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The definition of Collins dictionary of complex pendulum is a structure mounted so that it can swing freely under the influence of gravity. Complex pendulums can be highly non-linear and behave in a chaotic manner. Complex pendulums consist of multiple bobs and multiple arms attached to itself. Most of the complex pendulums have one pivot point. Now when the complex pendulum is let free in the air it will produce seemingly random but not random swings in the air. This report will explain how complex pendulums are chaotic. Report has multiple complex pendulums which demonstrate the chaotic effect. Report will provide illustrations and animations of how complex pendulums are sensitive to initial conditions.

Introduction

Chaos theory is a branch of mathematics foucing on the study of chaos - states of dynamical systems whose apparently random states of disorder and irregularity are often governed by the deterministic laws that are highly sensitive to initial conditions. Such sensitive dependence on initial position means that further one goes into the future the more inaccurate predictions become. Systems that are sensitive to initial conditions are said to be Chaotic. Chaos theory was summarized by Edward Lorenz. Because of the non linearity in the chaotic system it becomes difficult to make accurate predictions about the system over a time particular interval. Example of the application of chaotic theory is weather forecast. (Wikipedia)

Trajectory of double pendulum with small initial change

Sensitivity to initial conditions means that a small change in a variable can result in a large change in the trajectory of the pendulum. Even a change as small as 0.000001 degree can have a huge change on the later side of the pendulum that is the path and journey of the trajectory of the pendulum.

Graphical representation of difference in trajectory of double pendulum

Here the x-axis represents time and the y-axis represents the difference of trajectory. Here the scale of the system is shown at the top-left corner if values are two small, for values closer to the scale of the 1000th part, are shown simply with scale of 1. In this animation for sake of observation time moves at 5th of speed. This depicts the chaotic motion of two pendulums against each-other.

Here we can see that when the Pendulums start out their difference is in the sale of 1E-8 then 1E-6. After a short period their difference grows rapidly to the scale of 1E-3. And then again quickly goes as high as 2 (max distance possible). Then they don't have any correlation and seemingly move randomly.

Graph of two single pendulums in one and its difference of trajectories

Here x-axis represents time and y-axis represents difference. Here the scale of the system is shown at the top-left corner if values are two small, for values closer to the scale of the 1000th part, are shown simply with scale of 1. In this animation for sake observation time moves at 5th of speed. This depicts very correlated motion of two simple pendulums. Here we can see that when the Pendulums start out their difference is in the sale of 1E-8 then slowly their difference grows. Contrary to double pendulums their motion is much more correlated and their difference looks like a harmonic motion (because it is the difference between two harmonic curves). They stay very correlated and their difference grows predictable (more as time goes) and slowly.

Conclusion

Through the exploration of the double pendulum we can observe that chaotic systems are not random because if we have set the same initial degree for both of the pendulum the behavior will be excealty same. But we get differences in distance because of the difference in initial condition and for chaotic systems like double pendulum the difference can be very small and still produce big enough differences over time. That's why we can say that double pendulums or any chaotic systems are random but are very sensitive to the initial condition.