Multiple time scales for nonlinear systems (Q1820297)

This paper extends the results on the multiple time-scale structure for linear autonomous systems of the form \(\dot x=A(\epsilon)x\), where A(\(\epsilon)\) is analytic in the small parameter \(\epsilon \in [0,\epsilon_ 0]\), to the nonlinear case \(\dot x=q(z,\epsilon)\), where \(q\in C^ r\), r some large integer. The main result of the work is in obtaining conditions under which the linearized system and the nonlinear system around an equilibrium point have the same time-scale structure. It is proved that analytically the time-scale structure does not follow directly from the linearized equations around the equilibrium point. However, in practice the linearized equations predict the correct time- scale structure if we restrict our system to the domain where the linearized equations approximate the nonlinear system. The paper is clearly written.