Question 8.1: Finding phase paths for an autonomous system Sketch the phas...
Finding phase paths for an autonomous system
Sketch the phase diagram for the autonomous system of equations
\begin{gathered}\frac{d x_{1}}{d t}=x_{2}-1, \\\frac{d x_{2}}{d t}=-x_{1}+2.\end{gathered}
The phase paths of an autonomous system can be found by eliminating the time derivatives. The path gradient is given by
\begin{aligned}\frac{d x_{2}}{d x_{1}} &=\frac{d x_{2} / d t}{d x_{1} / d t} \\&=-\frac{x_{1}-2}{x_{2}-1}\end{aligned}
and this is a first order separable ODE satisfied by the phase paths. The general solution of this equation is
\left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2}=C
and each (positive) choice for the constant of integration C corresponds to a phase path. The phase paths are therefore circles with centre (2, 1); the phase diagram is shown in Figure 8.2.
The direction in which the phase point progresses along a path can be deduced by examining the signs of the right sides in equations (8.18). This gives the signs of \dot{x}_{1} \text { and } \dot{x}_{2} and hence the direction of motion of the phase point.
\begin{aligned}&\dot{x}_{1}=F_{1}\left(x_{1}, x_{2}\right), \\&\dot{x}_{2}=F_{2}\left(x_{1}, x_{2}\right),\end{aligned} (8.18)
