Unexpected behavior of nonautonomous linear dynamical systems in two dimensions (Q2570844)

Consider the two-dimensional linear system \[ {dz\over dt}= -nA(t) z(t),\tag{*} \] where \(n= 1,2,\dots\) is a parameter, \(t\to A(t)\) is continuous and \(A(t)\) is symmetric and nonnegative. Thus, \(|z(t)|\) is a nondecreasing function of \(t\). The author presents two systems of type \((*)\) such that \(|z(t)|\) is strictly decreasing in \(t\), but \(|z(t)|\) does not tend to zero as \(n\) tends to infinity.