5.8 Phase Space Diagrams

All variables of a system constitute its phase space. A selection of a few variables is referred to as a phase space projection. For a differential equation $Math content$ $Math content$ $Math content$ are three meaningful projections of the phase space.

The general characteristics divergent/ convergent/ oscillating of a differential equation, can be visualized well in a diagram that shows, in addition to the function $Math content$ and its derivative $Math content$ also the projection $Math content$.

In Fig.5.9 the phase space projection for the system $Math content$ with the differential equation $Math content$ is shown in the right window. The adjustable constant $Math content$ determines the number of periods in the interval $Math content$

For the case of the trigonometric function $Math content$ is for $Math content$ a circle that is transversed periodically; for $Math content$ the curve becomes because of the factor $Math content$ an ellipse and is not closed (why?). For $Math content$ this ellipse is transversed multiple times.

In this case the differential equation is particularly simple. More complex differential equations of order $Math content$ define families of more involved functions. One can however always differentiate between solutions that converge, diverge or oscillate with increasing variables, and the phase space projections $Math content$ make this difference particularly apparent.

Later we will visualize solutions of differential equations in more detail.