Lyapunov's direct method and parametric resonance in linear systems with delay (Q2738669)

Stability conditions are derived by the method of perturbed Lyapunov functions for the following linear system with delay NEWLINE\[NEWLINE y''+L^2 y+\mu P(\nu t)y(t-h)=0,NEWLINE\]NEWLINE where \(y=(y_1,y_2\ldots,y_m)\), \(\mu\) is a small parameter, \(L^2 = \text{diag}(\lambda_1,\ldots,\lambda_m)\), \(\lambda_i>0\), \(\lambda_i\neq \lambda_j\), \(i\neq j\), \(i,j=1,\ldots,m\), \(h,\nu>0\) and \(P(s)\) is a symmetric \(m\times m\)-matrix. It is shown that the introduction of the delay essentially changes the behavior of solutions and the conditions for the appearance of parametric resonance.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00044].