Chapter 2 – Overview of Fractal and Chaos Theories
2.1 – Prelude to Fractal Geometry
Euclidean geometry is about regularity – lines, planes, triangles, squares, cones, spheres,
etc. Nature generally isn’t Euclidean.
Definition – a fractal is a set that shows irregular – but self-similar – features on many or
all scales. Self-similarity = part of an object is similar to other parts or to the whole.
Mathematically, a fractal is characterized by a power-law relation  linear relation in a
log-log scale. A power law implies self-similarity.
Given a step size, 𝜖, and a number of steps, 𝑁(𝜖) then the length of a path is given by
𝐿 = 𝑁(𝜖) ∙ 𝜖
Equation 2.1
What’s remarkable is that typically 𝑁(𝜖) scales with 𝜖 in a power law fashion:
𝑁(𝜖) = 𝜖 −𝐷 , 𝜖 → 0
Equation 2.2
where 𝐷 noninteger and 1 < 𝐷 < 2. The noninteger 𝐷 is often termed the fractal
dimension to emphasize the fragmented and irregular characteristics of the object under
study.
In general, random fractals are more useful than fractal geometry.
The Mandelbrot set is defined as the set of complex 𝐶 such that for 𝑍0 = 0,
𝑍𝑛+1 = 𝑍𝑛2 + 𝐶
Equation 2.3
remains bounded.
2.2 – Prelude to Chaos Theory
Along with relativity and quantum mechanics, chaos theory is often considered one of the
three most revolutionary theories of the 20th Century.
Sensitive dependence on initial conditions – for true low-dimensional chaos, a small
separation between two arbitrary trajectories grows exponentially (the trajectories
diverge) according to
𝑑(𝑡) = 𝑑(0)𝑒 𝜆1 𝑡
Equation 2.4
where 𝜆1 is the largest Lyapunov exponent. This is the sensitive dependence on initial
conditions property of chaotic dynamics.
Boundedness – another fundamental property of a chaotic attractor is that it is an
attractor – trajectories in the phase space are bounded. Two things are going on here:
 Incessant stretching due to exponential divergence between nearby trajectories.
 Folding due to the boundedness
The result of these two processes is that the chaotic attractor is a fractal, characterized by
Equation 2.2, where 𝑁(𝜖) now represents the (minimal) number of boxes of linear
dimension ≤ 𝜖 required to cover the attractor in the phase space. Typically, 𝐷 is a
noninteral number and is termed the box-counting dimension of the attractor.
Consider the map (see c:\Teaching\Introductory Mathematical Biology\Lecture
Materials\chaotic_map.m):
𝑥𝑛+1 = 2𝑥𝑛 mod 1
Equation 2.5
where the mod 1 means that only the fractional part of 2𝑥𝑛 will be retained as 𝑥𝑛+1 . If
we represent an initial condition 𝑥0 in binary as
∞
𝑥0 = 0. 𝑎1 𝑎2 𝑎3 … = ∑ 2−𝑗 𝑎𝑗
Equation 2.6
𝑗=1
where each 𝑎𝑗 is either 1 or 0. Then
𝑥1 = 0. 𝑎2 𝑎3 𝑎4 …
𝑥2 = 0. 𝑎3 𝑎4 𝑎5 …
and a digit that was initial far to the right of the decimal, say the 40th digit (corresponding
to 2−40 ≈ 10−12 , and hence playing only a very minor role in determining the value of
𝑥0 , eventually becomes the first – and most important – digit  a small change in initial
conditions makes a large change in 𝑥𝑛 .
Routes to Chaos
 Period doubling – e.g., the logistic equation
 Quasi-periodicity – one basic periodicity two or more basic periodicities
sudden onset of chaos
 Intermittency – system operates between smooth and erratic modes, depending on
variation of critical (bifurcation) parameter
Chapter 3 – Basics of Probability theory and Stochastic Processes
3.1 – Basic Elements of Probability Theory
3.1.1 Probability System
Euclidean geometry is about regularity – lines, planes, triangles, squares, cones, spheres,
etc. Nature generally isn’t Euclidean.