Please read Notes and Credits before clicking the green flag.
This project is one in a series of projects designed to build understanding of the concept of a strange attractor. My project: y = (x + 6)/2 – A Simple Attractor https://scratch.mit.edu/projects/302717792/ demonstrates the existence of a mathematical single-point attractor. On page 39 of his book Chaos – The Making of a New Science by James Gleick the author says "The laboratory mouse of the new science was the pendulum:emblem of classical mechanics, exemplar of constrained action, epitome of clockwork regularity. A bob swings free at the end of a rod. What could be further removed from the wildness of turbulence?" The pendulum, with damping, is an example from the physical world of a fixed-point attractor. ********************************************************************* Set the 'damping' slider to 0. Click on the green flag. Click on the mouse to move the yellow pendulum bob to the right or left. release. The swinging pendulum will trace, in 'phase space' the trajectory of the pendulum that, without friction, oscillates in a closed loop. Click on the red STOP sign. ********************************************************************* Set the slider to '1'. Click on the green flag, and displace the pendulum bob. Release. The bob will now trace, again in case space, a spiral ending at a fixed-point angle = 0 and velocity = 0. ********************************************************************* The Point: attractors, including strange attractors, live in 'phase space' where the coordinates of any point in the phase space, on the attractor, describes the system at that point. ********************************************************************* Compare the results of this project with the diagrams on pages 136 and 137 of the book Chaos – The Making of a New Science by James Gleick. ********************************************************************* Set the damping slider to 0 for no damping. or 1, with damping. Consider damping to be the retarding force of friction. Click on the green flag, lift the pendulum (on the right side) to less than a horizontal height, and click. The pendulum will fall, and start to oscillate. Its phase space diagram will plot behind the pendulum. The phase space graph plots the velocity of the pendulum bob on the vertical axis and the distance (angle) from the rest position on the horizontal axis. Motion to the left is negative, to the right positive. As the graph starts, velocity is increasing and distance to the rest position is decreasing. Passing through the rest position, velocity decreases, distance increases. At the extreme left, distance is at a maximum, velocity is zero, and the pendulum starts its oscillation to the right. You should be able to look at the graph and tell the top half of the story. Compare this project with page 136 in Gleick's Chaos book. Many thanks to MalinC for the original project and for providing the link to the equation of motion. Added decimal slider from dapontes. The motion of the pendulum is side to side. This is the visual provided by the oscillating pendulum. The elliptical graph (damping = 0) or spiral graph (damping = 1) is the graph of the pendulum's motion in 'phase space'. In His book ' Chaos - Making a New Science, James Gleick uses the motion of a pendulum to illustrate the important concept of phase space. The strange attractors (Lorenz, Henon, Rossler, etc.) described in his book are all pictured as graphs in phase space.
