Refining and Optimizing Analytical Models
This section gives an example of how the model developed earlier could be extended to account for additional knowledge of the system's structure. Again, if you have not yet been exposed to this particular type of mathematical analysis, simply note that all models can be refined as we gain more knowledge about the system. Think back to the model you identified in the last section. How could you extend this model to make it a better reflection of reality?
Interaction (Euler's method for Systems)
Calculation (phase plane)
On the previous page, you have learned to approximate solutions of a system of differential equations with Euler's method. It is the same as for single differential equations, only now you work with a vector instead of a scalar. A graph of the solutions of the initial value problem is the following.
Both P and G have been plotted as functions of time t. Another way to visualise these solutions is to plot P and G against each other: to draw the solutions in the phase plane.
The phase plane is in this case the PG-plane. At any time t the solutions P and G constitute a point (P(t),G(t)) in the phase plane. The collection of points (P(t),G(t)) for parameter t in an interval form a curve in the phase plane.
A solution curve drawn in the phase plane is called a trajectory. If you take another initial value for P and/or G, you obtain a different solution, and you can add its trajectory to the phase plane as well. A few trajectories in a phase plane constitute a phase portrait of the differential equations. An example of a phase portrait is the following graph:
