What is chaos?
It is a seemingly random behavior exhibited in a system, with sensitive dependence on initial conditions. What we mean by ‘seemingly random’, is that this behavior is very unpredictable — so much so that it may appear random. This random appearance is due to the fact that the changes in behavior are extremely sensitive to the initial conditions of the dynamical system. In essence, chaos is the study of small changes that significantly change the future of a system.
What is a Dynamical System?
It is a system that can be described with a function with time dependence of a point in geometrical space. It is a system whose state evolves with time over a state space according to a dynamical law/rule. Simply put — it is the evolution of a system over time.
To understand the basic philosophy of what makes a dynamical system what it is, check out this article:
Examples of dynamical systems:
- a swinging pendulum
- the flow of water from a faucet
- the behavior of rational individuals in a negotiation game
- population growth over time
Chaos theory is aptly used to model dynamical systems that are highly sensitive to initial conditions.
Examples of systems that exhibit such behavior:
- Weather
- Two pendulums swinging together from a similar starting point
There are two kinds of dynamical systems chaos theory is used to model:
- Deterministic systems: a system where the same input always yields the same output.
- Non-deterministic systems: a system where same input does not always yield the same output. This is due to factoring in randomness.
The focus of this article is to delve into one of the most famous equations in deterministic chaos — The Logistic Map.
The Logistic Map
What you’re about to see is an impressive feat of complexity arise from something that may appear very simple — maybe even mundane. A side note — map is just another way of saying function.
We’ve all seen this simple quadratic equation:
y = ax-ax²
In factorized form,
y = ax (1-x)
Now let us rewrite this non-linear quadratic equation in this form:
Graphically represented, this equation depicts a concave down parabola:
The way we drew the graph of course was by finding out the coordinates of the maxima.
Two Levels of Understanding the Equation and its Graph
Let us begin by making an important connection that the quadratic nature of the xn+1 = λxn (1-xn) (eq. 1) would have been exactly the same for y = ax (1-x)(eq. 2).
However the distinction lies in how we can use the equations. Eq.1 is written symbolically in such a way that it signifies an iteration and a relationship between the current population xn and the future population xn+1.
It is therefore important to note that n is defined as n discrete steps/points of time. These discrete steps are the ‘states’ of the dynamical system.
In order to explore the iterative relationship between the current state and the future over discrete steps n, we will construct a final state diagram.
Final State Diagrams
To create a final state diagram for eqn. 1 we must follow this process:
- Fix the value of λ
- Choose the initial condition x0, (xn=x0)
- Iterate then plot for 40 times
The resulting graph is called a final state diagram.
I will use Matlab to carry out this process for four different values of λ but with the same value of x0=0.7.
Graph for λ=0.5
We can see that as n tends to infinity, xn tends to zero(the trivial solution). The point where the graph tends to is called a fixed point attractor. Everything before the fixed points (or periodic cycle) is called the transience. A fixed point attractor is independent of the initial conditions — meaning that if we were to vary the x0 the fixed point would not change (provided that x0 is not equal to 0).
Graph for λ=2.8
In this graph we can see that as n tends to infinity, xn tends to x* which is a value between 0.6 and 0.7. The behavior is called period 1 cycle.
Graph for λ=3.3
The behavior seen here is called period 2 cycle, because now there is a set of two fixed points that repeat.
Graph for λ=3.5
And finally, there is a set of 4 repeating fixed points, and hence this is a period four cycle.
Starting to notice a pattern here? As λ increases we get period doubling. We use a computer to generate the following results:
Realize how each time we increased λ by a smaller and then smaller increment the period doubled? We can visualize these ‘increments’ as intervals where λn eventually converges geometrically to λ∞.
To see where the ratio of the difference in consecutive intervals converges to, we take the limit as n approaches infinity:
Keep the result δ in mind as we will return to its significance.
Having come up with these intriguing results one question seems to still lay unanswered from the beginning —
What Happens After λ∞?
So far, we’ve seen as a λ increases the period doubled. However it is the fact that a smaller increase in λ than the previous interval is what lead to λ geometrically converging at a value of λ∞ = 3.569946… as it tended to infinity. It is this nature that makes us uncertain of what will happen after λ∞.
Once again, we will use an iterative process where we will plot a graph that shows:
- xn, the set of attractors for each λ only after the transience has been deleted
- λ in the x-axis
- xn in the y-axis
Essentially all we are doing is adjusting the λ parameter to see how its xn varies. From 0 to a little before 3 in λ we can expect a stable period 1 behavior. Between 3 and 3.569946… we can expect period doubling. What we will create is called the bifurcation diagram. Each vertical slice of a bifurcation diagram is a single final state diagram. Bifurcation is just a fancy way of saying splitting in two. Graphically, bifurcation looks like a fork.
Using Matlab we get:
Stunned? Well, you should be! What you are looking at right now is the most famous depiction in chaos theory. This is the simplest non linear equation in deterministic chaos. Lets dissect the intervals and go through them:
- [0,1): No growth
- [1,3): Period 1cycle
- [3,3.499…): Period 2 cycle
- [3.499,3.54409…): Period 4
- [3.54409…,3.5644…): Period 8
- [3.5644……,-…): Period 16
- ….
- [-…, 3.569946…): Period ∞
- After 3.569946… periodic behavior ceases and we have chaotic behavior.
Look back at the diagram. Can you see the white area amidst the chaos? This in fact is a transition from chaotic behavior back to periodic. If we were to extend the values on the x-axis you would be able to notice another pattern — there are transitions from periodic to chaotic to periodic to chaotic and so on.
What if we were to zoom in to the chaotic ‘mess’ in the diagram, what would we see?
All these discrete points almost look like fractals. If we were to add more iterations and hence more points it would seem even more obvious that this chaotic zone overlaps with the Mandelbrot set.
The Mandelbrot set is two dimensional. One of the dimensions is complex. The other is real. That real dimension (1D) is exhibited in the bifurcation diagram.
What Does All of This Mean?
Recall δ? Well that happens to be a very important constant in mathematics. It is up there with π and e. This constant is called the Feigenbaum constant which expresses the ratio in a bifurcation diagram. What makes it so impressive is its appearance in not only this single non-linear map (logistic map) but in all one-dimensional maps with a single quadratic maximum. The constant is therefore believed to be transcedental.
Summing Up
A logistic map is a simple, completely deterministic equation that when iterated, can display chaos depending on the value of λ.
The act of iteration is a very simple process (made simpler by using computers) that can be infinite. As we’ve come to see, iterating a very simple equation gave rise to strange behaviors which were previously unseen.
We saw that for a large range of initial conditions the system evolved into a steady state on the attractor/fixed points. Another observation was that there was sensitive dependence on initial conditions. Once we had defined the initial conditions the future became determined, albeit appearing unpredictable. These features are what make the logistic map the most interesting equation in chaos theory!