matrices and M matrices instead. Though there are a good monograph (Berman & Plemmons, 1994) and an article (Fiedler & PtA ak, 1962), these subjects are not covered by a standard linear algebra course. Notwithstanding the above points, I strongly recommend the book because it covers important properties of positive systems, and it includes wide range of applications. I particularly :nd the anno...

Most dynamic models of interest in machine learning, robotics, AI or cognitive science are nonautonomous and input-driven. In the last few years number of important innovations have occurred in mathematical research on nonautonomous systems. In understanding the long term behavior of nonautonomous systems, the notion of an attractor is fundamental. With a time varying input, it turns out that f...

In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to...

Here, it is shown that the path-integral representation of any stochastic or deterministic continuous-time dynamical model is a cohomological or Witten-type topological field theory, i.e., a model with global topological supersymmetry (Q-symmetry). As many other supersymmetries, Q-symmetry must be perturbatively stable due to what is generically known as non-renormalization theorems. As a resul...

THEOREM ([41], [42]). Let f,Λ, m and R be as above. Then: (a) If ∫ Rdm <∞, then f admits an SRB measure μ. (b) If, additionally, gcd{Ri} = 1, then (f, μ) is mixing. (c) If m{R > n} < Cθ for some θ < 1, then ∃ θ̃ < 1 s.t. ∀φ, ψ, Cn(φ, ψ) < Cθ̃ . (d) If m{R > n} = O(n−α) for some α > 1, then Cn(φ, ψ) = O(n −α+1). (e) If R is as in (d) and α > 2, then the CLT holds for all φ. Next, we argue that con...

These notes are about the dynamics of systems with hyperbolic properties. The setting for the first half consists of a pair (f, μ), where f is a diffeomorphism of a Riemannian manifold and μ is an f -invariant Borel probability measure. After a brief review of abstract ergodic theory, Lyapunov exponents are introduced, and families of stable and unstable manifolds are constructed. Some relation...

Usage of fuzzy differential equations (FDEs) is a natural way to model dynamical systems under possibilistic uncertainty. We consider second order hybrid fuzzy differentia

In this paper, we present an innovative non–linear, discrete, dynamical system trying to model the historic battle of Salamis between Greeks and Persians. September 2020 marks anniversary 2500 years that have passed since famous naval which took place in late 480 B.C. The suggested describes very well most effective strategic behavior two participants during a (or war). Moreover, compare result...