Nonautonomous equations with arbitrary growth rates: A Perron-type theorem (Q448530)

The paper considers a linear system NEWLINE\[NEWLINE x'=A(t)x,\quad x\in\mathbb C^n, \tag{1} NEWLINE\]NEWLINE under the assumption that the matrix \(A(t)\) is in block form and all Lyapunov exponents are limits. It is shown that the asymptotic exponential behavior of the solutions of the linear system (1) carries over to the solutions of the nonlinear equation NEWLINE\[NEWLINE x'=A(t)x+f(t,x) \tag{2} NEWLINE\]NEWLINE for any sufficiently small perturbation \(f\). This means that, for any solution \(x(t)\) of (2), the limit NEWLINE\[NEWLINE \lambda=\lim_{t\to +\infty}\frac{1}{t}\log ||x(t)|| NEWLINE\]NEWLINE exists and coincides with a Lyapunov exponent of the linear equation (1). The perturbation is required to satisfy NEWLINE\[NEWLINE \lim_{t\to +\infty}\int_t^{t+1}\text{e}^{\delta\tau} \sup_{x\neq 0}\frac{||f(\tau,x)||}{||x||}d\tau=0 NEWLINE\]NEWLINE for some \(\delta >0\) or certain more general condition. The approach is based on Lyapunov's theory of regularity.