Leonov, G. A.; Smirnova, V. B. Nonlocal reduction method in differential equations theory. (English) Zbl 0755.34045 Topics in mathematical analysis, Vol. Dedicated Mem. of A. L. Cauchy, Ser. Pure Math. 11, 658-694 (1989). [For the entire collection see Zbl 0721.00014.]The paper is devoted to a new method of stability investigation for dynamical systems. The authors give a short historical review of basic concepts of global qualitative study of differential equations and describe the sources of the new method. These are the concept of surfaces without tangency and the comparison principle. The new Leonov’s method which is called nonlocal reduction method proposes that a Lyapunov function constructed for the initial system should contain information about a certain comparison system which is called a reduction system. The latter is of lower order than the initial one but preserves all its qualitative properties. The nonlocal reduction method gives the opportunity to prove new Lyapunov-type theorems. In particular, the paper contains a theorem about Lagrange stability which is “wider” than the well-known Yoshizawa theorem. The paper contains also frequency-domain stability and oscillations theorems for indirect control systems with periodic nonlinearities, a new frequency-domain criterion of absolute stability, different from the Popov criterion. The new method may be extended to distributed parameter systems. Reviewer: V.Yakubovich (St.Petersburg) Cited in 3 Documents MSC: 34D20 Stability of solutions to ordinary differential equations 34D99 Stability theory for ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory 93D10 Popov-type stability of feedback systems Keywords:stability; dynamical systems; historical review; global qualitative study of differential equations; surfaces without tangency; comparison principle; nonlocal reduction method; Lyapunov function; Lagrange stability; frequency-domain stability and oscillations theorems; indirect control systems; periodic nonlinearities; absolute stability; Popov criterion Biographic References: Cauchy, A. L. Citations:Zbl 0721.00014 PDFBibTeX XMLCite \textit{G. A. Leonov} and \textit{V. B. Smirnova}, in: Analytic and Gevrey regularity for linear partial differential operators. . 658--694 (1989; Zbl 0755.34045)