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