Description
Question 1 (2.5 marks)
Consider a classical particle subject to the effective 1D potential,
V (x) = x
2
2
−
x0
p
1 + gx2
p
1 + gx2
0
, (1)
where x0, g > 0 are constants.
(a) Determine the extrema of V (x) and comment on their nature as a function of g and x0.
(b) Using your answer to (a) as a guide, sketch/plot the potential for various values of g and assuming
x0 = 1. You may use any numerical tools at your disposal if that is useful.
(c) Imagine that the particle is placed at the point x(0) = x0 > 0 and is initially at rest. Without explicitly
solving the dynamics, comment on the expected behaviour of x(t) as a function of g and x0.
Question 2 (4 marks)
Consider the nonlinear iterative map,
xn+1 = µ sin(πxn) (2)
where x ∈ [0, 1] and 0 ≤ µ ≤ 1.
(a) Construct a bifurcation diagram of the map with x0 = 0.4. To do this you will need to write a short code
(in whatever language you choose) and evaluate the iterative map for a range of 0 < µ < 1. An example
of what to expect is shown in Fig. 1. Note: While you should compute values x0, x1, x2, …., xN , in the
bifurcation diagram you should only plot the last handful of values of xn you evaluate. For example,
for N = 103 one might only plot x900, …, x1000 in the bifurcation diagram.
(b) This map can feature up to two fixed points depending on the value of µ. Determine the fixed points
(you may leave one of them as a transcendental equation – it will be obvious which this is!) and thus
the special value µ0 that delineates maps featuring two fixed points (µ > µ0 from those with only one
(µ < µ0).
(c) Assume µ > µ0. Perform a stability analysis of the fixed points of the map by studying the derivative
of the function that defines the map, e.g., xn+1 = f(xn) with f(x) = µ sin(πx). For the critical point
defined by a transcendental equation you will find the stability depends on the value of µ. You may
use numerical methods to determine for what values of µ this point is stable/unstable.
(d) Explain how your results for (b) and (c) relate to the features of the bifurcation diagram in Fig. 1.
1
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
n Figure 1: Bifurcation diagram described in Q2 part (a).
Question 3 (2 marks)
Consider the map defined by:
xn+1 =
(
2αxn, if xn < 1/2
2α(1 − xn), if xn ≥ 1/2
where 0 ≤ α ≤ 1. For what values of α is the map chaotic?
Question 4 (2 marks)
For this question, you need to read the paper “Deterministic chaos in the elastic pendulum: A simple
laboratory for nonlinear dynamics”, R. Cuerno, A. F. Ranada, and J. J. Ruiz-Lorenzo, Am. J. Phys. 60, 73
(1992).
You can access it at https://aapt.scitation.org/doi/10.1119/1.17047. To download the pdf you will
need to be on campus, use the University VPN or access the journal via the library website.
(a) Give a one paragraph summary of the manuscript including their methods and main results.
(b) Outline the procedure used to generate Fig. 3 (e.g., what “recipe” did the authors follow to obtain
this figure). Discuss and comment on what is plotted in each panel of Fig. 3 (e.g., what information is
being conveyed about the system and what are the authors trying to report).
