On the stability of periodic solutions to nonlinear differential systems (Q2761391)

The author considers the system of differential equations NEWLINE\[NEWLINE \frac{dx}{dt} = f(t,x),\tag{1} NEWLINE\]NEWLINE with \(x\in \mathbb{R}^n\), \(f(t,x)\in C_{tx}^{(0,1)} (\mathbb{R}\times \mathbb{R}^n)\), \(f(t+\omega,x) = f(t,x)\), and the corresponding first approximation system NEWLINE\[NEWLINE \frac{du}{dt} = A(t)u,\tag{2} NEWLINE\]NEWLINE with \(A(t) = f_x'(t,\eta(t))\) and \(\eta(t)\) is an \(\omega\)-periodic solution to system (1). General coefficient criteria for the asymptotic stability of system (1) are established in terms of the matrix of fundamental solutions to system (2). Some corollaries are presented.