Diagonal perturbation of regular extensions of dynamical systems on torus (Q2782063)

The author considers a system of differential equation NEWLINE\[NEWLINE \frac{d\varphi}{dt} = a(\varphi),\qquad \frac{dx}{dt} = A(\varphi)x, NEWLINE\]NEWLINE where \(\,\varphi\in \mathcal T_m\,\) is an \(m\)-dimensional torus, \(\,x\in \mathbb R^n\), \(\,A(\varphi)\in C^0(\mathcal T_m)\), \(\,a(\varphi)\in C_{\text{lip}}(\mathcal T_m)\,\) is a linear extension of the dynamical system on torus. In the paper the conditions are established under which the block-triangular linear extensions of the dynamical systems on torus preserve regularity under diagonal perturbations of some classes of linear extensions of the dynamical systems on torus.