A generalized normal form and formal equivalence of systems of differential equations with zero characteristic numbers (Q1768958)

Normal forms (analytic or formal) are important in the theory of systems of ordinary differential equations (ODEs). Systems in normal forms are sometimes referred to as ``resonance normal forms''. The existence of transformations that reduce a differential equation to a simple form, a normal form, has been investigated by many authors. In particular, following a previous work by \textit{A. D. Bryuno} [Tr. Mosk. Mat. Obshch. 25, 119--262 (1971; Zbl 0263.34003) and Math. Notes 14, 844--848 (1973); translation from Mat. Zametki 14, No. 4, 499--507 (1973; Zbl 0298.34004)], in the paper under review, divided in two parts, the first one being devoted to a general study on the autonomous system of ODEs: \[ \dot x_i= \lambda_i x_i+ \sigma_i x_{i-1}+ x_i(x),\quad i= 1,2,\dots, n,\tag{1} \] whose right-hand sides do not contain free terms, the matrix of the linear part is reduced to a Jordan form and the nonlinearities are power series, either formal or convergent at zero. By suitable changes of variables (or substitutions) system (1) can be reduced to other more simplified normal forms. The author considers a singular case occurring when all eigenvalues \(\lambda_1,\lambda_2,\dots, \lambda_n\) of the matrix of the linear part of (1) are zero and by virtue of the classification of normal forms depending on the arrangement of the eigenvalues \(\lambda_i\) on the complex plane realized by A. D. Bryuno, it is derived that any system (1) has in this case a form which is a normal form itself and that can also be rewritten as \[ \dot x_i= P_{\gamma i}^{[k]}(x)+ X_i(x), \quad(i=1,2,\dots, n,\tag{2} \] where \(P_{\gamma i}^{[k]}(x)\) stands for a form \((\not\equiv 0\) for any \(i)\) of degree \(k\) with weight \(\gamma\), referred as to the zero approximation or not perturbed part of system (2) and \(X\) being the perturbation of this system. A new formal substitution reduces (2) to another system which is referred to as a generalized normal form (GNF). The simplification of the preceding system leads to new equations referred to as resonance equations (see the reference above due to Bryuno). The following two results are established: I) System (2) is formally equivalent to any (GNF) system, provided that under certain conditions, the resonance equations are satisfied; and II) There exists a change of variables reducing an arbitrary system (2) to a GNF. The knowledge of the resonance equations permits one to write out an arbitrary GNF and to represent explicitly various structures of the GNF for which it is formally equivalent to the original system. According to the various classes of the zero approximations, the final structure of the system is found and the resonance equations can now be written out in a closed form. In the second part of this paper, the above theory is applied to a two-dimensional system with zero approximation \(P= (x_2,\alpha x_1x_2+\beta x_1^3)\). One of the normal forms for system (2) with the above mentioned nonperturbed part is constructed. The author concentrates on a system (2), where \(n=2\), \(k=1\), the variable \(x_1\) is of first infinitesimal order and \(x_2\) is of second order, i.e., \(\gamma=(1,2)\) and \(P_\gamma^{[k]}(x)= (x_2,-x_1^3)\) being the quasihomogeneous polynomial of zero approximation. Therefore, system (2) acquires the form \[ \dot x_1= x_2+ X_1(x), \quad \dot x_2= -x_1^3+ X_2(x). \] This system realizes the critical case of two zero roots of the characteristic equation with a nonsimple elementary divisor. Finally, some applications are given, too.