Revision Sheet II

The quiz will just require one word or one sentence answers and will just cover the basic facts summarized below. The intention is to (mostly) cover material between lectures 5 through 11.

Quiz date: Tuesday Dec 4.

Origin of fractals in nature: the motion of a chaotic dynamical system typically lies on a fractal - a so-called strange attractor. Examples:
- The logistic map - values of x in a region of chaos do not cover the line between 0 and 1 uniformly - there are gaps in which the motion never goes. These gaps come in all sizes -- there is structure at all length scales. This is a characteristic feature of fractals.
- The Henon map - 2 variables now - x and y. When the values of (x,y) are plotted see that they do not cover the (x,y)-plane. Furthermore, under magnification see that this attractor has structure down to very small scales - fractal again!
- Regular fractals - know how to construct Cantor's Set and the Sierpinski Triangle.
- Fractal dimension -- know how to calculate the fractal dimension by covering the fractal with boxes of a certain size s and counting the number needed N(s). Plot log(N(s)) against log(1/s) - the fractal dimension is just the gradient of the (straight) line.
- For regular fractals there is a short cut to calculate D -
```
D=log(number of pieces needed at kth stage of construction)/log(1/size)
```
  eg. for Cantor's Set the number of pieces after k construction steps = 2^k (2 to the power k). The size of each piece = (1/3)^k
```
So D=log(2)/log(3)
```
- Similarly, know how to compute D for Sierpinski's Triangle.
- What are Julia sets ? Give the map which is used to generate Julia sets
```
z=z*z+c
```
  For fixed complex number c, a Julia set is the set of complex numbers z which do not get infinitely big under repeated iteration of this map. They make up a fractal which can best be visualized by plotting the real and imaginary parts of z in the (x,y)-plane.
- Know how to add and multiply complex numbers.
- Be able to give some examples of real world fractals eg
  - Coastline of Norway
  - Snowflakes
  - Motion of stars within globular clusters (Henon map)
  - DLA - diffusion limited aggregation.
  - Motion of Hyperion (a moon of Saturn).
Note that copies of the lecture slides are available from PHY307 homepage under PHY307/Lecture Slides