Boundary effects in the stepwise structure of the Lyapunov spectra and corresponding wavelike structure of the Lyapunov vectors are discussed numerically in quasi-one-dimensional systems of many hard disks. Four different types of boundary conditions are constructed by combinations of periodic boundary conditions and hard-wall boundary conditions, and each leads to different stepwise structures...

Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinosson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always s...

We consider strongly connected networks of input-to-state stable (ISS) systems. Provided a small gain condition holds it is shown how to construct an ISS Lyapunov function using ISS Lyapunov functions of the subsystems. The construction relies on two steps: The construction of a strictly increasing path in a region defined on the positive orthant in R by the gain matrix and the combination of t...

Lyapunov functions for general systems are difficult to construct. However, for autonomous linear systems with exponentially stable equilibrium, there is a classical way to construct a global Lyapunov function by solving a matrix equation. Consequently, the same function is a local Lyapunov function for a nonlinear system. In this paper, we generalise these results to time-periodic and, in part...

This paper studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods tha...

We present a framework to transform the problem of finding a Lyapunov function of a Chemical Reaction Network (CRN) in concentration coordinates with arbitrary monotone kinetics into finding a common Lyapunov function for a linear parameter varying system in reaction coordinates. Alternative formulations of the proposed Lyapunov function is presented also. This is applied to reinterpret previou...

Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a verification that the constructed function...

For any N ×N monodromy matrix we define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of...

We consider a finite number of nonlinear systems interconnected in an arbitrary way. Under the assumption that each subsystem is input-to-state stable (ISS) regarding the states of the other subsystems as inputs we are looking for conditions that guarantee input-to-state stability of the overall system. To this end we aim to construct an ISS-Lyapunov function for the interconnection using the k...