Chaos Theory and Fractals

1. Chaos Theory and Fractals By Tim Raine and Kiara Vincent

2. Chaos Theory About finding order in disordered systems �(Math.) Stochastic behaviour occurring in a deterministic system.� Initial Conditions Butterfly Effect E.g. x2+1, 2x2+1 Definition translates into ordinary English as �Lawless behaviour governed entirely by law.� For pic, Edward Lorenz did an experiment, with a starting value of .509127. He then wanted to repeat his experiments, but for quickness only typed .509 into his computer, but the results he got were completely different. This shows the significance of initial conditions. starting values on .000127 difference but produce totally different results. Butterfly effect � a butterfly�s wing flapping in Japan could cause or prevent a tornado or hurricane happening in America � this is basically what the butterfly effect is � a small change in initial conditions can cause a big change in the results. X2+1 is not chaotic � it settles into a pattern of �1, 0, -1, 0, -1, 0, -1, 0 2x2+1 is chaotic � there appears to be no pattern in the results, but there is an underlying law that controls the results. Weather is a chaotic system, so is the flight of a meteorite.Definition translates into ordinary English as �Lawless behaviour governed entirely by law.� For pic, Edward Lorenz did an experiment, with a starting value of .509127. He then wanted to repeat his experiments, but for quickness only typed .509 into his computer, but the results he got were completely different. This shows the significance of initial conditions. starting values on .000127 difference but produce totally different results. Butterfly effect � a butterfly�s wing flapping in Japan could cause or prevent a tornado or hurricane happening in America � this is basically what the butterfly effect is � a small change in initial conditions can cause a big change in the results. X2+1 is not chaotic � it settles into a pattern of �1, 0, -1, 0, -1, 0, -1, 0 2x2+1 is chaotic � there appears to be no pattern in the results, but there is an underlying law that controls the results. Weather is a chaotic system, so is the flight of a meteorite.

5. Chaos in the Real World There are plenty of chaotic systems in the real world: Weather Flight of a meteorite Beating heart Electron flow in transistors Dripping tap Double pendulum Heart beats are chaotic, and don�t appear to follow a pattern Electron flows in transistors can be chaotic, with spontaneously appearing vortices, possibly caused be impurities in the semiconductor.Heart beats are chaotic, and don�t appear to follow a pattern Electron flows in transistors can be chaotic, with spontaneously appearing vortices, possibly caused be impurities in the semiconductor.

7. Sierpinski�s Triangle Simple fractal Formed by cutting out equilateral triangles Has 1.58496 dimensions Sierpinski & Pascal

8. The Menger Sponge Fractal made using cubes � divide a cube into 27 smaller cubes (3x3x3) then remove the middle one and the one in the centre of each face Has 2.72683 dimensions

9. Made by folding a strip of paper in half, always the same way, then opening up With each iteration, the area gets less, yet the length of the line is the same By the 20th iteration, a 1km long piece of paper would cover less area than a pin point Jurassic Park Fractal

12. Cantor Set Produced with a line and removing the middle third If we add up the amount removed to infinity, we get 1 (using geometric series), this tells us the whole of the line has gone But the endpoints are never removed � there must be something left!

13. More Complex Fractals Part of the Mandelbrot set A fractal that is easy the see the order in A shell-shaped fractal Another part of the Mandelbrot set The Julia Set Fractal . . . . . . And again! A computer-generated �3D� fractalPart of the Mandelbrot set A fractal that is easy the see the order in A shell-shaped fractal Another part of the Mandelbrot set The Julia Set Fractal . . . . . . And again! A computer-generated �3D� fractal