Question 1.10.3: Calculate the equilibrium positions of the pendulum of Figur......

The pendulum equation in state-space form is given by

\begin{aligned} & \dot{x}_1=x_2 \\ & \dot{x}_2=-\frac{g}{l} \sin \left(x_1\right) \end{aligned}

so that the vector equation F(x) = 0 yields the following equilibrium solutions:

x_2=0 \text { and } x_1=0, \pi, 2 \pi, 3 \pi, 4 \pi, 5 \pi \ldots

since sin(x_1) is zero for any multiple of π. Note that there are an infinite number of equilibrium positions, or vectors x_e. These are all either the up position corresponding to the odd values of π [Figure 1.47(c)], or the down position corresponding to even multiples of π [Figure 1.47(b)]. These positions form two distinct types of behavior. The response for initial conditions near the even values of π is a stable oscillation around the down position, just as in the linearized case, while the response to initial conditions near odd values of π moves away from the equilibrium position (called unstable) and the value of the response increases without bound.