Understanding Dynamical Systems: Introduction to Chaos Theory and Its Real Life Applications

Hi everyone. Welcome to my science post for this week. Today I will introduce you to dynamical systems, under which I will discuss chaos theory. Actually, chaos theory is not a theory limited to physicist or mathematicians alone, other areas of science has tapped into the beauty of this theory in explaining so many hidden behaviors of systems. It is been used by engineers, social scientists, biologist, medical researchers, geologists, and many other professionals. Perhaps you have not heard of chaos in science before, or you have but it sounded more like just an English word. I will be introducing you to what the theory is and its many application in our everyday life. Chaotic motion has been identified a kind of motion under nonlinear dynamical systems. First, let us understand what dynamical systems are so we can appreciate chaotic motion. Please come with me…

Dynamical Systems

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The first point I want you to grab here is that a dynamical system can be linear or nonlinear. When you hear of dynamical system, it is just a system that evolves with time. Simply put, it is a system whose future state can be determined from its present configuration. That is, the system follows a set of defined rules whereby its condition at an instant can be used to determine its condition at the next instant of time. Example is the swinging of a pendulum clock. You would notice that the future state of the pendulum is governed by the Newton’s equation of motion F=ma at a particular instant.
Since a dynamical system evolves with time, it can be thought of as a case in which an initial state at an instant can evolve into a state or a collection of state at another instant. Therefore, dynamical system describes an initial value problem.


Dynamical system can be deterministic or stochastic (random) depending on the state of the system. When the system is unique, such that there is a direct relationship between the present state and the future state, then we say it is deterministic. However, when probability is involved in determine the future state from the present state, then we say it is random or stochastic. A typical example is the tossing of a coin. Say you want to toss a coin (initial state). The outcome could either be a head or a tail, and each has equal probability of occurrence (future state).


Types of Dynamical systems
Since dynamical system evolves with time, there are basically two types of dynamical systems classification according to time. These are:

1. Discrete-time dynamic system: These systems update the future state by taking the present state as input, and producing an output with the governing rule applied at discrete times.
   
   
2. Continuous-time dynamic system: these systems produce an output (future state) by updating the rules at a much smaller updating time which makes it appear continuous.
   
   

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In determining a dynamical system, we can either describe it using its state variables or by means of the relationship between the input and the output. The first thing I pointed out is that dynamical system can be linear or nonlinear. Let me explain what that is.

Linear Dynamical Systems

Apart from evolving with time, linear system has properties that are much simpler than nonlinear systems. A linear dynamical system is usually described by polynomial functions of degree one, in which separate solutions can be combined by means of simple arithmetic. This can however become very complicated in that its equation of motion as describe by the polynomial function can involve many variables. You will also be right to say that a linear system will evolve with a linear process. That is, there is a one-one one correspondence between changes in the initial condition and the output. Linear modeling finds application in areas such as signal processing, automatic control theory, propagation medium for wireless communication systems etc.

Nonlinear Dynamical Systems
Nonlinear system shows some more complicated properties than their linear counterparts. First, they do not obey the superposition principle. That is, the output of the system is not directly related to the input. There is no one-on-one correspondence between the input and the output.
Let’s talk mathematics again, it simply means that the equation of motion for nonlinear systems will be a set of simultaneous equation in which the unknown are variables of higher degrees other than one.
Nonlinear systems are further divided into four types namely: periodic systems, quasiperiodic systems, chaotic systems, and hyperchaotic systems.

The first two types are somewhat common. But the last two are often not understood. I will introduce you now to chaotic systems and chaos theory.

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Chaos Theory

A Brief of History

It was Sir Isaac Newton who introduced the use of differential equation to describe nonlinear dynamical systems. The solution to these differential equations mainly predicts a very regular kind of motion of the system. There is also a problem, although the differential equation is sufficient to determine the behavior of a system, it is not sufficient in telling us what that behavior would be. Therefore, the solution either describe a steady state system or an oscillating system that will either be periodic or quasiperiodic e.g. the motion of the moon and the planets in their orbits.
However, these types of motion are not fit for some systems that have a somewhat complicated behavior. Example is the collision of air in a room or a boiling pot of water. The behavior of the particles in these systems cannot be simply periodic or quasiperiodic. There kind of motion is describes as chaotic motion.


So what is chaos?

I’m sure you probably have heard this term in some everyday conversation. It could mean confusion, disarray, a state of unpredictable happenings etc. Yes it is not so far from what I am describing here. It is an erratic movement, not necessarily due to large number of interacting particles in a confined space as you might be tempted to assume.
Other mathematicians and Physicists alike have also studied and observed chaotic motion. James Clerk Maxwell in 1860 studied the motion of gases. He realized that a system of two colliding particles in a box would display chaotic behavior sensitive to a change in the initial condition. This is to say, a minor alteration in the initial condition of the system could lead to a very big change in the eventual outcome.
Another notable scientist and often referred to as the father of chaos theory is Henri Poincaré. In1980, he studied a solar system of three highly simplified bodies and concluded that the motion were sometimes complicated.

Let’s Talk About Chaotic Systems
Chaotic systems are nonlinear dynamical systems that have some unique characteristics. The first one is their sensitivity to initial conditions. Have you heard of the butterfly effect? A butterfly flapping its wing in a continent could lead to a hurricane in another continent! That is how sensitive chaotic systems can be to initial conditions.


They are also deterministic. Deterministic in the sense that they follow a set of rules from which the future of the system can be predicted from the present configuration. Being deterministic does not give room for randomness. When the system is simply random (when probability is involved), the model will be random or stochastic and the system is simply a linear dynamical system as I have described above.
Deterministic chaos provides a rich account for the sequential irregular behavior and anomalies in systems which are not inherently random.
I believe every system has the tendencies to become chaotic. Long ago, chaotic behaviors were attributed to external factors. But because we cannot observe it readily, doesn’t mean it cannot be inherent.

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Examples of Real Life Data Sets

1. Human breath rate: nonlinear dynamics (chaos theory) has been used to study the rate of fluctuation in human breath. Analysis of time series data on the breath of real human subjects has shown that, although it seems to be somewhat periodic, it has an embedded chaotic behavior (fluctuation) which is very unlikely to represent just a noisy harmonic oscillation [1]. This will lead to better administration and development of devices such as pacemaker and enhance our understanding of how the human heart works.

2. Foetal Electrocardiogram: nonlinear dynamics also hold promise in signal processing problem of extracting a foetus from the abdomen of a pregnant woman. Although the foetus and the mother have broad band power spectra, the smaller and faster spectral signal from the foetus cannot be easily distinguished from that of the mother purely by classical linear methods. Using a technique of nonlinear dynamics called phase space projection; the distinction and eventual separation can be done in a simple automated manner [1].

3. Ionospheric TEC variation: nonlinear dynamics could also hold the key to understanding and eventual nearly accurate predictions of the hidden nonlinearity in the behavior of the ionosphere and its irregularities as it affect radio communication.

4. NMR: The use of nonlinear dynamical tools has also been extended to the study of Nuclear Magnetic Resonance laser. It was observed that there is a level of irregularities associated with the output, which is not just due to the input noise. It is therefore possible to accurately predict future values with nonlinear prediction algorithm [1].

5. Vibrating String Data: vibrating string data when investigated show chaotic behavior. Depending on the initial condition, it could display oscillatory(periodic motion) motion or chaotic motion [1].
   
   

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Summary
Nonlinear time series analysis has shown increasing application in recent years. It has provided a robust way of viewing data set unlike the traditional linear methods. It has also provided an explanation to the perceived behavior of many systems which were thought to be mainly due to external factors. It was initial just a mathematical tool, but it has now gained applications in various aspects and can be used to analyze the human heartbeat, stock market. Chaos theory now has extensive use in economics, physiology, geophysics, biology and also in secure communication. I hope to soon be able to use this tool to analyze various crptocurrencies fluctuations, and maybe we can determine the present from the past.
Knowing the inherent nonlinear behavior of systems will provide a better understanding of the system and also proffer solutions to so many problems which traditional linear method could not provide.

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References

1. Holger Kanz and Thomas Schreiber:Nonlinear Time Series Analysis.Second Edition. Cambridge University Press. ISBN 0-521-52902-6
2. Chaos Theory
3. Wikipedia: Dynamical system
4. Dynamical systems
   
   
   Thank you for reading my post. This is just the introductory post. I will be discussing more on nonlinear dynamical systems in my subsequent post. Stay tuned...
   
   
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