Calcul des invariants de Birkhoff des systèmes d’ordre deux. (Calculation of Birkhoff invariants of second order systems). (French) Zbl 0719.34016
After some theoretical remarks concerning the formal Birkhoff transformation and the Birkhoff’s invariants, the author gives an effective method for the numerical calculation of the Birkhoff’s invariants for a second order system \(x^{r+1}dX/dx=AX\) with \(A\in {\mathcal M}_ n(2,C\{x\})\) for arbitrary range of Poincaré. One, also, shows effectively the choice of the basis in \(H^ 1(S^ 1;St(A_ 0))\) and the form of the Cauchy-Heine coefficients in three particular cases
\(r=2:\) \(A= \begin{pmatrix} 1+4x/3 & x/2 \\ 2x & 2+x-x^ 2 \end{pmatrix},\)
\(r=3:\) \(A= \begin{pmatrix} x-x^ 3 & 0 \\ (2+3x+6x^ 2+x^ 3)/2 & 2x+x^ 2 \end{pmatrix},\)
\(r=1:\) \(A= \frac{1}{4}\begin{pmatrix} x^ 2 & 1 \\ x & 2x \end{pmatrix}\).
\(r=2:\) \(A= \begin{pmatrix} 1+4x/3 & x/2 \\ 2x & 2+x-x^ 2 \end{pmatrix},\)
\(r=3:\) \(A= \begin{pmatrix} x-x^ 3 & 0 \\ (2+3x+6x^ 2+x^ 3)/2 & 2x+x^ 2 \end{pmatrix},\)
\(r=1:\) \(A= \frac{1}{4}\begin{pmatrix} x^ 2 & 1 \\ x & 2x \end{pmatrix}\).
Reviewer: P.Craciunas (Iaşi)
MSC:
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
