At the edge of chaos - making sense of nonsense

At the edge of chaos — making sense of nonsense

 (previously published on LinkedIn dated January 2018)

This catchy phrase ‘At the edge of chaos’ has been in my mind ever since I became aware of fuzzy logic and chaos theory during my engineering. I had been thinking about this topic and overlaid with my general ways of ‘pattern finding’ to see how best we can model various events, people, behavior, etc. I am sharing here few of my thoughts on the same through this blog.

Today the world seems more chaotic than ever before. There seems to be uncertainty everywhere. Newsmakers and media can take the blame partially to abet this madness by focusing their story on disorder, chaos and lack of governance. It doesn’t really matter which side they are trying to be — ‘left’ or ‘right’ or ‘center’ — the newsmaker’s business is all about grabbing attention by ‘breaking’ news, apparently for us. Why am I saying this? My close ones will say I watch ‘news’ all the time. That is a fact. :-)

Not just the news, in our business world and work environment too, there has been an increasing degree of craziness, the rush in which everyone seems to be apparently busy, apparently doing the ‘right’ thing and aiming to scorch the sky. But we also see half-hearted, disconcerted efforts. Some of the business models are breaking down, even those that gave success a decade back. There are disruptive technologies that has changed the landscape of customers, business processes, and the lives of people all over. There is uncertainty everywhere — more than ever before!

Few years back when I was working in Switzerland, I had felt there was too much change in the air and the rate at which this change was happening was also very frequent. Too many things changing too fast. Now it appears the ‘change’ has moved to a stage, probably a transformative one, where it is difficult to make sense of anything. The narrative in business moved from ‘adaptation to change’ to ‘sustaining the business’ to ‘reskill to exist’. We, in our personal situation, family situation, business and political world, and in other areas, are really making multiple attempts to wade through this process of ‘change’ and ‘transformation’ to move knowingly or unknowingly to an intended or targeted objective of our collective conscious.

My objective here is not to give a sense of craziness and to call out ‘Yo, men! where are you heading?’ and preach some philosophy. The intention is to draw parallels to a mathematical model called ‘At the edge of chaos’ and to make sense out of what appears nonsense in a shortened time frame of perception. So, if you are still interested continue reading. If not, thanks for reading up to this point. I do revert back to the topic of current news at the end, so would urge such readers to skip the mathematics to the tail end sections and share your comments.(you will have to really scroll long for that. This blog appears longer than it really is. Thanks to LinkedIn, that provides only very basic features available for posts)

I don’t know where I caught this term ‘At the edge of chaos’, but I had a liking to this term when I was studying mathematics in my engineering courses, especially on probability and random process and differential calculus.

Just a caution for mathematic lovers and thinkers: I would like to state that this blog is not intended to be full-fledged paper or a research material, but my own thoughts and intuitions over the years, coupled with an attempt to give a simplistic view of the complex topics and to make it an interesting read. I hope I will succeed. So read through and feel free to share your feedback, comments, suggestions, criticism, etc.

Simplistic / Deterministic Models — Linear and Differential Equations

Mathematics has been one of my favorite subjects. Most of my engineering science has been mathematics too! To model a real-world situation in terms of linear, non-linear and differential equations and to be able to predict and understand the degree and order in which they operate, gave a sense of satisfaction. Please excuse for a little bit of educative language now for the benefit of all. Equation such as: 

Y = ax + b

mx + c  = 0

ax + by + cz + d = 0

where a, b, c, m and d are non-zero are linear equations. They are called linear because if we plot them in a graph for various values of x and y, we will see them as straight lines. Very simple, straightforward and predictable, and can be used as a starting point model in most real-life scenarios. We parameterize the unknowns as variables x, y, etc. of degree 1 (exponents of variables are 1).

A non-linear equation would be where at least one of the unknown variables occur in more than 1 degree (exponents of variables are more than 1). The graphs formed by such equations would not be straight lines, but curved lines or shapes. For example:

y = ax² + b

ax² + by² + c = 0 

would all be non-linear equations. They find usage in modeling many real-world situations, for instance involving curved geometrical shapes in as simple as areas and volumes of closed shapes to varied uses in architecture, engineering, etc. Still things are simpler and can be used in many cases to model the problem domain. Bring in the concept of change and rate of change now. The concept of calculus and derivatives, immediately comes to our rescue. Derivatives are a measure of rate of change. In geometrical or graphical terms, it is the slope of the curve (or straight line) at a given point and is generally represented as:

 

(for those not used to mathematical notation — “dy” means “differential y”. So, read them together “dy” by “dx”. When we say, “tends to”, it means when the value of y₁ comes closer to y₂ or x₁ to x₂. This means if the points of the graph for the equation are as close as possible, you get the “slope” value at that point. For straight lines, slope is the same throughout. For curved lines, slope can vary at each and every point. See graph below for illustration.)

 or in function terms

and is generally denoted as:

(again — for those not used to mathematical notation — ”lim” means “limit” when h tends to zero. f(x) is read as “f  of x or function of x” and f ’(x) as “f dash of x” and f ’’(x) as “f double dash of x” which is “d square y by d x square)

Thus, geometrically differentiation is the slope of the tangent of the curve at the given point. Hence note that not all functions may be differentiable. The immediate example that is commonly used in physics is for finding the rate of change of any physical parameter. For instance, speed is the rate of change of distance over time, and acceleration is the rate of change of speed. Thus, the use of derivatives helps us form linear differential equations (derivatives of degree 1) and non-linear differential equations (derivatives of degree more than 1).

y’ = ay + x²

y’’ – xy’ + y = 0

ay’’ + b sin y = 0

y_x’ + c y_t’ = 0

Further differential equations can be ordinary differential equations (ODE) or partial differential equations (PDE). Without going into further details, suffices to say differential equations are able to greatly help model more complex problems and has wide applications in science and engineering.

However complex, we are still dealing with deterministic models so far i.e. given a set of variables, we can model the system using simple linear, non-linear and/or differential equations to map them as close as possible to predict the outcome given the initial conditions and set of input values.

Complex / Non-Deterministic — Probabilistic Models and Random Process

When input variables become non-deterministic or random, probabilistic models are used to provide a statistical distribution of outcomes. Even in cases, where input variables are deterministic, if there are multiple variables, and there are possibilities of dependencies, different permutation and combinations, probabilistic models can be used for better predictability. Probabilistic model uses random variables and standard probability distributions such as Gaussian/normal, binomial, Bernoulli, etc. Maximum likelihood, Bayesian (prior or posterior) parametric models in combination with the distribution models are used to further allow us to optimize or integrate to suit our needs of approximations. Mean, Median, Mode, Standard Deviation, Variance are used as measures of spread of the data.

• A normal or Gaussian distribution, also called the bell curve, is typical in most situations.
• A Bernoulli’s distribution is a discrete one with only two outcomes — ‘success’ or ‘failure’

A random or stochastic process is where the outcome result distribution depends on the possible values of input parameters (generally time) and can be considered as a set of random variables. For example, the price of a stock, the number of customers arriving at a retail store, etc. There are multiple graphical and stochastic models like Bayesian network, Markov random field, Bernoulli, Poisson process, etc.

Bernoulli random process is considered as a sequence of independent Bernoulli trials (similar to tossing a coin), where:

P(x_i=1) = p, where x_i being a trial at a discrete time i

P(x_i =0) = 1 – p 

(P(x=1)  — this is read as “probability of x equals 1”)

Another frequently used process is Poisson process, where time interval is considered continuous (like number of waves hitting a sea shore). Both Bernoulli and Poisson processes outcomes are discrete state.

P(x) = e^-m m^x / x!  where m is the mean, m > 0, e = 2.71828 
(The symbol x! means “x factorial” which is a mathematical way of getting the multiplication value of x multiplied by (x-1) multiplied by (x-2) etc. until you hit 1)

Another popular one is a Markov chain, where each event depends only on the state attained in the previous event. They can be discrete time interval or continuous time interval as well their states can be discrete or continuous.

P(X_t= x_t| X_t-1= x_t-1 ,…, X_t-k= x_t-k) = P(X_t= x_t| X_t-1= x_t-1) for all k, t > 0 and 1 ≤ k < t 

Simply put, Markov’s assumption is “Future depends on the past only through the present”. Markov chains may be modeled by “finite state machines” (computational model), and “random walks”(simulation paths) and useful in economics, weather predictions, queueing, game theory, genetics, finance and many more. PageRank used by Google to order the search results, is a type of Markov chain. A Markov chain is generally shown by a state diagram or transitional matrix. For example, consider a ball being picked out of a bag and you can get any number of red, blue, or green colored ones. The following illustrates this. The table on the left indicates the probability. The first row indicates, that you currently got a red color and the probability of the next one being red is 0.25, blue is 0.5 and green is 0.25. The state diagram gives a visual representation of the same and so does the transitional matrix.

I am not sure how further to give a brief about such models and its applications in simple terms that makes it a good read here. If you have any suggestions on how to do so, do share your suggestions. There are enough materials on the internet though. I would suggest these two sites which give a very good visual representation of this process.

[ Markov-Ref 1 ][ Markov-Ref 2 ]

For many situations, the non-deterministic models can be approximated to a deterministic model to simplify and work within certain acceptable error margin. There are other complex situations that require more accurate estimation of the situation and might require the use of probabilistic or a combination of deterministic and probabilistic model. However, there are situations where even probabilistic models may not work well.

Modeling a Waterfall

One of my earliest wonder when studying these concepts and its applications to various problems in science and control systems was — when a model broke down under a particular condition and there arose the need to start all over again to evolve the model in the best possible way to deal with the changed situation. Perhaps, it is the way we evolve and find new models.

via media.giphy

However, in some cases, there was a sudden change in the model parameters, after which it cannot be turned back, that is the earlier model no longer worked and cannot be reversed. The example of waterfall always lingers in my mind to explain this. The water flow before the water reaches the waterfall point (edge) can be modeled in a particular way. But at the edge, the flow is neither taking the model based on which it was flowing earlier, nor the vertical fall model that has taken course after the water falls. See the illustration diagram below.

The water particles at the edge of waterfall point are actually moving in a horizontal direction, but also have ‘started’ falling. There seems to be a breakdown of ‘time’ factor when the water is at the same time seeming to move in both direction, at least mathematically.

Pulse Signal

Another example is in pulse signals, which moves from ‘lower’ level to ‘higher’ level on one side and moving from ‘higher’ level to ‘lower’ level, at the falling side.

At the point (edge) of switching to another level, it could move into an unpredictable state, when the frequency of the signal gets higher as the slope of the pulse gets steeper.

Fuzzy logic

I heard about ‘Fuzzy logic’ during my days in college, more as a marketing term used by some ‘Washing machine’ companies. I did not do much research on this topic at that time, except that I understood it as using some sort of heuristic and intuitional approach to problem solving. That also introduced me to the words ‘heuristics’ and ‘intuition’, which did not make any logic to me on the ways we thought of coding and programs using logic. Fuzzy logic deals with more than the two binary values of ‘true’ and ‘false’ and has ‘partial truths’ between ‘completely true’ and ‘completely false’ (linguistic variables). Lotfi Zadeh proposed this logic considering human decisions are not based on binaries but a range. For example, the temperature in a control system can be modeled as:

Temperature, t = { very cold, cold, warm, very warm, hot }

Fuzzification helps map mathematical inputs to such variables, form rules and logic to model the problem scenario and de-fuzzification to get the continuous variables from fuzzy values.

Fuzzy logic and probability help in different forms of uncertainty — the former one to define an input/observation in a fuzzy set, and later assumes subjective-ness and bias based on likelihood of a certain condition. The conversion can use ‘hedges’ which are adverbs such as ‘very’, or ‘somewhat’, which modify the meaning of a set. But not all sets/mappings can be assumed to form a fuzzy function/mapping.

Chaos Theory

Chaos theory, first came to my knowledge, probably from the novel/movie Jurassic Park, where the character Ian Malcolm mentions about the same to explain that small changes can have big, unpredictable effects in a complex system. The ‘Butterfly effect’ is the famous way to explain this:

A butterfly flapping its wings in Beijing causes a storm in a city in USA.

Edward Lorenz was the first to use this term when studying weather patterns and predictability. Weather system is non-linear, complex and extremely sensitive to the initial condition. He demonstrated that limit of predictability of such systems. Although the term chaos in the normal sense means disorder, chaos theory provides for patterns that influence the system in closed loop way in the randomness of complex dynamic systems.

Chaos theory states that adjacent points in a complex system will eventually end up in different positions in different times. In order to mathematically model chaos, certain properties have been proposed.

via upload.wikimedia

A double rod pendulum would end up tracing a different shape each time we release the pendulum, depending on the initial position. Chaos, in a sense can be defined as a phenomenon when supersymmetry of the stochastic model breaks down spontaneously. I will not delve much into the mathematical expressions here. But it is an interesting aspect that needs to be understood based on one’s understanding of the earlier models mentioned here. They help explore the transition between order and disorder, and interconnectedness of apparently disconnected events and systems.

Fractals are infinitely complex pattern that are ‘self-similar’ across different scales.

via media.giphy

Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, architecture and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals!

[ Fractals-Ref1 ] [ Fractals-Ref2 ]

Chaos Theory, Fractals, Self-Organizing behaviors help to understand complex systems across multiple science disciplines as these are found in living and non-living systems.

Edge of chaos

Now we come to the subject of this blog. Edge of chaos refers to the transition between order and disorder, where there is complete randomness or chaos, where complexity is extremely high. We can as well term it “edge of order”.

Christopher Langton in the process of his study of self-reproducing cellular automata, discovered a threshold value, which is a transition between the state when the automata eventually repeats itself and state where there is completely random generated states that never repeat. Later Doyne Farmer termed this as the “edge of chaos”. Jim Crutchfield found that there is a peak at the “edge of chaos” where there is a maximum of information.

As the system and our model of the same moves from predictable models, to complex non-deterministic model and random process, we move towards the edge of chaos, the very thin border line towards the chaotic or completely disordered state.

Transformation rather than change

Our mathematical models consider how to incorporate change and rate of change. But I believe in the context of the edge of chaos theory, we need some mechanism to identify and differentiate between normal change, increased rate of change and what could potentially be a state of transformation, where change could be at it’s peak with a maximal degree of complexity. This is my own thought, that within the region called the edge of chaos, probably lies a transformation zone which could bring together certain uncorrelated events to move the complex system out of chaos to a targeted ordered state. There possibly, my guess, is where we can find answers to many of our questions around evolution, epigenetics, quantum weirdness and what causes them.

Out of chaos comes order

Does order come out of chaos? It is clear from the edge of chaos model, that this is the case. Friedrich Nietzsche quote might have come from a philosophical or philological perspective, or from the original Latin phrase ‘Ordo ab Chao’ invented by Freemasons (the motto of the 33rd Degree of Scottish Rite Freemasonry) or a similar term ‘lux e tenebrious’ (meaning Light in Darkness) from the Latin translation of Gospel of John. But definitely it seems true in our mathematical and scientific context. Physicist have also seemed to have proved this in an experiment of introducing disorder to bring in order. [ ChaosExp-Ref ]

Theory of Everything(ToE)

I think it is not possible, not to mention about this theory in this context of finding models and patterns, chaos and order. But I am not going into details as it is a different topic in itself. ToE is a unifying hypothetical framework that should explain every physical aspect of our universe. Although at present there seems to be none, ‘String theory’ seems to come closest. Suffice to say human mind has been thinking on this for eons of time and the quest will continue. I can’t but use the below quote which is very relevant in the context of this subject:

At the same time, we should give due considerations to certain metaphysical aspects that would need a wider acceptance of such subjective experience that consciousness influences; biology; our environment and our interconnectedness. With quantum physics, coming close to certain philosophies, with telepathic research proving true in certain experiments, a larger ToE must consider encompassing all aspects.

[ ToE-Ref1 ] [ ToE-Ref2 ] [ ToE-Ref3 ] [ ToE-Ref4 ]

Why I believe these subjects are important now?

Einstein might have had one of the most complete understanding of the laws governing the universe, than probably anyone else in the scientific history, when he mentioned the quote -

in his famous speech on “Geometry and Experience” at Berlin in 1921. But it does not take away the sheen in which mathematics has helped shape our thinking and perception of the world. In fact, I remember reading somewhere that natural geniuses are found only in mathematics and music. I consider music also as a form of harmonic mathematics. As Rabindranath Tagore, the mystic poet interacting with Einstein said:

First of all, let me confess here, that while I just started writing this blog a month back with my intuitive, back of the mind thoughts about ‘edge of chaos’, ‘model of the world’ and ‘transformation’, to bring these topics and connect with real research in the areas of mathematics and science, I had to dig deep. And in this process, my fascination only has grown more on this topic, with a little madness — yes :-) — that it infuses due to the complexity involved in understanding the mathematics and science behind it. And with my practice of spirituality and interest in philosophy, I also noticed certain pointers as to why these studies are going to be important in the days to come.

1. Revolution of Artificial Intelligence (AI)
2. In the current context of the world and it’s problems

Revolution of Artificial Intelligence (AI)

It is already being hyper-marketed, but there is no smoke without fire. There are very real transformational unit changes happening in AI, that are going to not only revolutionize the way we do our stuff, but human life in a dramatic way.

It is critically important for working folks to put topics relating to AI in their career roadmap to be relevant and contribute positively. The mathematical topics mentioned above are very relevant to development of computing technology and neurological simulations that are used in AI. I believe a singularity event is waiting to happen to dramatically change the ways AI is going to change our course. There is also the fear of some of the sci-fi concepts coming real, but we could use those concepts to avoid pitfalls early. A world governing council for AI, I believe is needed to ensure purpose of AI and principles are adhered to, lest it goes into a state of anarchy detrimental to our human civilization.

In the current context of the world and it’s problems

I revert to this topic of the current sense of craziness in this world, which is being made as business by the media world. As I mentioned in the beginning, my own observation over the past decade, seemed to indicate an increasing trend towards changes that seem to accelerate (similar to the water drops at the edge of a waterfall). If we apply the mathematic models to this scenario, there are very many influencing factors that probably have gone beyond the natural order that existed before. Although, what we see may be in a shortened time scale, the rate of change could be an indication moving towards an uncertain state in the shorter cycle itself, apparently into a dramatic shift to a singularity that will make it impossible to reverse. The singularity can move us in to a transformational way for most of us and humanity at large or in a downward way to our own decimation. I have not looked into yet at the details of what our great thinkers have predicted, except certain news about predictions of a war for water, Hawking’s premonition about the end of the world and other doomsday predictions that appear in the frontline now and then. There has also been independent work based on web bots to predict the future events. But what is essential, is to take a scientific mindset, and to realize it is time to think about this, think right and shape our thinking with an openness, and target an achievable state that might be balanced. For order to come out of this chaos, there are few very important things (out of possibly many more) that comes to my mind that humanity needs to focus upon:

• Food and potable water scarcity and ways to address them
• Economies based on need rather than wants, new ways of supply chain that covers all
• Division among humans on any basis, and the elements that foster it, and what we do about it
• Importance of mental health and scientific ways to address them
• Mindfulness & Heartfulness practices and Yoga among students and working professionals
• Climate change and how to deal with what might come, due to our inaction in the past decade
• Anti-biotic resistant super bugs and our approach to medicine and health issues

While at the outset this may not seem related to concepts of probability and Chaos theory, it is actually very pertinent to use such a model with some of the parameters of order and disorder in such a model. I have provided a list of sites, which I had referred to for my own research on these topics and hope you find them useful to further any interest my ramblings would have incited.

Self-Marketing

In Summary, the quest of mankind has been to understand the nature of this universe in it’s entirety. Be it scientific, mathematical, philosophical or spiritual, the quest will continue. It is also time for the mergence of various approaches for the larger benefit of humanity and our ecosystem so the future human being can indeed look back at this pivotal period as a transformative one that laid the foundation for their existence.

In view of the butterfly effect, mentioned above, it would not have been possible for me to arrive at this blog without the major contribution from the below, for which I am thankful. While this may be taken as a self-marketing of those I am associated with, it is also with a feeling of sharing what I think as good to my network.

• Patience and Endurance from my near and dear while drafting this
• Reviews and inputs from few friends
• My previous blogs and the comments I received
• Some who transformed my life [ Heartfulness ] [ Chariji ] [ Daaji ]
• My wonderful colleagues, stakeholders whom I have worked with so far
• The institutes and universities I have been associated with
• My larger family I grew up with, and everyone who reads this post patiently up to this point!

Follow My blogs at:
Sense And Balance @ Blogspot and Medium.com

References

Few references which I had used as well as for furthering your interest in these topics, which I recommend. This list is huge, but I can’t help but keep them here.

1. Linear equation — Wikipedia
2. Basic Algebra/Systems of Linear Equations — Wikibooks
3. Points on Straight lines
4. Nonlinear system — Wikipedia
5. Differential equation — Wikipedia
6. Taking Derivatives and Differentiation | Wyzant Resources
7. Derivative — Wikipedia
8. Differential Equations — Definitions
9. Probability Models
10. Probabilistic Theory
11. Probabilistic and deterministic models
12. Deterministic and probabilistic models
13. Graphical model — Wikipedia
14. Multiple Discrete Random Variables
15. Probabilistic_models
16. Nondeterministic vs. Probabilistic Models
17. Normal Distributions: Definition
18. Standard Normal Distribution Table
19. Probability
20. Normal Distribution
21. Stochastic Model
22. Bernoulli Distribution
23. Binomial Distribution
24. An Introduction to Stochastic Modeling
25. Introduction_to_Stochastic Processes
26. What is a stochastic process?
27. Probabilistic Modeling in Physics
28. Quantum Probabilistic Models Revisited
29. Quantum probability — Wikipedia
30. Probabilistic Systems Analysis
31. Markov Chains
32. Markov Chains explained visually
33. PageRank — Wikipedia
34. Standard Deviation and Variance
35. Random walk — Wikipedia
36. Finite State Machines
37. Stoachastic Process — notes
38. Random Processes: Basic Definitions
39. Fuzzy logic — Wikipedia
40. Control theory — Wikipedia
41. Fuzzy set — Wikipedia
42. Fuzzy logic — Wikipedia
43. Fuzzy set — Wikipedia
44. Artificial Intelligence Fuzzy Logic Systems
45. What is Chaos Theory? — Fractal Foundation
46. Chaos theory | Jurassic Park wiki
47. Chaos theory — Wikipedia
48. Self-organization — Wikipedia
49. Fractal — Wikipedia
50. Fractal — from Wolfram MathWorld
51. Butterfly effect — Wikipedia
52. Chaos in an Atmosphere Hanging on a Wall
53. Does the river run wild? Assessing chaos in hydrological systems — ScienceDirect
54. Modeling the Pulse Signal by Wave-Shape Function
55. Decomposition Analysis of Digital Volume Pulse Signal
56. Acceleration — Modeling a waterfall
57. Edge of Chaos — YouTube
58. Edge of chaos — Wikipedia
59. Complexity: Life at the Edge of Chaos — Roger Lewin — Google Books
60. Edge of Chaos
61. The Edge of Chaos
62. James P. Crutchfield — Wikipedia
63. The Edge of Chaos — The Imaginative Conservative
64. Theory of everything — Wikipedia
65. Laplace’s demon — Wikipedia
66. From Eternity to Here: The Quest for the Ultimate Theory of Time — YouTube
67. A Theory of Everything
68. Matter Energy and Consciousness — Heartfulness Blog
69. Prominent Scientist Says Consciousness Is Key to a ‘Theory of Everything’
70. From chaos comes order? Physicists make baffling discovery
71. Stephen Hawking’s Six Wildest Predictions
72. Inside the algorithms that are predicting the future
73. Web Bot — Wikipedia
74. Glossary of artificial intelligence — Wikipedia
75. Physicists Are Closing the Bell Test Loophole | Quanta Magazine
76. Quantum weirdness has been tested beyond the particle scale for the first time
77. Tagore and Einstein — School of Wisdom®
78. Albert Einstein — Laws of mathematics refer to reality — context of quote
79. Einstein: “Geometry and Experience”
80. Lotfi A. Zadeh — Wikipedia
81. Quotations from Friedrich Nietzsche
82. Friedrich Nietzsche — Wikipedia
83. ‘From chaos, comes order’
84. Ordo ab Chao in Freemasonry
85. ORDO AB CHAO | GnosticWarrior