This monograph presents a collection of results observations and examples related to
dynamical systems described by linear and nonlinear ordinary differential and difference
equations. In particular dynamical systems that are susceptible to analysis by the Liapunov
approach are considered. The naive observation that certain diagonal-type Liapunov functions
are ubiquitous in the literature attracted the attention of the authors and led to some natural
questions. Why does this happen so often? What are the spe cial virtues of these functions in
this context? Do they occur so frequently merely because they belong to the simplest class of
Liapunov functions and are thus more convenient or are there any more specific reasons? This
monograph constitutes the authors' synthesis of the work on this subject that has been jointly
developed by them among others producing and compiling results properties and examples for
many years aiming to answer these questions and also to formalize some of the folklore or cul
ture that has grown around diagonal stability and diagonal-type Liapunov functions. A natural
answer to these questions would be that the use of diagonal type Liapunov functions is frequent
because of their simplicity within the class of all possible Liapunov functions. This monograph
shows that although this obvious interpretation is often adequate there are many in stances
in which the Liapunov approach is best taken advantage of using diagonal-type Liapunov
functions. In fact they yield necessary and suffi cient stability conditions for some classes
of nonlinear dynamical systems.
