"Does God Play Dice? The New Mathematics of Chaos" is a beautiful book by Ian Stewart. It provided me with a valuable introduction into the concept of chaos and chaos theory. Motivated by the book I decided to experiment with Chaos and write a dynamics simulator on MATLAB to simulate a 2 degrees of freedom pendulum (double pendulum).
In chaos theory 'Chaos' does not necessarily mean random (or a state of disorder). "Chaos occurs when a deterministic system behaves in an apparently random manner". Chaos is actually all around us. Chaos is the rule rather than the exception.
It is only when you think that everything is under 'control' that you wake up one morning to see that yesterday's weather forecast was all wrong and there is a huge storm outside, you check the stock market only to find out that you have lost thousands of dollars. HELL! you turn on the news only to realize that suddenly the Libyan people rose against a crazy tyrant who ruled them for 42 years. Chaos.
An important characteristic of Chaotic systems is their high sensitivity to initial conditions. Even for a deterministic system (with no random elements), this high sensitivity to initial conditions is what causes the impossibility of long term prediction of the system's behavior. A slight change in the initial conditions would yield completely different and diverging end results.
To experiment with Chaos, I wrote a dynamic simulation of a double pendulum (2DOF pendulum) on MATLAB. The double pendulum is a classical example of Chaos theory. I used the "Simple Forward Kinematics library for Robotic Chains" that I developed back in 2010 (click here to download), and I developed a new library to compute the joint-space dynamic equations and to perform the simulation for any kinematic chain (click here to download). So back to our chaos theory. To experiment, I ran two simulations:
- The first (on the left in the video below) is a simulation with a base joint initial position of 130 degrees
- The second (on the right in the video below) is a simulation with a base joint initial position of 129 degrees
Notice that although the initial conditions are almost the same, a difference in the trajectory begins to be noticeable only after 7 seconds. This difference increases as time passes. And after 40 seconds we notice that each pendulum is located on a totally different position in space.
We've all heard of the vivid and beautiful term "the butterfly effect". Here it is. The impossibility of predicting the long term behavior of the pendulum.
In 1972, Muammar Al-Gaddafi, while having diner in his tent and surrounded by his beautiful amazonian guards, decided to eat an apple rather than a banana. Imagine he ate the banana instead and it turned out the banana was contaminated with a deadly bacteria. He would have died after a month of suffering. Libya would have been 'inherited' by a crazier person (as is always the case). This person would have changed the entire political scene in the region, maybe causing the Lebanese civil war to be more intense. My father would have decided to flee the war and immigrate to Canada and he wouldn't have met my mother. I wouldn't have existed. I owe my existence to Gaddafi's apple.
- Simple Forward Kinematics library for Robotic Chains
- Dynamics Simulator for Kinematic Chains