Linear Version of the Anti-Perron Effect of Change of Positive Characteristic Exponents to Negative Ones
N. A. Izobov, А. В. Ильин · Differential Equations · 2022
A linear version of the anti-Perron effect of changing all positive Lyapunov characteristic exponents to negative ones is realized. For arbitrary numbers $$\lambda _n\geq \ldots \geq \lambda _1>0$$ and $$\mu _1\leq \ldots \leq \mu _n<0 $$ , the existence of the following $$n $$ -dimensional linear systems is proved: an original system $$\dot {x}= A(t)x$$ , $$t\geq t_0 $$ , with characteristic exponents $$\lambda _i(A)=\lambda _i $$ , $$i={1,\dots ,n}$$ , and a perturbed system $$ \dot {y}= A(t)y+Q(t)y$$ with perturbation matrix $$Q(t)\to 0 $$ as $$t\to \infty $$ and characteristic exponents $$\lambda _i(A+Q)=\mu _i $$ , $$i ={1,\dots ,n}$$ . Moreover, the coefficient matrices of the original and perturbed differential systems are bounded and infinitely differentiable on the half-line $$[t_0,+\infty )$$ .