# Lyapunov functions

Lyapunov Functions and Threshold CA Rules [c.274]

Lyapunov Functions and Threshold CA Rules [c.277]

Lyapunov Functions and Threshold CA Rules [c.279]

Lyapunov Functions and Threshold CA Rules [c.281]

Lyapunov Functions and Threshold CA Rules [c.283]

Lyapunov Functions and Threshold GA Rules [c.285]

Lyapunov Functions and Threshold CA Rules [c.287]

Lyapunov functions are discussed in section 5.5. [c.521]

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. [c.19]

Classical Lyapunov functions are an important tool for determining the stability of solutions to continuous dynamical systems [sanchfiS]. An analogous discrete construct, which is the subject of this section, has also proven to be enormously useful in charac terizing the behavior of certain kinds of CA rules most notably the class of threshold (or voting see section 3.4.3) rules. In particular, Lyapunov functions help to both characterize the long-term behavior of a finite system by specifying the possible cycle lengths, and provide bounds on the length of transients. [c.274]

Threshold automata date back to McCulloch and Pitts [mccul43] interconnected networks of binary valued neurons. In their networks, a given neuron becomes active if and only if the number of active neighboring neurons exceeds some threshold. In order to emphasize the broad applicability of Lyapunov functions, we will actually deduce both the long-term and transient behavior for slightly more generalized (but still binary valued) versions of threshold rules than those originally introduced by McCulloch and Pitts. In particular, we will prove results for rules for which different sites may have different thresholds, and the underlying lattice structure (or, more correctly, the neuronal interconnectivity pattern) is essentially arbitrary. Properties of simpler one and two dimensional systems, such as the class of voting rules, will then be seen as immediate consequences of much more general theorems. [c.275]

See pages that mention the term

**Lyapunov functions**:

**[c.274] [c.275] [c.276] [c.283] [c.341] [c.509] [c.522] [c.752] [c.756] [c.756] [c.528]**

See chapters in:

** Cellular automata
-> Lyapunov functions
**

Cellular automata (2001) -- [ c.521 ]