Nonlinear Dynamical Systems: Chaos Theory, Models and the Butterfly Effect

Hi and welcome. Today I will continue my discussion of nonlinear dynamical systems and the chaos theory. I hope you will find it worthwhile. I have been writing posts in relations to my PhD research work for the past one month. Recently I took a detour to talk about the tools I am exploring for use in my analysis. I am investigating the nonlinear behavior of ionosphere during solar eclipse. I am writing updates on what I read about. Why am I doing this? Well for two primary reasons which are not really far-fetched.
Firstly, I realized a long time ago that if you claim to have learnt something, but you cannot write it down, or explain it to others to learn also, then you have not learnt. Secondly, by writing whatever I am doing/reading in my own way, it will best serve my memory in retaining the knowledge. Any other reason you might think of is actually secondary to me. It’s like the icing on the cake.

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Let us begin right? Before I go too far, I will like to make a quick distinction between real and ideal systems. Since I will also be talking about models. I’m sure you hear of real systems and ideal systems, and you might be wondering what the difference between the two is really. Well, I want to tell you that ideal systems really do not exist. That is why idealizations are called models. Models are predictions of how real physical process should behave. The main aim of models is to capture the features and behavior of real-world processes.


Models depict perfects systems. Systems that conform to all the theoretical predicted behavior. Well I don’t know any of such system. However, what I know exist in the vast universe are real systems. When real systems are approximated, they become an ideal system. Nature itself is an approximation of the ideal. For example, the ideal gas equation represents how an ideal gas is expected to behave, sorry to disappoint you by telling you that what we have are real gases. I am making this clarification because of the physical systems we deal with. Of course I am a physical scientist, and I deal with real systems. At best, some simplifications have to be made for the systems to be an approximation of what is predicted by the models. Therefore, the model either captures the behavior due to its simplicity, or the systems behavior conforms to the model by its simplification.

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Let’s talk about nonlinear dynamical systems now

_{[image:Kalman filtering (prediction and update) credits: Wikimedia commons under the Creative Commons CC0 1.0 Universal Public Doamin Dedication]}
In my last post I talked about dynamical systems being of two types. They are linear and of course nonlinear dynamical systems. Of particular interest are nonlinear systems which I will be dwelling on here.
As a reminder also, nonlinear systems can be periodic, quasiperiodic, chaotic and hyperchaotic. I think it is good for us to know when a system can be classified into any of these groups. Of course, a major characteristic of nonlinear dynamical system is their sensitivity to initial conditions. So, all nonlinear dynamical systems show sensitive dependence on initial conditions.
Generally, a periodic system or quasiperiodic system must have the following properties:

1. At least one nonlinear factor (term) must be present in the system
2. It must be at least one dimensional
   
   

One major quantifier used in determining where a dynamical system belongs in these four categories is the Lyapunov exponent. Nonlinear dynamical systems are said to be periodic if the Lyapunov exponent is negative. Also, the number of the attractors has to be countable. For quasiperiodic systems, Lyapunov exponent will be zero.
What of chaotic and hyperchaotic system?
For chaotic systems, there must be at least one nonlinear term in the system, it must be at least two dimensional, and the system must also have at least one positive Lyapunov exponent.
Hyperchaoticity demands that the systems have at least one nonlinear term as well, but in terms of dimension, it must be at least four dimensional, and the system must have at least one positive Lyapunov exponent.


Okay, Lyapunov exponent all the way. I have just been making a distinction between nonlinear dynamical systems on the grounds of their Lyapunov exponent. Let me explain that now.

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Lyapunov Exponent

The Lyapunov exponent also called the Lyapunov Characteristics Exponent (LCE) is named after Aleksandr Lyapunov. It describes the rate at which the system diverges for a change in the initial conditions. i. e. it quantifies the rate at which close trajectories or path of a system are separated for different initial conditions.
Consider it this way, different initial conditions of a system will follow different trajectory of evolution in time. If two very close trajectories are considered, the Lyapunov exponent is what quantifies the rate of divergence. The rate of this separation can be different for different configurations of initial separation vectors. What this mean is that there would be several of such Lyapunov exponent. The number of Lyapunov exponent of a systems depends on the system's dimension in state (phase) space. However, one of the Lyapunov exponents will be zero since there won't be divergence for a perturbed path in the direction of the unperturbed path. The maximum one is often referred to as the Maximal Lyapunov Exponent. The larger the value of the Maximal Lyapunov exponent, the more the divergence. I will talk more of the Lyapunov exponent in fine details in my other post on the subject matter. For now, I will maintain my trajectory on chaos theory.

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Chaos theory

In physics, chaos theory is a field of study that deals with dynamical systems that shows strong dependence on initial condition. Such system will have at least one positive Lyapunov number. The more positive it is, the more chaotic the system is. Remember we say dynamical systems are deterministic? I.e. the future state can be determined from the present state with a set of rule governing the way the system will evolve. However, when we talk about chaotic systems, generally the future predictability is somewhat impossible on a long term basis due to its strong sensitivity to initial conditions. What I mean is that a small change in the initial condition will lead to a purely divergent outcome even though the system is inherently deterministic without randomness. The predictability of the system fails after a while. The uncertainty in predicting the future state of a chaotic system increases exponentially with time. When a meaningful prediction becomes unattainable, the system will appear to be somewhat stochastic.

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Chaotic behavior are being exhibited by many physical systems such as weather and climate and many other physical systems, some of which I highlighted in my previous post.


Chaotic behavior of systems is being studied with the use of mathematical models and various analytical techniques. These includes phase space methods, Approximate entropy (ApEn), Sample Entropy (SampEn), False Nearest Neighbors, Recurrence plot etc. I will discuss some of these techniques in my subsequent post.
The high sensitivity of chaotic system to initial condition is often referred to as the butterfly effect. Although the phrase “butterfly effect” has been used in many instances, to imply cause and effects, the location of the aforementioned have varied widely. One notable use was in 1972 when Edward Lorentz presented a talk at the 139th gathering of the American Association for the Advancement of Science in Washington, D.C titled:

Predictability: Does the Flap of a Butterfly’s wing in Brazil set off a Tornado in Texas?[1]

He explained that the flapping of the wings of the butterfly depicts a small change in initial condition which could lead to a change in the trajectory of a system. This in fact explains why chaotic systems such has weather forecast cannot be done for more than few days, as there are some tiny details which will make a remarkable difference as time goes on. The butterfly effect was used to explain the unpredictability of weather, but has now become a metaphor in everyday science. I guess it is logical to call it the butterfly effect, as the computer model that led to its discovery looks more like a butterfly.
Well in actual sense, you might want to explore the possibility of this happening. For a change in weather to occur, there must be fluctuations in air pressure which is achieved when a butterfly flapped its wings. But how plausible will the effect of that be on the occurrence of tornado in Texas? Well, the fluctuation in air pressure generated by the butterfly dies off a few inches away.

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Needless to say, the butterfly effect is just a metaphorical statement to illustrate the effects small changes in initial conditions will have on a chaotic system. In the case of the weather forecast from which the butterfly effect originated, some researcher have debunked the plausibility of this on the challenges of accurate predictions over long period of time. However, many believe that other factors such as individual/ location of clouds poses a big limitation in weather models, as these factors hinders the establishment of a perfect model.[2]

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Summary

Chaos theory has become a fascinating field of study in contemporary science. Although it will not be wise to hide under this theory in explaining changes or difficulties in predictability of further future state of a system unless one has rigorously investigated and found that the conditions for chaos are met. The better understanding of chaotic behavior of systems will lead to better formation of models and help in further closing the gap between real systems and ideal systems.
On the butterfly effect, it will be well appropriate to know that although the input of butterfly wing-flapping scale changes into the existing weather models might not necessarily cause it to diverge, but a collection of these small changes could make a significant effect.

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References:

1. The Butterfly Effect

2. Can a butterfly in Brazil really cause a tornado in Texas?

3. Important Notes on Lyapunov Exponent

4. Lyapunov exponent

5. Chaos Theory

6. Holger Kanz and Thomas Schreiber:Nonlinear Time Series Analysis.Second Edition. Cambridge University Press. ISBN 0-521-52902-6

7. CHAOS: Introduction to Dynamical systems. Kathleen T. Alligood, Tim D. Sauer, James A. Yorke. Springer: ISBN 0-387-94677-2

Thank you for reading my post. I will be writing more on nonlinear dynamical systems in my subsequent post. Stay tuned...