On the ``freezing'' method for nonlinear nonautonomous systems with delay (Q5952371)

A fundamental aspect of nonlinear ordinary differential equations (ODEs) with delay is omitted in the paper: any stable solution has an influence domain of initial functions (``functional'' basin). There are two additional limitations: nonlinear terms must possess linear majorants, and linear terms must have a slow \(t\)-dependence. When the ``extent'' of the basin is unknown, only local stability can be claimed. The existence of hard-type oscillation near the reference solution \(x(t)\equiv 0\) is thus implicitly excluded.NEWLINENEWLINENEWLINEWhen \(x(t)\equiv 0\) is stable, the term ``absolute stability'' stands for structural stability with respect to small changes of the form of the ODEs.NEWLINENEWLINENEWLINEThe stability argument is based on Lyapunov's first method, as extended to ODEs with delay mainly of A. Halanay (not cited). According to a proof of Schmidt (1911, also not cited) stability is determined by the absolutely smallest (generalized) eigenvalues. The stability condition (equation 3.5) of the illustrative example defines the transition of \(x(t)\equiv 0\) to soft-type oscillations.