Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convec...

In this paper, the structure of several convex cones that arise in the study of Lyapunov functions is investigated. In particular, the cones associated with quadratic Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases, some t...

The noninvariance of Lyapunov exponents in general relativity has led to the conclusion that chaos depends on the choice of the space-time coordinates. Strikingly, we uncover the transformation laws of Lyapunov exponents under general space-time transformations and we find that chaos, as characterized by positive Lyapunov exponents, is coordinate invariant. As a result, the previous conclusion ...

This paper studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. A number of versions of converse control-Lyapunov function theorems are proved and their connections to the switched LQR problem are derived. It is shown that the system is exponentially stabilizable if and only if there exists a finite integer N such...

This paper deals with a general class of discrete 2D systems based on the Roesser model. We first discuss the existing definitions of (Lyapunov) stability in the area of multidimensional systems and provide strong incentives to adopt new definitions in order to be coherent with the ones usually adopted in the 1D case. Once this background has been carefully designed, we develop different Lyapun...

The concept of control Lyapunov function has been proven a useful tool for designing robust control laws for nonlinear systems. Recently, this concept has been extended to time-delay systems in the form of Control Lyapunov Razumikhin functions. In this paper we further develop this extension by introducing Control Lyapunov Krasovsky functionals and show their use for robust stabilization of tim...

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform globa...

We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily exponentially stable. This complements our previous Lyapunov function constructions for rapidly time-varying dynamics. We also explicitly construct input-to-stat...

The paper presents a new method for higher order sliding mode control using control Lyapunov function for chain of integrator system with nonlinear uncertainties. The stability proof of the suggested scheme is analyzed in terms of two Lyapunov functions using appropriate switching function. Using these Lyapunov functions and switching scheme it is proved that, output and its higher order deriva...

We show that there is a Lyapunov functional for the neural field equation, a neural network model which represents highly dense cortical neurons as a spatially continuous field, and that the system necessarily converges to an equilibrium point as far as the length of the field is finite. We also show that the Lyapunov functional is a natural extension of the Lyapunov function of the Hopfield mo...