Asymptotic stability criterion for nonlinear monotonic systems and its applications (review)

Abstract

Introduction. The comparison method in the qualitative theory of equations is of paramount importance in studies of large-scale systems. This method is based on differential inequalities such as the Chaplygin-Wazewski inequality and Lyapunov functions (scalar, vector, or matrix-valued), which play the role of nonlinear transformation of the original system to an equation (a system or a matrix system) of lower dimension. The comparison method states that if for the system in question, there exists a Lyapunov function (scalar, vector, or matrix-valued) satisfying appropriate conditions, then various dynamic properties of the solutions of the original system follow from the respective dynamic properties of the comparison system.Thus, the stability analysis of a dynamic system involves setting up a Lyapunov function and establishing stability criteria for the zero solution of the comparison system.The present review discusses some approaches to solving such problems, establishes a uniform asymptotic stability criterion for a nonlinear comparison system, and exemplifies its applications in modern nonlinear dynamics.1. Problem Formulation. Let x t ( )be an n-dimensional state vector of some mechanical system with a finite number of degrees of freedom. Its behavior is described by the perturbed equations of motionwhere a, b are positive constants and j( ) t is a bounded nonnegative function. This equation cannot be integrated analytically, but using it leads to a number of interesting results for both system (8) and uncertain systems (see [30] and the references therein).3. System of Comparison with Vector Lyapunov Function. The curse of dimensionality in the sense of Bellman (see [8]) in the qualitative theory of equations makes it impossible to set up a scalar Lyapunov function for a large-scale system. In these conditions, Kron developed diakoptics [22] for the analysis of complex systems by parts. This method is based on the concept of a vector Lyapunov function and corresponding comparison system.Recall the following concept from the theory of monotonic systems (see [98]).

On Periodic Motion and Bifurcations in Three-Dimensional Nonlinear Systems

“…This result can be obtained by using, e.g., the method of comparison [9][10][11]. According to the method of comparison, we write the equations of comparison (Ważewski-type equations) with the property of quasimonotonicity.…”

“…According to the method of comparison, we write the equations of comparison (Ważewski-type equations) with the property of quasimonotonicity. The main sources and results of the investigations of stability of monotone systems can be found in [9,10] (see also the survey [11]). …”

Bifurcations and Multistability of the Oscillations of a Three-Dimensional System

“…The comparison method involves setting up comparison equations (Wazewski-type equations) that are quasimonotonic. Major references and results on the stability of monotonic systems are presented in [3,8].…”

“…Theorem on Instability of Wazewski System in a Cone K (see [3,8]). If the Wazewski system satisfies Assumptions 1 and 2 and there exists a sequence of points J Î m K, J ® m 0as m ® ¥, such that the following inequalities hold for each m:…”

“…Thus, the real part of the two complex-conjugate roots reverses sign at the point C. This facilitates the plotting of the separatrix on the plane xz. We use the characteristic equation (8) to plot curves that divide the plane xz into domains with different behavior of points. Figure 1a shows the separatrix that divides the plane xz of an exponentially nonlinear generator.…”