Comprehending Complexity: Data-Rate Constraints in Large-Scale Networks

This paper is concerned with the rate at which a discrete-time, deterministic, and possibly large network of nonlinear systems generates information, and so with the minimum rate of data transfer under which the addressee can maintain the level of awareness about the current state of the network. While being aimed at development of tractable techniques for estimation of this rate, this paper advocates benefits from directly treating the dynamical system as a set of interacting subsystems. To this end, a novel estimation method is elaborated that is alike in flavor to the small gain theorem on input-to-output stability. The utility of this approach is demonstrated by rigorously justifying an experimentally discovered phenomenon. The topological entropy of nonlinear time-delay systems stays bounded as the delay grows without limits. This is extended on the studied observability rates and appended by constructive upper bounds independent of the delay. It is shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics

Review on contraction analysis and computation of contraction metrics

Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit. Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy). This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics

Stability analysis via averaging functions

\u3cp\u3eA new class of Lyapunov functions is proposed for analysis of incremental stability for nonlinear systems. This class of Lyapunov functions allows to establish input-dependent incremental stability criteria. Two substantially different situations are considered: when incremental stability is guaranteed by the inputs of sufficiently small amplitude and when, similar to the excited van der Pol oscillator, the stability is induced by sufficiently large inputs.\u3c/p\u3

Stability analysis via averaging functions

A new class of Lyapunov functions is proposed for analysis of incremental stability for nonlinear systems. This class of Lyapunov functions allows to establish input-dependent incremental stability criteria. Two substantially different situations are considered: when incremental stability is guaranteed by the inputs of sufficiently small amplitude and when, similar to the excited van der Pol oscillator, the stability is induced by sufficiently large inputs