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).


PIC
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 xrange 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) = sinx with the differential equation y = dydx = ncosnx = n1 - sin 2 nx = n1 - y2 is shown in the right window. The adjustable constant n determines the number of periods in the interval 0 x 2π

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.