A continuous time system with a minimum of three independent dynamic variables can exhibit chaotic motion under certain initial conditions.
A damped and driven pendulum is one of the simplest systems to use in the study of chaos. One of the important characteristic of a chaotic system is its extreme sensitivity to initial conditions. This Demonstration implements a number of methods used in the analysis of such systems: bifurcation plot, Poincare map, phase portrait, time series, and power spectrum. The Lyapunov exponents method will be included in a future version.
Viscous damping is assumed at the point where the pendulum is suspended.
A torque at that point is then applied as the pendulum swings. The details section contains the theory and description of the use of this Demonstration.
Is a second-order nonlinear differential equation that models the motion of a simple pendulum subject
to a forcing function 
, where 
is the damping coefficient, 
is the length of the pendulum,
and 
is its angular position.
For ease of analysis this differential equation is first converted to a nondimensional form as follows. Let
where 
is the pendulum's undamped natural frequency given by 
and 
is the
gravitational constant.
Applying this transformation results in 
, and using
results in
Substituting these back into the original differential equation and dividing throughout by 
gives
Where the derivatives are now with respect to 
is called the quality factor. This is the nonlinear differential equation that is used in this
Demonstration, not the original one.
The driver frequency 
is given by 
where 
is the phase angle. This system is now
converted to state form 
which contains three first-order differential equations as
follows.
Let 
, and 
. Taking derivatives gives
Therefore viewing the system in state form gives
NDSolve[] is used to solve this system. The initial conditions are 
, and
. Therefore, the three dynamic variables (or state-space variables) in this model are
, and 
.
The top buttons are used to run and control the simulation. You can vary the simulation time up to a
maximum of 
units of time. Use the pause button to pause the simulation at any time. Use the step
button to advance the time by one unit. Use the reset button to bring the system back to initial time.
Use the button labeled fast to advanced the system to the end of the simulation period immediately
without going through the animation steps.
Use the slider labeled 
to adjust the size of the time step. The smaller the step size,
the more accurate and more smooth the motion will appear but it will take longer time to
run.
The parameters of the system 
are varied using the sliders. Making 
zero will
remove damping from the system, while making 
zero will remove the forcing function. This
effectively makes the system run as a standard simple pendulum with no damping and no forcing
function.
There are five main plots on the display. The Poincare map is always plotted to reflect the state at
the end of the simulation period. The 
axis represent the values of the angular velocity
of the pendulum sampled at time intervals equal in magnitude to the period of the forcing
function.
The phase plot is updated as time is advanced and at each step. Similarly for the time series plot and the power spectrum plot.
Switch between the time series plot and the power spectrum plot using the option selection shown in
the left part of the display. The phase portrait was drawn such that the angle of the pendulum is kept
between 
and 
to make it easier to draw the plot.
The display of the actual physical pendulum is also shown with a small arrow attached to the bob. The length of the arrow indicates the relative magnitude of the current angular velocity. The inner arrow represents the relative strength of the external torque.
The directions of the arrows represent the sign of the respective values. Anticlockwise arrow direction indicates a positive sign.
The bifurcation plot is only updated when you click the button labeled ”generate”. This lets you choose
your parameters before viewing the final result as it takes time to generate the plot. The parameters used
to select the bifurcation plot lets you select the range of values of the parameter 
(the torque
amplitude). The more intervals you ask for, the clearer the plot will be, but it takes longer to generate as
the system is numerically solved for each interval. Sampling of values starts after the time has
reached 
periods of the driver frequency, in order to allow transient effects to attenuate
sufficiently.
The solution is then run for an additional 
periods to collect the data for the plot. This is done at
each interval. Therefore, the 
axis for the bifurcation is similar to the y axis of the Poincare
map; however, the 
axis represents 
and not 
as in the phase portrait and Poincare
map.
The current time the simulation is at is displayed on the right side of the title of the bottom plot. When the system reaches the end of the simulation, the reset button can be used bring it back to the initial time.
Simulation will not advance beyond 500 time units.
The Demonstration comes with a number of test cases preconfigured to show different aspect of the behavior of the system. These test cases are selected from the popup menu shown. Once a test case is selected, the user interface parameters are updated automatically, and the final simulation result is shown based on the use of these new parameters.
was fixed in the program for efficient
implementation. Hence this value should not be increased without adjusting the corresponding
internal variable in the software that depends on this value.
