A criterion for conditional instability by the first approximation for solutions of differential systems (Q2461818)

The author of this paper considers the linear system \[ x' = A(t)x \tag{1} \] of \(n\) equations defined for \(t \geq 0\) and a nonlinear perturbation \(f(t,x)\) defined in the domain \(\{ t,x:t \geq 0,\| x \| < \rho \}\) and vanishing for \(x = 0\), where \(\| . \|\) is the Euclidean norm. The author established some conditions on the system and the perturbation guaranteeing that to a \(k\)-dimensional space of solutions of system (1) exponentially decaying as \(t \to \infty \) there corresponds a \(k\)-dimensional manifold of solutions of the system \[ x' = A(t)x + f(t,x) \] tending to zero as \(t \to \infty\).