Chaos theory

Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems whose apparently-random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas.

compound pendulum

A animation of a double-rod pendulum at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a vastly different trajectory. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.

Gif Credits: Wikipedia

Chaos is typically understood as a mathematical property of a dynamical system. A dynamical system is a deterministic mathematical model, where time can be either a continuous or a discrete variable. Such models may be studied as mathematical objects or may be used to describe a target system (some kind of physical, biological or economic system, say).
A simple example of a dynamical system would be the equations describing the motion of a pendulum. The equations of a dynamical system are often referred to as dynamical or evolution equations describing the change in time of variables taken to adequately describe the target system (e.g., the velocity as a function of time for a pendulum). A complete specification of the initial state of such equations is referred to as the initial conditions for the model, while a characterization of the boundaries for the model domain are known as the boundary conditions.

Principles of Chaos

The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle. Who knows what the long-term effects of teaching millions of kids about chaos and fractals will be?
Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc) in the World, accurate long-range weather prediction will always remain impossible.
Order / Disorder Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.
Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. Examples: Two neighboring water molecules may end up in different parts of the ocean or even in different oceans. A group of helium balloons that launch together will eventually land in drastically different places. Mixing is thorough because turbulence occurs at all scales. It is also nonlinear: fluids cannot be unmixed.
Feedback: Systems often become chaotic when there is feedback present. A good example is the behavior of the stock market. As the value of a stock rises or falls, people are inclined to buy or sell that stock. This in turn further affects the price of the stock, causing it to rise or fall chaotically.
Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

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Equations for Chaos: A Mathematic Paradox

Equations are ordered, elegant mathematical constructs used to describe specific patterns. Can you imagine some formulas depict the very opposite: chaos and randomness? What's more intriguing, they also underpin several complicated theories behind natural phenomena.

A video presented on Veritasium used the following equation to describe an example of this type of miraculous mathematical paradox.

Xn+1 = rXn(1-Xn)

In this logistic model that describes how the change of an animal population, "r" denotes the growth rate, "Xn" the percentage of the maximum population at a certain year, and "Xn+1" the population of the year after. Given the population has access to a limited amount of space and natural resources, the equation uses "(1-Xn)" as a constrainer for how big or small the population can get.

Like many real-life species in nature, no matter what the size of the starting number is, the model predicts that the population fluates in the beginning but eventually reaches a stable equilibrium. If one plots the growth rate (r) against equilibrium (e), the curve would split into two paths once "r" reaches three. As "r" gets bigger, the original fork would split again and again. (This periodic doubling pattern is also know a bifurcation.) Once it reaches 3.75 and above, the value of "e" becomes random.

Why is this type of equations so important? Scientists have found the periodic doubling pattern in many places, such as water dripping from a loosely closed faucet, random patterns of spontaneous neural firing, arrhythmic hearts' response to electrical stimuli, and so much more.

Simple as these equations may look, the resulted mathematical dynamics can be immensely complicated. By taking a closer look at them and accepting chaos as a part of nature, we can have a better understanding of the nature of our world.

Source: Veritasium via Youtube

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