State scalar function defined on the state space of a system in order to obtain some qualitative properties - stability of equilibria, oscillatory behavior etc. - using a single function instead of several i.e. system’s state trajectories. A Lyapunov function is usually positive definite and, along system’s trajectories, is at least nonincreasing. The definite sign condition may also be relaxed for the generalized Lyapunov functions in the LaSalle sense. The basic physical model for the Lyapunov function is system’s energy - a state function that is nonincreasing along the state trajectory being at the same time positive definite. The strength of the Lyapunov function is exactly its independence of the physical concepts since writing down the stored energy of a system is not an easy job except possibly such standard cases as mechanical systems or electrical circuits. The energy like concepts may be nevertheless inspiring when “guessing” a Lyapunov function. In the infinite dimensional cases e.g. time delay or propagation systems, the Lyapunov function is replaced by a Lyapunov functional defined on the infinite dimensional state space.
Published in Chapter:
Neural Networks and Equilibria, Synchronization, and Time Lags
Daniela Danciu (University of Craiova, Romania) and Vladimir Rasvan (University of Craiova, Romania)
Copyright: © 2009
|Pages: 7
DOI: 10.4018/978-1-59904-849-9.ch178
Abstract
All neural networks, both natural and artificial, are characterized by two kinds of dynamics. The first one is concerned with what we would call “learning dynamics”, in fact the sequential (discrete time) dynamics of the choice of synaptic weights. The second one is the intrinsic dynamics of the neural network viewed as a dynamical system after the weights have been established via learning. Regarding the second dynamics, the emergent computational capabilities of a recurrent neural network can be achieved provided it has many equilibria. The network task is achieved provided it approaches these equilibria. But the dynamical system has a dynamics induced a posteriori by the learning process that had established the synaptic weights. It is not compulsory that this a posteriori dynamics should have the required properties, hence they have to be checked separately. The standard stability properties (Lyapunov, asymptotic and exponential stability) are defined for a single equilibrium. Their counterpart for several equilibria are: mutability, global asymptotics, gradient behavior. For the definitions of these general concepts the reader is sent to Gelig et. al., (1978), Leonov et. al., (1992). In the last decades, the number of recurrent neural networks’ applications increased, they being designed for classification, identification and complex image, visual and spatio-temporal processing in fields as engineering, chemistry, biology and medicine (see, for instance: Fortuna et. al., 2001; Fink, 2004; Atencia et. al., 2004; Iwahori et. al., 2005; Maurer et. al., 2005; Guirguis & Ghoneimy, 2007). All these applications are mainly based on the existence of several equilibria for such networks, requiring them the “good behavior” properties above discussed. Another aspect of the qualitative analysis is the so-called synchronization problem, when an external stimulus, in most cases periodic or almost periodic has to be tracked (Gelig, 1982; Danciu, 2002). This problem is, from the mathematical point of view, nothing more but existence, uniqueness and global stability of forced oscillations.