The Perron effect of change of values of characteristic exponents for solutions of differential systems (Q630263)

In this short note, the two-dimensional linear system \[ x' = A(t)x \] and its perturbation \[ y' = A(t)y + f(t, y) \] are considered and the following result is obtained: for arbitrary parameters \(\lambda_1 \leq \lambda_2 <0\), \(\beta_2 \geq \beta_1 \geq \lambda_2\) and \(m>1\), there exist a linear system with bounded infinitely differentiable matrix \(A(t)\), \(t\geq t_0\) with characteristic exponents \(\lambda_1(A)=\lambda_1 \leq \lambda_2(A)=\lambda_2 <0\), and an \(m\)-perturbation \(f: [t_0, +\infty)\times \mathbb{R}^2 \to \mathbb{R}^2\) infinitely differentiable in time and continuously differentiable with respect to \(y\in\mathbb{R}^2\) such that the nontrivial solutions \(y(t, c)\) of the perturbed system have Lyapunov exponents \(\lambda[y(\cdot, c)] = \beta_1\) for \(c_1 =0\) and \(\lambda[y(\cdot, c)] = \beta_2\) for \(c_1\not= 0\). Therefore, the Perron effect is fully realized that the characteristic exponents of exponentially stable linear systems change their sign under perturbations of higher-order smallness. This extends the authors' previous results [Differ. Equ. 46, No.~10, 1395--1408 (2010); translation from Differ. Uravn. 46, No.~10, 1388--1402 (2010; Zbl 1215.34062); a detailed version of the paper can be found in Differ. Equ. 46, No.~11, 1537--1551 (2010); translation from Differ. Uravn. 46, No.~11, 1536--1550 (2010; Zbl 1217.34090)].