Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions [7]. Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in [17] and further impro...

The Lyapunovmethod for stability analysis of an equilibrium state of a nonlinear dynamic system requires a Lyapunov function v(t, x) having the following properties: (1) v is a positive definite function, and (2) v̇ is at least a negative semi-definite function. Finding such a function is a challenging task. The first theorem presented in this paper simplifies the second property for a Lyapunov ...

An artificial neural network is proposed as a function approximator for empirical modeling of a Lyapunov function for a nonlinear dynamic system that projects stable behavior as potentially observable in its state space. Theoretical framework for the methodology of designing the so-called Lyapunov neural network, which empirically models a Lyapunov function, is described. Algorithms for trainin...

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...

The analysis of potentially chaotic behavior in biological and biomedical phenomena has attracted great interest in recent years (1–6). Although no universally accepted mathematical definition of the term chaos exists, Strogatz (7) provides a working definition as ‘‘aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.’’ Aperiodic long-...

We introduce the concept of sos-convex Lyapunov functions for stability analysis of both linear and nonlinear difference inclusions (also known as discrete-time switched systems). These are polynomial Lyapunov functions that have an algebraic certificate of convexity and that can be efficiently found via semidefinite programming. We prove that sos-convex Lyapunov functions are universal (i.e., ...

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a subsystem by a suitable rescaling in a very cost effective way. We compute the Lyapunov spectrum for the subsystem by truncating the original Jac...

We introduce a definition of a "localization width" whose logarithm is given by the entropy of the distribution of particle component amplitudes in the Lyapunov vector. Different types of localization widths are observed, for example, a minimum localization width where the components of only two particles are dominant. We can distinguish a delocalization associated with a random distribution of...