Averaging method for multifrequency systems with lag (Q1822261)

The author considers the multifrequency system with delay \[ dx/dt=\epsilon X(x,x_{\Delta},\phi,\phi_{\Delta}),\quad d\phi /dt=\omega (x)+\epsilon Y(x,x_{\Delta},\phi,\phi_{\Delta}), \] where \(x\in R^ n\), \(x_{\Delta}=x(t-\Delta)\in R^ n\), \(\phi_{\Delta}=\phi (t-\Delta)\), \(\phi =(\phi_ 1,...,\phi_ m)\), X and Y are \(2\pi\)-periodic functions with respect to \(\phi\) and \(\phi_{\Delta}\). It is proved that the norm of the difference between the solution of a two-frequency system and the solution of the corresponding averaging system is of order \(\sqrt{\epsilon}\). This result is extended to multifrequency system.