Extension of Floquet's theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows. (Q2784131)

The authors consider a time-periodic system of differential equations NEWLINE\[NEWLINE x'=Ax+v(x,t), \tag{1}NEWLINE\]NEWLINE where \(v\in C^{\infty}\) and \(v(0,t)=0\). They show that if \(A\) is a hyperbolic matrix such that its eigenvalues are weakly nonresonant, then system (1) is \(C^{\infty}\)-equivalent to an autonomous system in a neighborhood of the origin. This result is applied to give a sufficient condition under which a smooth diffeomorphism near a fixed point can be embedded into a smooth flow generated by an autonomous system of differential equations. These conditions are applicable to a germ of a diffeomorphism \(f\) near its fixed point \(p\) (i.e., to a family of diffeomorphisms that are smoothly conjugate with \(f\) near \(p\)).