
This chapter corresponds to a brief introduction to some of the most important concepts in
dynamical systems. It starts by presenting the basic formulation in term of systems of
differential equations,
including the nice graphical interpretation of the phase velocity field (see [Arnol'D
]) in terms of the instantaneous
velocity at each point in the state space. Next, linear systems are presented and illustrated in terms of examples,
and the general solution to systems of linear differential equations in terms of
the fundamental matrix is developed
and also illustrated. The important role of the eigenvectors of the coefficient matrix in determining the long term
tendency of the dynamic evolution of such linear systems is then conceptually discussed and illustrated. As a matter
of fact, it is rather hard to verify that the long term evolution of the situation in Figure 5.4c will tend to the
eigenvector, since this is not even illustrated in the picture. This chapter concludes with a brief discussion of
nonlinear systems,
outlining the stability issue, linearization, and
Lyapunov functions. This last concept, however,
is only approached in an abstract fashion, not being illustrated in terms of an example.


