5 Dynamics
    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.