {C}^{ \mathcal {T}\cup \{\emptyset \}}$ with δ _∅=1 (that is $\delta\in \mathcal {G}$, so that B(δ,y) is a near-identity B-series), and an arbitrary B-series B(η,y), the composition B(η,B(δ,y)) is a B-series of the form B(ζ,y), where ζ _∅=η _∅ δ _∅ and for $u\in \mathcal {T}$ the coefficient ζ _u =(δ∗η) _u , $u\in \mathcal {T}$ equals η _∅ δ _u +η _u δ _∅=η _∅ δ _u +η _u plus a sum of products of the form $\eta_{u_{0}}\delta_{u_{1}} \cdots\delta_{u_{m}}$ with |u ₀|+|u ₁|+⋯+|u _m |=|u|, $u_{0},\ldots,u_{m} \in{ \mathcal {T}}$. For instance, for the tree in the last row of Table 1 we have (we use brackets rather than subscripts for typographical convenience):

(80)

Multiplication of the formula (80) by $\exp(i(\mathbf {k}+\mathbf {l}+\mathbf {m})\cdot\theta)$ establishes the validity of (52) at

In the general case, the coefficient (δ∗η) _u , $u\in \mathcal {T}$ can be written as

(81)

where, for each fixed $\delta\in \mathcal {G}$, $L_{\delta}: \mathcal{V}\rightarrow\mathcal{V}$ is a linear map (defined below) in the vector space $\mathcal{V}=\mathrm{span}( \mathcal {T}\cup \{\emptyset \})$ of linear combinations (with complex coefficients) of rooted trees, and η (originally defined as a map $\mathcal {T}\cup \{\emptyset \}\rightarrow \mathbb {C}$) has been extended by linearity to $\mathcal{V}$ (i.e. $\eta(\sum_{j}\lambda_{j} u_{j}) = \sum_{j} \lambda_{j} \eta_{u_{j}}$ for each linear combination ∑ _j λ _j u _j of trees).

In order to define the linear map L _δ , we first extend the notation $[u_{1} \cdots u_{m}]_{\mathbf {k}}$ (used to recursively represent trees in $\mathcal {T}$), to define for each m≥1 and each multi-index $\mathbf {k}\in \mathbb {Z}^{d}$, an m-linear, commutative m-ary operation on $\mathcal{V}$ by setting

Now, L _δ (u) is defined recursively by

(82)

One can check that for any $u \in \mathcal {T}$, L _δ (u)=δ(u) ∅+u+v where v a sum of rooted trees with fewer vertices than u.

Observe that, pictorially, in the right-hand side of (80), the tree associated with the right factor η ranges in the set of trees obtained by pruning some edges in the tree in the left-hand side (‘pruning’ includes complete uprooting, as in the first term, or no pruning at all, as in the last). In each term, the trees associated with the left factor δ are precisely the parts that have been chopped off. It can be shown that (81)–(82) imply that the same pruning procedure is valid to write the formula for (δ∗η) _u for arbitrary $u \in \mathcal {T}$.