III.4 Phase Trajectories and Limit Cycles

In this section the concept of phase trajectories is explored. The following two equations describe a so-called  limit cycle:

=+y +

=-x +

With initial conditions: and   and .
In the following  the system will be studied over a period of 10 seconds with the following sets of initial conditions:

Let's start  by setting up the equations:

Next  a series of parametric plots is produced for each initial condition, showing the trajectories of and for different .

Finally   all the results  are plotted  together in one graph:

OPTIONAL
III.4.1a Explain in your own words the results obtained in the graphs above. What will happen if you choose two initial conditions very close to one another, inside and outside the limit cycle. What happens in point (0,0)?