Phase Space Diagrams

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 $y^{'} = y^{'} (y, x)$ $y (x), y^{'} (x)$ $y^{'} (y)$ 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 $y (x)$ and its derivative $y^{'} (x)$ also the projection $y^{'} (y)$ .

Figure 5.9: Phase-space projections for

n = 1

in the figure. The left window shows

y (x)

in blue and

y^{'} (x)

ins green. The zero line is marked in magenta. The right window shows

y^{'} (y)

. The parameter

x r a n g e

determines the size of the interval, the parameter

n

the number of periods in the interval. The blue point in the phase space is the end point of the interval.

In Fig.5.9 the phase space projection for the system $y (x) = sin x$ with the differential equation $y^{'} = d y ∕ d x = n cos n x = n \sqrt{1 - {sin}^{2} n x} = n \sqrt{1 - y^{2}}$ is shown in the right window. The adjustable constant $n$ determines the number of periods in the interval $0 \leq x \leq 2 π$

For the case of the trigonometric function $y^{'} (y)$ is for $n = 1$ a circle that is transversed periodically; for $n < 1$ the curve becomes because of the factor $n$ an ellipse and is not closed (why?). For $n > 1$ this ellipse is transversed multiple times.

In this case the differential equation is particularly simple. More complex differential equations of order $n$ 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 $y^{(n)} (y)$ make this difference particularly apparent.

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