Numerical Methods  

Chaotic solutions to the driven pendulumSome differential equations have analytic solutions that can be expressed in terms of simple functions like $\sin(t)$ or $\exp(t)$. The solutions to nonlinear differential equations are typically more difficult to express numerically although sometimes an analytic expression for an approximate solution can be found. There are however some differential equations that exhibit chaotic solutions. It is not possible to find an analytic expression for a chaotic solution. One system that exhibits chaotic solutions is the driven pendulum. The differential equation that describes the motion of the pendulm can be written in a normalized form, [Fitzpatrick 2006], \[ \begin{equation} \large \frac{d^2\theta}{dt^2}+\frac{1}{q}\frac{d\theta}{dt}+\sin(\theta)=A\cos(\omega t), \end{equation} \]where $q$ describes the damping, $A$ measures the torque that is used to drive the pendulum at frequency $\omega$ and $\theta$ is the angle measured from vertical. At $\theta=0$ the pendulum hangs down and at $\theta=\pi$ the pendulum stands up. The left panel below shows a simulation of the motion of the driven pendulum. The center panel shows a phase portrait where $\sin\theta$ is ploted horizontally and $\frac{d\theta}{dt}$ is plotted vertically. The right panel is the Poincaré map. A red point at $(\sin\theta,\frac{d\theta}{dt})$ is plotted in the Poincaré map every time $\omega t = 2\pi n$ where $n$ is an integer.
For the parameters $q=2$ and $\omega=0.67$:
