I find chaos theory fascinating, the idea that something can be both deterministic

Double pendulum diagram

showing

A simulation of a double pendulum was constructed to investigate the predictability of its final position after a set amount of time, for different initial conditions. Certain ranges were found to be predictable while others were chaotic. The boundaries of these regions showed a direct correlation to where the lower pendulum inverted.

Many natural systems exhibit chaos, such as the weather. Despite their deterministic nature these systems can be unpredictable due to their extreme sensitivity to initial conditions. Knowing what defines the chaotic regions is important in many fields.

In the simplest (*Ref1*)
double pendulum (Figure1) *m* << *M*, thus the motion (amplitude) of *M* can be approximated by (*eq.*1).

(*eq.*1)

where *a _{0}*
and

(*eq*.2*&*3)

where *ω* is
the angular velocity of *m*. Constants *r*,*
g* are arbitrarily chosen to be 1. Initial values of *ω*, *θ* are chosen
to be 0.0 and 0.1, respectively.

The first simulation evolves for t_{max}=100 seconds
and records the *θ*_{final}.
This simulation is repeated through a range of values for *a _{0}* and

Values of θ for 0.21 < a

Values of θ for 0.21 < a

A phase-space plot (*ω*
verses *θ*) for a single pendulum
would be a circle. Figures 4&5 show phase-space plots for simulations
inside and outside the chaotic boundary. The chaotic pendulum’s plot stretching
to the left demonstrates *θ*
leaving its normal range, i.e. it made a full rotation. Investigating this as
the reason for the chaotic region, the program was modified to identify regions
where the lower pendulum inverted. This was achieved by measuring the
proportion of time *θ* escaped the
range (-π,π) and highlighting the graph in red to represent it. (Figures6&7)

Phase-space plot with a

Phase-space plot with a

Values of θ for 0.21 < a

Values of θ for 0.21 < a

A deeper hue means θ spent a bigger fraction of the simulation outside the region ( -π , π )

As time evolves, the chaotic regions become clearer and spread asymptotically. The red highlight follows the same area exactly, as can be seen by comparing Figures 2&3 to Figures 6&7. Zooming out, the boundary of chaos surrounds the regular, predictable pattern in a parabola (Figure8). Outside this region the pendulum behaves like a single pendulum as the amplitude of M is too small, or the period of M is too large, to have an effect. The base of the region marks the resonant frequency of the lower pendulum (ref3).

Values of θ for 0.0 < a

Same as figure 8 but with θ

Same as figure 8 but with θ

Investigating the dependency on initial conditions, Figures 9&10
show how the same area changes when
and
(*ω _{init}* is kept constant). The chaotic region grows
as the pendulum can escape the normal region much easier. At
the graph becomes pure
chaos. Changing

Choosing levels of accuracy is a compromise with computing power and available time. Reducing Δt makes the regions clearer but takes longer to render. Throughout the simulations Δt=0.1, chosen after examining Figure12.

Throughout, all decimal variables were stored as double
floats (8 bytes). This gives around 16 places of precision. The limitations of
the program are visible when simulations are made in steps of ~10^{-17}
(Figure11). When investigating chaos, rounding errors can have a huge effect on
the output (*ref2*). However, the
graphs in this simulation were plotted in steps of ~10^{-5} hence
rounding errors should not have effected the results significantly.

Extremely high resolution plots become “pixilated” as the limitations of variable accuracy become evident.

Three test plots of the same area for different values of Δt. The upper and centre plots show noticeable differences, whereas the lower and centre plots have negligible difference, justifying the use of Δt=0.1 as a compromise.

The model itself is only an approximation, assuming the
angle of *M* remains small. This has shown the basic concepts of a double
pendulum but a more relevant result would be achieved if *m* ≈ *M*. A plot of
initial angles for both pendulums on either axis would produce a result similar
to Figure13.

Figure 13.

- David Clements,
*First Year Computing Laboratory*, Version 2.9 - Raymond Sneyers (1997) "Climate Chaotic Instability: Statistical Determination and Theoretical Background",
*Environmetrics*, vol. 8, no. 5, pages 517–532. - Meirovitch, Leonard (1986).
*Elements of Vibration Analysis*(2nd edition ed.). McGraw-Hill Science/Engineering/Math.

The output in this file was then fed into and plotted using MS Excel.