Chaos Theory

Chaos Theory is a mathematical sub-discipline that attempts to explain the fact that complex and unpredictable results can and will occur in systems that are sensitive to their initial conditions. Some common examples of systems that chaos theory helped understand are earth’s weather system, the behavior of water boiling on a stove, migratory patterns of birds, or the spread of vegetation across a continent. The Butterfly Effect is one of more famous examples of chaos theory. I have discussed more about it here. Chaos occurs in nature and it manifests itself in various forms. Chaos-based graphics show up all the time, wherever flocks of little space ships sweep across the movie screen in highly complex ways, or whenever amazing landscapes are displayed in some dramatic movie scene. It is used a lot in movies to generate obscure background using computer-generated chaos art. So what exactly is chaos? How does it work?  

What do you mean by “initial conditions”?

Let’s consider a simple system described by the equation given below:

                                      y = f(x+1000)

This is a simple non-linear system. Here, ‘x’ is the input and ‘y’ is the output. The term ‘1000’ refers to the initial condition. As mentioned earlier, chaotic systems are very sensitive to their initial conditions. Let’s say the value of ‘x’ is 10. If you substitute the value for ‘x’ and compute ‘y’, you will get some number as the result depending on the function ‘f’. Now if we change 1000 to 1000.000001, the output will change by an enormous value, say 6,306,060,416! What on earth happened here? The input ‘x’ remained the same and we changed the initial condition by just 0.000001, but the change in output went into billions! This is what chaotic systems are all about. A very small change in initial conditions will cause a huge change in the output.

In real life, this small change manifests itself in the form of errors in measurement, uncertainties, constraints etc. Some physical quantities just cannot be measured to such precision. Hence even if you give the exact same input, the output will be different every time. This is the reason we cannot predict the weather for more than a few days into the future.

Linear vs Non-linear Systems

Before we proceed further, it’s important to understand the difference between linear and non-linear systems. In linear systems, the magnitude of the output is controlled by that of the input according to simple equations of the form y=ax+b. Two central features of linear systems are proportionality and superposition. Proportionality means that if you change the input, then the output will change accordingly and it has a direct relation to the input. Superposition means that if you give sum of two inputs as an input, then the output will be the sum of the respective outputs. The overall output will simply be a summation of these constituent parts. This means that there are no surprises or anomalous behaviors.

In contrast, nonlinear systems violate the principles of proportionality and superposition. An example of a nonlinear equation is y = x²+5. This is an equation for a parabola. The non-linearity of this equation arises from the quadratic term (x²). As we can observe, a change in ‘x’ will not cause a proportionate change in ‘y’. Why are we talking about this? Well, basically because chaotic systems are complex non-linear systems.

What exactly is chaos?

Chaos Theory is used to model and understand complex systems. In this case, complex systems are systems that contain so many moving elements that computers are required to calculate all the various possibilities. Although the focus of much recent attention, chaos actually comprises only one specific subtype of non-linear dynamics.

A famous French mathematician, Henri Poincaré, starting working towards this in the early 1900s, Prior to that, science was dominated by the belief that the behavior of systems for which one could write out explicit equations should be fully predictable for all future times. But most of the natural occurring phenomena in the universe seem to violate this rule. This is where chaos comes into picture. Chaos refers to this particular anomalous behavior. Even if you keep every parameter the same, the system will give different outputs because it’s way too sensitive to the initial conditions. Chaos Theory helped people understand why these things happen and how we can form a mathematical model for it.

Determinism

Determinism is the philosophical belief that every event or action is the inevitable result of preceding events and actions. Thus, in principle, every event or action can be completely predicted in advance. According to the deterministic model of science, the universe unfolds in time like the workings of a perfect machine, without a shred of randomness or deviation from the predetermined laws. The problem with chaotic systems is that they don’t seem to obey this law. The theory of determinism fails to explain many natural phenomena. That’s why people started looking into chaos theory for answers.

One of the most interesting issues in the study of chaotic systems is whether or not the presence of chaos may actually produce ordered structures and patterns on a larger scale. Some scientists have speculated that the presence of chaos operating through the deterministic laws of physics on a microscopic level, may actually be necessary for larger scale physical patterns to arise.

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4 thoughts on “Chaos Theory

  1. Pingback: The Butterfly Effect | Perpetual Enigma

  2. Great blog!
    I dont understand when you say thay if you change the intial condition from 1000 to 1000,00001 the output increment is 6,306,060,416. For x = 10, y = (1010)^2 = 1020100 and y = (1010.000001)^2 = 1020100.002020… so that the difference is only around 0.002. Where does the 6,306,060,416 come out?

    Thank you very much

    • Thanks for pointing it out. It was supposed to be a demonstrative example for chaotic systems, but I now realize that it was confusing. I have updated the explanation to clarify that.

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