Exercises

1. Practise plotting the state-space plots for the map $x_{n+1} =x_{n}^{2}$ for different initial values of $x_0$. Use your own software or that in Ref.[1] or a modified form of the MAPLE program mf35.mws from Ref.[3].
2. Consider the one-dimensional map $x_{n+1} = \sqrt{x_n}$ for $x_0 \geq 0$.
   (a) Explore the above with a calculator, or guess its general behaviour. Where are the fixed points, are they stable, and what is their basin of attraction ?
   (b) Confirm your results in (a) analytically.
   (c) Confirm your results above with a graphical analysis using the same software as you used in the previous problem.
3. (a) Give one example of positive feedback and one of negative feedback from your personal experience.
   (b) From your experience, give one example of a linear change and one example of nonlinear dynamics or change.
4. With regards to the logistic map,
   (a) Complete the steps leading to its derivation in the text.
   (b) Show that if the control parameter is restricted to $0 \leq r \leq 1$, then any initial $0 \leq x_{0} \leq 1$ leads to a value of $x_{n}$ that is bounded between the same values.
   (c) Show alegbraically that for a reproduction rate $0 \leq r \leq 1/4$, the population of species will eventually (at infinite time) become extinct no matter what the initial population is.
   (d) Illustrate the result in (c) above graphically using a state-space plot.
   (e) Confirm the results in (4c,4d) analytically by showing that for $0 \leq r < 1/4$, the only period-one fixed point of the system is the 'trivial' one at the origin and that it is stable (atractive). What happens when $r$ is exactly $1/4$?
5. With regards to the logistic map,
   (a) Show analytically that for $r > 1/4$ the trivial fixed point at the origin becomes unstable.
   (b) Show analytically that for $r > 1/4$ there is a period-one fixed point at $x_f = 1-1/4r$ and that this point is stable for $1/4 < r < 3/4$.
   (c) Verify the result in 5(b) graphically, using your favourite software, for different values of $r$ and $x_0$.
6. (a) With regards to the logistic map, show that as $r$ is increased beyond $3/4$ the fixed point at $x_f = 1- 1/(4r)$ becomes unstable.
   (b). Explain why the values of the two new period two fixed points that result for $r>3/4$ are obtained by solving the equation
   

   $$x_{n+2}=x_{n}.$$ (3.17)

   
   
   (c) Write out the above equation explicitly in terms of the single variable $x_n$. Without actually solving it, explain why two of the roots of the equation must be $x_f=0$ and $x_{f}= 1 - 1/4r$, the two fixed points that already exist for lower values of $r$.
   (d) Verify explicitly that $x_f=0$ and $x_f = 1-1/4r$ are roots of the above equation.
   (e)Find the two new roots of the above equation. Show that these roots exist only for $r>3/4$.
   (f)What is the condition for stability of period-two fixed points ? Show that the two new roots in (e) above represent stable (attracting) fixed points.
   (g) Convince yourself of the period-2 orbits using your favourite software.
7. (a) Use the critical values of the control parameter for the logistic map, $a_k$, given in the text to evaluate
   

   $$d_k \equiv \lim_{k-> \infty} {a_k - a_{k-1} \over a_{k+1} - a_{k}}$$ (3.18)

   
   
   for $2 \leq k \leq 7$.
   (b) Compare the result of (7a) with the Feigenbaum constant $\delta$.
   (c) Use the values of $a_6$, $a_7$ and the Feigenbaum constant $\delta$ to estimate the value of $a_8$. Compare the result with the known value.
8. Use your favourite software (see for example the Maple file mf36.mws of Ref.[3]) to explore the bifurcation diagram for the logistic map. Look closely at the periodicity windows that open up in the chaotic region and convince yourself that period three bifurcations exist there.
9. Explore the sine-map, $x_{n+1}= r \sin (\pi x_n)$ for $0 \le x_0 \le 1, 0 \le r \le 1$, by plotting the state space plots and the bifurcation diagram. Compare with the logistic map.
10. Explore the map $x_{n+1} = \mbox{frac}( 2 x_n)$. Here 'frac' means 'keep only the noninteger part'.
    (a) Make a state-space plot.
    (b) Determine the fixed points.
    (c) Show that there are infinitely many periodic and aperiodic orbits.
    (d) Show that the map displays sensitive dependence on initial conditions.
    Hint: It is useful in this problem to consider the binary representation of numbers.
11. Use your favourite software (e.g. the maple file ross.mws) to explore the Rossler system (time series and phase plots) for different values of the control parameter $c$. Note that in the Maple file you can easily view different perspectives of three-dimensional plots (especially that of the strange attractor) by clicking the mouse on the plot and dragging it. Note also that by plotting only later points of the iteration, the transient part fo the orbit is neglected. Practise plotinng also the transient parts fo the orbits to see the approach to the attractors.
12. (a) Repeat the above excercise for the Lorenz system using your favourite software (for example the Maple file mf10.mws of Ref.[3]).
    (b) Obtain the Lorenz map for the Lorenz system using its $z(t)$ time series.
    (c) Reconstruct the Lorenz attractor using one of its time-series, (using for example the file mf42.mws of Ref.[3]).
13. Give an example of a situation where Chaos might be useful. Inform yourself about some applications of chaos as mentioned in, for example, Ref.[10].
14. Check out the resources in Refs.[7, 8, 9] for self-study.