In the most general case we will study systems of the type

where

is an element in a vector space for every fix choice of *t*.

In this course we will mainly study systems on the form

where *x=x(t)* and *y=y(t)* are regular real valued functions. In this case

The solutions are defined on some interval

and can be drawn in the *state space*.

Alternatively they can be represented as a parametric curve in the *xy*-plane, also known as the *phase plane*.

The arrows in the figure show how the system is developed in time. Here

*The equilibrium points* are obtained by solving the system

Since

we obtain

A *phase portrait* of the system is all orbits and equilibrium points in the phase plane. A quick way to get a phase portrait is to:

*
1)* Find all equilibrium points by solving the system

*2)* Let a standard software (e.g. MATLAB) plot the solutions of

In part 9 we describe a more careful way to create phase portraits of linear dynamical systems, which we can get as approximations of general nonlinear systems by a first order taylor expansion. We give several concrete examples of continuous dynamical systems in the next part.