Please read Notes and Credits before clicking the green flag.
This project is to help build understanding of the concept of a 'strange attractor'. At the beginning of Chapter IX of the 'Chaos' movie http://www.chaos-math.org/en/film the following set of differential equations are given: dx/dt = –y – z dy/dt = x + ay dz/dt = 2 + z(x – 4) This project implements this set of equations. ******************************************************************** 'Bifurcation' is a word commonly found in the attractor and chaos literature. It means 'to split in two'. This 'doubling' can be seen in the map of the equation y = Rx(1 – x). https://scratch.mit.edu/projects/305779100 The doubling leads to chaos but there are parameters where the trajectory stabilizes at periods 2 or 4. ********************************************************************* Examine the behavior of the set of equations for the following values of parameter 'a'. Now click on the green flag and activate turbo. Explore the following settings. a = 0.3 – period 1 a = 0.335 – period 2 a = 0.37 – period 4 a = 0.40 – period 2 a = 0.45 – period 4 a = 0.46 - chaotic ********************************************************************* The map of y = Rx(1 – x)—as shown in the project linked to above surprisingly appears (like π) in many seemingly unrelated situations. If the intersection of the trajectory of the above set of equations with the x-y plane (a Poincaré map) is examined, a map similar to the logistic map is obtained. To see this, view the first 4 minutes of Chapter IX of the Chaos movie linked to above (this is the only graphic example I know of that shows how a 2-D attractor is derived from a 3-D attractor). *********************************************************************
