Bifurcation of the point of equilibrium in systems with zero roots of the characteristic equation (Q1889610)

A real autonomous system of \(2d\), \(d\geq 2\), differential equations with small positive parameter \(\varepsilon\) is considered, when the matrix of its linear part has \(d\) pairs of zero eigenvalues with not simple elementary divisors \[ \begin{aligned} \dot{x}_i&=x_{i+d}+X_i^{[n+1]}(x,\varepsilon)+X_i^{[n+2]}(x,\varepsilon)+X_i^{\ast[n+3]} (x,\varepsilon),\\ \dot{x}_{i+d}&=-x_i^{2n-1}+X_{i+d}^{[2n]}(x,\varepsilon)+X_{i+d}^{[2n+1]}(x,\varepsilon)+ X_{i+d}^{\ast[2n+2]}(x,\varepsilon), \quad i=1,\dots,d\geq 2. \end{aligned} \] Here, \(X_j^{[k]}(x,\varepsilon)\) are homogeneous polynomials of \(k\)th degree, \(j=1,\ldots,2d\), with first order of smallness for \(x_i\), second order for \(\varepsilon\) and \(n\)th order for \(x_{i+d}\); the functions \(X_j^{\ast[k]}\) are continuous on the set \(H=\{(x,\varepsilon)\mid \|x\|<x_0, 0\leq\varepsilon<\varepsilon_0\}\) and \(X_j^{\ast[k]}(\varepsilon^{1/2}x',\varepsilon^{n/2}x'',\varepsilon)= \varepsilon^k\widehat{X}_j^{\ast[k]}(x',x'',\varepsilon)\), where \(x'=(x_1,\dots,x_d)\), \(x''=(x_{d+1},\ldots,x_{2d})\), \(\widehat{X}_j^{\ast[k]}\) are sufficiently smooth functions in \(H\). The author gives explicit conditions on the coefficients of the homogeneous polynomials \(X_j^{[k]}\) guaranteeing the birth of \(d\)-dimensional invariant tori for this system for any sufficiently small \(\varepsilon>0\).