Almost-periodic solutions to systems of differential equations with fast and slow time in the degenerate case (Q1280694)

The author deals with existence and stability of almost-periodic solutions to the system \[ {dx \over dt}=\varepsilon X(t,\tau ,x,\varepsilon),\quad \tau =\varepsilon t, \tag{1} \] with \(x\in \mathbb{R}^n\), \(\varepsilon >0\) is a small parameter, the vector function \(X\) is smooth, \(\omega\)-periodic in \(\tau\), and almost-periodic in \(t\) uniformly with respect to \(x\), \(\tau\), and \(\varepsilon\). It is suggested that the averaged system \[ {dy \over d\tau}=Y(\tau ,y),\quad \tau =\varepsilon t, \quad Y(\tau ,y)=\lim _{T\to\infty}{1\over T}\int _{0}^{T} X(t,\tau ,x,0)dt, \] has a nonhyperbolic \(\omega\)-periodic solution \(y(\tau)\). Conditions are found under which for small \(\varepsilon\) system (1) has a unique almost-periodic solution \(x(t,\varepsilon)\) near to \(y(\tau)\). Stability properties of \(x(t,\varepsilon)\) are discussed.