PH316  Non-Linear Phenomena and Chaos



 Further Exercises  


1. Does the function f(x) = x^2 + 1 have an attracting fixed point
   at the origin?

   
2. Find and classify the fixed points of the functions:  
  
   (i)   f(x) = sqrt(x)
   (ii)  f(x) = A arctan(x)   A a constant
   (iii) f(x) = |x|


3. Does the point x = 1 lie on a cycle of period-2 for the map:

   f(x) = x(n+1) = x(n)^2 - 1 ?


4. Describe the orbits of the mapping for the Bernoulli shift map,

   x(n+1) = 2 x(n)  (mod 1)   when starting at (a) x(0) = 1/3
                                               (b) x(0) = 1/7 .

5. Consider the function of the form

          f(x) = A (exp(x) - 1)   when A > 0.

   For which values of A is the fixed point at the origin attracting?
   For which values is it repelling?
   Describe the bifurcation that occurs at A = 1.
   Sketch the phase portraits for these functions before, at and after
   the bifurcation.


6. The length of a Cantor set can also be found by considering what has been
removed from the Cantor set at each stage and summing the result.

Form the infinite geometric series that this process describes and show that
its sum approaches unity ... thus showing that the Cantor set has no length.


7. Construct a fractal that is similar to the middle thirds Cantor set, but
instead remove the middle half from each previous section. 

Show that its dimension is equal to 1/2.


8. Start with an equilateral triangle and construct a Koch curve on each side
of the triangle to form a Koch Snowflake. 

Show that the boundary of has infinite length but that it encloses a finite
area. Can you estimate an upper bound to this area? 




MW - 07/03/97