On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters (Q2821740)

For a real analytic linear quasi-periodic system of dimension \(2\) NEWLINE\[NEWLINE \dot x=(A+Q(t,\varepsilon))x, \quad x\in\mathbb R^2, NEWLINE\]NEWLINE with \(A\in sl(2,\mathbb R)\), \(|\varepsilon|\ll 1\), and \(Q\in C^{a,m}(\mathbb T_s^r\times [0,\delta],\, sl(2,\mathbb R))\) the set of all matrix functions \(Q(t,\varepsilon)\in sl(2,\mathbb R)\) which are analytic quasi-periodic in \(t\) on \(\mathbb T_s^r\) with frequency vector \(\omega\) and \(C^m\)-smooth in \(\varepsilon\) with \(m=1\) or \(0\), if \(Q(t, \varepsilon)\) closes to constant for \(|\varepsilon|\ll 1\), then under some non-resonance assumptions on the basic frequencies and the eigenvalues of \(A\), they prove that the system is reducible for many of the sufficiently small parameters.NEWLINENEWLINEHere, reducible means that for a linear quasi-periodic system NEWLINE\[NEWLINE \dot x= A(t)x,\quad x\in \mathbb R^n, NEWLINE\]NEWLINE with \(A(t)\in gl(n,\mathbb R)\) an analytic quasi-periodic matrix or a \(C^r\)-smooth one, there exists a non-singular quasi-periodic \(\Phi(t)\in \mathrm{GL}(n,\mathbb R)\), which is at least \(C^1\)-smooth, such that the transformation \(x=\Phi(t)y\) sends the system to a system \(\dot y= By\) with \(B\) a constant matrix.