By Knebelman M. S.

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**Example text**

4. Time Scale on the Phase Plane. The phase-plane curve involves time implicitly so that a time scale can be set up along a solution curve. GRAPHICAL METHODS 37 M=0 FIG. 5. Isocline construction and phase-plane diagram for van der Pol equation with e = 5. Only the upper half plane and the limit cycle are shown. This process requires a step-by-step integration and can be performed in several ways. The most obvious way is based on the fact that, for small increments Ox and Or, the average velocity is A small increment Ox can be measured on the phase-plane curve and the corresponding va determined.

It should be noted that m = dx/dt is a quantity having dimensions and that the choice of numerical scales for the axes used in the construction is important. In Fig. 2, the scales along the x and t axes are the same, so that m can be interpreted directly as the slope of the solution GRAPHICAL METHODS 31 curve in the geometrical sense of the word. If it were desired to expand the x scale, say, so as to display the shape of the solution curve to better advantage, this would not be so. In other words, with an expanded x scale, the particular value m = f(x,t) = 1 would not correspond to an angle of 45 degrees with respect to the horizontal axis.

2. Solution curves are sketched in the figure, starting at each of the specified initial points. There are no singularities for this simple equation. It should be noted that m = dx/dt is a quantity having dimensions and that the choice of numerical scales for the axes used in the construction is important. In Fig. 2, the scales along the x and t axes are the same, so that m can be interpreted directly as the slope of the solution GRAPHICAL METHODS 31 curve in the geometrical sense of the word. If it were desired to expand the x scale, say, so as to display the shape of the solution curve to better advantage, this would not be so.