### 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
${y}^{\prime}={y}^{\prime}\left(y,x\right)$
$y\left(x\right),{y}^{\prime}\left(x\right)$
${y}^{\prime}\left(y\right)$ 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
$y\left(x\right)$ and its
derivative ${y}^{\prime}\left(x\right)$ also
the projection ${y}^{\prime}\left(y\right)$.

In Fig.5.9 the phase space projection for the system
$y\left(x\right)=sinx$ with the differential
equation ${y}^{\prime}=dy\u2215dx=ncosnx=n\sqrt{1-{sin}^{2}nx}=n\sqrt{1-{y}^{2}}$
is shown in the right window. The adjustable constant
$n$ determines the number
of periods in the interval $0\le x\le 2\pi $

For the case of the trigonometric function
${y}^{\prime}\left(y\right)$ 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 ${y}^{\left(n\right)}\left(y\right)$
make this difference particularly apparent.

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