On the existence of slow solutions for highly oscillatory problems (Q1325753)

The existence of a slow solution for the system \[ \varepsilon y' = \bigl[ A(t) + \varepsilon C(y,v,t) \bigr] y + f(v,t),\;v' = h(y,v,t),\;y(0) = y_ 0,\;v(0) = v_ 0, \tag{1} \] is studied, where \(A(t)\), \(C(y,v,t)\), \(f(v,t)\) and \(h(y,v,t)\) are bounded \(C^ \infty\) functions. A solution pair \((y,v)\) of (1) is said to be slow to order \(p \geq 1\) on a time interval \(0 \leq t \leq T_ 0\), if both \(y(t)\) and \(v(t)\) have \(p\) derivatives bounded independently of \(\varepsilon\) on \([0,T_ 0]\). The existence of slow solution for system (1) is investigated for two cases: a) when \(A(t)\) is invertible, b) when \(A(t)\) is singular with constant rank. A class of physical problems to which this theory may be applied is presented.