Linear systems of ordinary differential equations

Abstract

The present survey deals chiefly with the Lyapunov theory of characteristic exponents of the general form of linear differential systems,with bounded piecewise continuous coefficients and particularly with its most recent advances. A sufficiently complete bibliography on such other independent branches of the theory of linear systems as "Linear Systems with Periodic Coefficients" (already surveyed in part [469] in the series General Mechanics) and the "Analytical Theory of Linear Systems" can be found in Sec. 15.

Stability of retarded systems with slowly varying coefficient

“…We extend the "freezing" method for ordinary differential equations cf. [1,7,12] and [3], Section 3.2, to equation (1.1). Besides, explicit exponential stability conditions are derived.…”

The freezing method for abstract nonlinear difference equations

“…Improvements of this method were developed by Izobov [11], Vinograd [16] and Gel'fand and Shilov [5]. Recently, this method was extended to linear difference systems by Gil' and Medina [9,14].…”

Decomposition of models of special classes of complex dynamical systems

“…In what follows we will also use some concepts from the general theory of linear dynamical systems [24,32,34].…”

“…Definition 1 [32][33][34]. The Lyapunov characteristic exponent of the solution x(t) = 0 of the linear differential system from the space LS is the number λ ≡ λ(x(t)) computed by the formula…”

“…A. M. Lyapunov has established that a linear differential equation system (1) can have at most n nonzero solutions with pairwise different Lyapunov exponents. The characteristic exponents of system (1) are determined in the following way [32][33][34]. Among all the bases in the solution vector space of system (1), chose those for which the sum of Lyapunov exponents of the basis solutions takes the least value.…”