1. Lyapunov Exponents for Differential Equations 2. Abstract Theory of Lyapunov Exponents 3. Regularity of Lyapunov Exponents Associated with Differential Equations 4. Lyapunov Stability Theory 5. The Oseledets Decomposition 6. Dynamical Systems with Nonzero Lyapunov Exponents. Multiplicative Ergodic Theorem 7. Nonuniform Hyperbolicity. Regular Sets 8. Examples of Nonuniformly Hyperbolic System...

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...

This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sections contain some very general results in nonuniform hyperbolic theory. We consider ( f , μ), where f is an arbitrary dynamical system and μ is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical...

This chapter presents a new face recognition system comprising of feature extraction and the Lyapunov theory-based neural network. It first gives the definition of face recognition which can be broadly divided into (i) feature-based approaches, and (ii) holistic approaches. A general review of both approaches will be given in the chapter. Face features extraction techniques including Principal ...

chaos theory discusses unstable and non-periodic behavior of non-linear fluctuating dynamicsystems. since evaporation level variations of lakes have a dynamic identity, chaos theory can have a uniquerole in collecting data of these variations. therefore it is an important and challenging problem in the field ofplanning and qualitative and quantitative management of urmia lake to verify whether ...

This paper discusses the application of Lyapunov theory in chaotic systems to the dynamics of tracer gradients in two-dimensional flows. The Lyapunov theory indicates that more attention should be given to the Lyapunov vector orientation. Moreover, the properties of Lyapunov vectors and exponents are explained in light of recent results on tracer gradients dynamics. Differences between the diff...

This book provides a systematic introduction to smooth ergodic theory, including the general theory of Lyapunov exponents, nonuniform hyperbolic theory, stable manifold theory emphasizing absolute continuity of invariant foliations, and the ergodic theory of dynamical systems with nonzero Lyapunov exponents. The book can be used as a primary textbook for a special topics course on nonuniform hy...