We provide the main features of Lyapunov theory when it is formulated for fractional order systems. We give consistent extensions of Lyapunov, LaSalle and Chetaev classical theorems to the case of fractional order systems. We give examples to illustrate the applications of the concepts and propositions introduced. © 2016 Elsevier Inc. All rights reserved.

We generalize the concept of Lyapunov exponent to transformations that are not necessarily differentiable. For fairly large classes of repellers and of hyperbolic sets of differentiable maps, the new exponents are shown to coincide with the classical ones. We also discuss the relation of the new Lyapunov exponents with the dimension theory of dynamical systems for invariant sets of continuous t...

In this paper we build on the spectral theory for linear fractional differential equations and prove that the fractional Lyapunov spectrum of solutions starting from a unit sphere is the union of a compact interval in R<0 and at most d distinct fractional Lyapunov exponents. AMS Subject Classifications: 34A08, 34D08.

Motivated by their important role in smooth dynamical systems, Lyapunov exponents have been conceived decades ago as a means to study the stability of cellular automata (CAs). More precisely, they quantify their sensitive dependence on initial conditions. As a next step towards the establishment of a dynamical systems theory of CAs that is inspired by its analogue for smooth dynamical systems, ...

A class of global Lyapunov functions is revisited and used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multi-group models in mathematical epidemiology. We show how the group structure of the models, as manifested in the derivatives of the Lyapunov function, can be completely described using graph theory.

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the inputto-state stability of a system. Then for the case of the systems described by abstract equations in Banach spaces we develop two methods of construction of local and global I...