1. Dynamical Systems

The class of problems concerned with determining the time dependence of a certain parameter is called dynamical systems. In particular, this class includes Cauchy problems for systems of ordinary differential equations (ODEs). We have already discussed solving ODEs in detail, and now we will consider specific methods for examining their subclass, dynamical systems.

As you recall, a dynamical system is a system of ODEs of the following type:

In addition to the argument x (t), which is of a vector type, any of the functions f _i can be dependent on any x _i (t). Moreover, there can be other arguments; in particular, functions f _i may depend implicitly on time t (in which case the dynamical system of equations is called nonautonomous):

Autonomous systems (1), which do not contain the time argument t as an explicit dependence, are more frequently the subjects of dynamical system theory analyses.

Additionally, functions f _i may contain different parameters as arguments, denoted as vector , which is very characteristic of most dynamical systems of practical interest:

A dynamical system can be parametrically dependent on one, two, or more of such parameters. Thus, dynamical systems involve formulating a system of ODEs to find the unknown functions x _i (t) as Cauchy problems initial condition problems. In addition to the equations themselves (of which there should be N in the general case: i=1, , N), N