Estimates for normal forms of differential equations near an equilibrium point (Q1826180)

The paper deals with the normalization to a finite order of vector fields around an equilibrium point, including some estimates for convergence radii and remainders. The content is the following. {\S} 1. Introduction. One gives historical and current references. {\S} 2. The Lie transform. Basic definitions and notations on Lie series theory, as well as an existence theorem are given. One denotes \(H_ k\) the space of the \(C^ d\)-valued homogeneous polynomials of degree k on \(C^ d\). Let \(V_{\epsilon}=\sum_{k\geq 0}V_ k\epsilon^ k\), \(X_{\epsilon}=\sum_{k\geq 0}X_ k\epsilon^ k\) be two vector fields on \(C^ d\) depending on \(\epsilon\). Denote by \(F_{\epsilon}\) the flow generated by \(V_{\epsilon}\) and consider the time-one Lie transform given by \(\phi_*=F_{1*}\) where \(F_{\epsilon *}X_{\epsilon}=\sum_{k\geq 0}X^{(k)}\epsilon^ k.\) {\S} 3. The normal form theorem: Let \(X=\sum_{k\geq 0}X_ k\), with \(X_ k\in H_{k+1}\) be an analytic vector field and let S be the diagonalizable part of \(X_ 0\). For every integer \(r\geq 2\), there exists a nonautonomous polynomial vector field \(V_{\epsilon}\) of degree r without constant and linear terms, such that \[ \phi_*X=X_ 0+\sum^{r-1}_{k=1}Z_ k+R_ r,\quad Z_ k\in H_{k+1},\quad L_ SZ_ k=0 \] the analytic remainder \(R_ r\) having no terms of degree smaller than \(r+1.\) {\S} 4. The main theorem. Let X,V and \(r\geq 2\) as in {\S} 3. Assume that \(\| X_ k\| \leq M/\rho^ k\) for \(k\geq 1\), with M, \(\rho \in R_+\), and \(X_ 0\) in Jordan canonical form. Then \(R_ r\) is analytic in the domain \(D_{\rho_ r}=\{z=(z_ 1,...,z_ d)\in C^ d\), \(\| z\| =(\sum^{d}_{j=1}z_ j\bar z_ j)^{1/2}<\rho_ r\}\) with \(\rho_ r=\rho /1+M_ r(3+K_ r)\), for some constants \(M_ r,K_ r\). In the closed domain \(\bar D_{\rho '}\), \(\rho '<\rho_ r\) a bounding condition for \(\| R_ r\|\) is given. {\S} 5. The Poincaré-Dulac theorem. It is a Corollary of the main theorem. {\S} 6. The exponential estimates for reversible systems of coupled harmonic oscillators. The stability of these systems is discussed, applying the estimates given in the main theorem.