2.10. Chaotic orbits
mulae which incorporates some non-linearity
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t 32 x + x2 ,
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f (x) =
t 32 (1 − x) + (1 − x)2
55
(x ≤ 1/2),
(x ≥ 1/2).
(i) Examine the Lyapunov exponent for this map using the Lyapunov
Exponents window of Chaos for Java.
(ii) Derive a formula for L(x0 ), where x0 is very close to 0.5, in the range
0 ≤ t < 12 , and also for L(x0 ) when t = 12 + 0, that is, limt→ 12 + L(x0 ).
Does this explain what you observe by numerical experimentation?
2.35 In this exercise we investigate some properties of the Logistic map
with r = 4, using Chaos for Java.
(i) Use the Graphical Analysis window of Chaos for Java to find
the values of fn0 (x∗ ) for a few of the fixed points for n = 1, 2, · · · , 6.
(ii) From these results, form a conjecture for a general formula for the
derivative, and hence about the Lyapunov exponents of these unstable
orbits.
(iii) Use the Iterate(1d) window of Chaos for Java to obtain numerical estimates of L(x0 ) for a couple of values and compare with the
results you obtained for the periodic orbits.
2.36 Use the Iterate(1d) window of Chaos for Java to calculate L(x0 )
for the Logistic map with r = 4 at a few initial points x0 , and verify
that for large enough sample sizes, they appear to give the value ln 2.
2.10
Chaotic orbits
Definition 2.13 (Chaotic orbits) A chaotic orbit of a bounded system
is one which is not periodic or eventually periodic, and which has positive
Lyapunov exponent.
A dynamical system will be said to be chaotic when it is in a regime
with chaotic orbits.36 Actually showing that the Lyapunov exponent is
positive, or even that it exists, may be a difficult problem for any particular
system. Often numerical calculation is used as the main evidence, but it
is important to have a theoretical analysis of some simple test cases. Note
36 Some definitions of chaos include a requirement that there be a dense set of unstable
periodic orbits. Here I follow the simpler path of just requiring chaotic orbits.
56
Chapter 2. Orbits of one-dimensional systems
0
-20
(dB)
-80
-100
-0.1 0.0 0.1 frequency 0.4 0.5 0.6
Figure 2.16: Fourier amplitudes for the logistic map, r = 3.9615,
x0 = 0.4. Sample size 2 × 103 points, initial 104 points discarded.
There is a strong period 4 component, but the orbit is not periodic.
that the property of having a positive Lyapunov exponent implies that the
orbit never falls within the basin of attraction of any periodic orbit. In case
the orbit is periodic, or eventually periodic, it must be unstable.
Given a chaotic orbit starting from position x0 , then for any distance,
no matter how small, there are starting points within this distance of x0 for
which the two orbits eventually separate from each other on a large scale.37
Numerical evidence for chaos
Fourier analysis can provide strong evidence of non-periodic behaviour.
Given this, together with a positive numerically estimated Lyapunov exponent, one can be rather certain that a particular system is exhibiting chaos,
even though a theoretical demonstration is not feasible.
As an example which is not completely trivial, consider the Fourier
spectrum shown in figure 2.16, for the logistic map with r = 3.9615. The
spectrum shows clear peaks at frequencies of 1/4 and 1/2, suggesting the
possibility of a period 4 orbit. But the sample size, and the time allowed
for the system to reach a stable situation, are both large, despite which
there is a significant amount of what looks like noise. Nor is the noise
caused by a bad choice of sample size: 2000 is divisible by 4. It is easy to
estimate the Lyapunov exponent of this exact orbit using the Iterate(1d)
window of Chaos for Java. Using the same initial value and sample size,
37 The actual scale relates to the long-term accessible states of the system. For example,
for the tent map with t = 0.6 the allowed states might be taken as the interval [0, 1], but
all orbits which start in (0, 1) are drawn to an interval whose end points are a ≈ 0.48,
b ≈ 0.51, which therefore contains an attractor (see exercise 2.43). For any initial
separation, no matter how small, there are neighbouring initial states in this interval
which lead to separations of the order of 0.03; this is the appropriate large scale.