On linear perturbations decaying at infinity and altering the Lyapunov exponents of regular linear differential systems (Q2027307)

In this paper, the author addresses a classical problem on the instability of Lyapunov exponents, \(\lambda_i(A)\), \(i=1,2,\cdots,n\) of linear system of ordinary differential equations \[ \dot x = A(t) x, \quad t\ge 0, \ x\in {\mathbb R}^n.\tag{S} \] The main result states that for any positive, monotone increasing function \(\theta(t)\) such that \(\lim_{t\to+\infty} \theta(t)= +\infty\) and \(\lim_{t\to+\infty} \frac{\theta(t)}{t}=0\), there is some example of regular system \((S)\) such that one has some perturbation \(Q(t)\) of the order \(O(\exp(-\theta(t)))\) as \(t\to +\infty\) so that \(\lambda_n(A+Q)> \lambda_n(A)\). Such a decay condition on perturbations is almost optimal, because it has been known that a regular system has stable Lyapunov exponents when the perturbations \(Q(t)\) decay like \(\limsup_{t\to +\infty} \frac{1}{t}\ln \|Q(t)\| <0\).