Chaotic solutions to the driven pendulum

Some 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.

Simulation

Phase Portrait

Poincaré map

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$q$

= 2

$A$

= 1.15

$\Omega$

= 0.67

For the parameters $q=2$ and $\Omega=0.67$:

  • At , a stationary solution at $\theta=0$ appears. Let the transients decay then press 'Clear all'. The phase portrait is a single point and the Poincaré map is a single point.
  • At , a periodic solution appears that has the same period as the drive force. Let the transients decay then press 'Clear all'. The Poincaré map is a single point.
  • At , there are two symmetry broken solutions. They both have the period of the drive. Start from different initial conditions (Change $A$ to a large value and then back to $A=1.04$). Let the transients decay then press 'Clear all'.
  • At , there are period doubled solutions that have a period that is twice as long as the drive frequency. Be patient to let the transients decay then press 'Clear all'.
  • At , there are chaotic solutions that never repeat. If you wait a long time, the Poincaré map will display a self similar structure. This is called a strange attractor.