I.3.4 Chaos Theory

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. During the 1960 s, the meteorologist Edward Lorenz was trying to simulate weather patterns on a computer. He then discovered that small deviations greatly changed the simulations, and he also discovered what came to be called the Lorenz Attractor, which is an area that pulls points towards itself.

Traditional science deals with supposedly predictable phenomena, such as gravity, electricity, or chemical reactions, while Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, such as turbulence, weather, the stock market, our brain states. The reason why such phenomena are chaotic is because they are described by fractal dimensions, which could have infinite complexity. Nevertheless, recognizing the chaotic nature can provide invaluable tools and knowledge for forecasting or understanding how complex systems evolve.

The Butterfly Effect is a very good example that can explain how small and unnoticeable actions could produce enormous results in the future, by driving the system slowly towards unexpected destiny. A butterfly flapping its wings in some jungle in Africa may, in principle, cause a hurricane in China. It may take a very long time, but if the butterfly flapped its wings at just the right point, it may cause a series of actions that could build the power needed for the hurricane. Therefore, the small changes in the initial conditions could lead to drastic results.

However, because we can never determine all the initial conditions of a complex system, we cannot hope to predict its ultimate fate. Any slight errors in measuring the initial state could be amplified dramatically as the system evolve, rendering any prediction useless. This is why it is impossible to offer any accurate long-range weather predictions.

Chaos Theory explores the transitions between order and disorder, which is often unpredictable and may lead to completely different scenarios due to the turbulence that is inherently present in complex systems. For example, two neighboring water molecules may end up in different parts of the ocean after a short time. Every emerging state of the system opens up many new possibilities that could then diverge dramatically. This continuous feedback is what makes the system described by fractal geometry. As we have seen above, fractals are created by repeating a simple process based on ongoing feedback loop. For this reason, the chaotic behavior is more evident in long-term systems than in short-term systems. Fractals are related to chaos because they are complex systems that are recursively defined and infinitely detailed and dynamic process. One way to think of fractals for a function  is to consider , , , , , etc, so every new state becomes the initial parameter of subsequent states. In principle, this process is completely deterministic, but because every small deviation may lead the system in different ways, it becomes impossible to trace all possible changes and theoretical behavior.

According to the laws of Classical Mechanics, it is possible to predict the evolution of any system very accurately if we can determine the initial conditions, because everything that could occur can be calculated from the laws of motion. The universe is often depicted as a billiard game in which the outcome unfolds mathematically from the initial conditions in a pre-determined fashion, like a movie that can be run forwards or backwards in time. However, although chaotic systems are mathematically deterministic, but they are nearly impossible to predict.

The phenomenon of chaotic motion was considered a mathematical oddity at the time of its discovery, but now physicists know that it is very widespread and may even be the norm in the universe. The fact that complex structures could come from simpler ones, is like order coming from chaos. The presence of chaotic systems in nature seems to place a limit on our ability to apply deterministic physical laws to predict motions with any degree of certainty. Nevertheless, tt has been found that the presence of chaos may actually be necessary for larger scale physical patterns, such as mountains and galaxies, to arise. The presence of chaos in physics is what gives the universe its arrow of time, the irreversible flow from the past to the future.

An early proponent of chaos theory was Henri Poincar , who was studying the three-body problem in the 1880s. He found that there can be orbits that are non-periodic, and yet not forever increasing nor approaching a fixed point. In 1898, Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature. Hadamard was able to show that all particle trajectories diverge exponentially from one another. Chaos theory began in the field of ergodic theory. Later studies, also on the topic of nonlinear differential equations, which were directly inspired by physics problems, such as the three-body problem, turbulence and astronomical problems, and radio engineering. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and non-periodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple noise was considered by chaos theorists as a full component of the studied systems.

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research, involving many different disciplines.