Motivated by a recently developed backward error analysis for QR methods, we consider the error in the Lyapunov exponents of perturbed triangular systems. We consider the case of stable and distinct Lyapunov exponents as well as the case of stable but not necessarily distinct exponents. We illustrate our analytical results with a numerical example. 1. Introduction. Lyapunov exponents are often ...

A new direct method for the computation of Lyapunov quantities (Lyapunov values or coefficients, Poincare-Lyapunov constants, focus values) for the second-order dynamical system, permitting us to narrow the requirements on a smoothness of system, is obtained.

Lyapunov equations that dependon communication graph topologies provide building blocks of Lyapunov functions, which play an important role in controller design and stability analysis of multi-agent systems. However, construction of such a Lyapunov equation in some published works has a flaw or requires unnecessarily strong graph conditions. This paper presents several choices of Lyapunov equat...

The recently developed technique for computation of piecewise quadratic Lyapunov functions is specialized to Lyapunov functions that are piecewise linear. This establishes a unified framework for computation of quadratic, piecewise quadratic, piecewise linear and polytopic Lyapunov functions. The search for a piecewise linear Lyapunov function is formulated as a linear programming problem, and ...

In this paper, we introduce Lyapunov measure to verify weak (a.e.) notions of stability for an invariant set of a nonlinear dynamical system. Using certain linear operators from Ergodic theory, we derive Lyapunov results to deduce a.e. stability. In order to highlight the linearity of our approach, we also provide explicit formulas for obtaining Lyapunov function and Lyapunov measure in terms o...

We show that existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided an additional mild assumption is fulfilled. For UGAS infinite-dimensional systems with external disturbances we derive a novel ’integral’ construction of non-coercive Lipschitz continuous Lyapunov functions. F...

This paper introduces the notion of Lipschitz stability for nonlinear th order 8 matrix Lyapunov differential systems and gives sufficient conditions for Lipschitz stability. We develop variation of parameters formula for the solution of the nonhomogeneous nonlinear th order matrix Lyapunov differential system. We 8 study observability and controllability of a special system of th order nonline...

In this paper, we introduce a new hyperchaotic complex T-system. This system has complex nonlinear behavior which we study its dynamical properties including invariance, equilibria and their stability, Lyapunov exponents, bifurcation, chaotic behavior and chaotic attractors as well as necessary conditions for this system to generate chaos. We discuss the synchronization with certain and uncerta...

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...