Nonlinear Systems & Chaos: An Introduction
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This course is a voyage into the extraordinary world of nonlinear systems and their dynamics, the primary focus of the course is to provide you with a coherent understanding of the origins and product of nonlinearity and chaos.
The course is designed as an intuitive and non-mathematical introduction, it explores a world of both extraordinary chaos where some small event like a butterfly flapping its wings can be amplified into a tornado, but also a world of extraordinary order in the form of fractals, self-similar structures that repeat themselves at various scales, one of nature's most ingenious tools for building itself.
Like quantum physics the world of nonlinearity is inherently counter intuitive,
it's a world where our basic assumptions start to break down and we get extraordinary results, once the domain of obscure mathematics, the concepts from nonlinear systems theory are increasingly proving relevant to the world of the 21st century.
This course covers all the key concepts from this domain, starting by looking at the origins of how and why we get nonlinear phenomena, we go on to talk about exponential growth, power laws, chaos theory, the butterfly effect, bifurcation theory, fractals and much more.
The course requires no prior specific knowledge of mathematics or science, it is designed as an introduction presenting concepts in a non-mathematical and intuitive form that should be accessible to anyone with an interest in the subject.
Nonlinear Systems Overview
In this module we start the course by giving an overview to the model of a system that will form the foundations for future discussion, we talk about linear systems theory based upon what is called the superposition principals of additivity and homogeneity. We will go on to talk about why and how linear systems theory breaks down as soon as we have some set of relations within a system that are non-additive, we also look at how feedback loops over time work to defy the homogeneity principle with the net result being nonlinear behavior.
Feedback Loops & Relations
In this section we introduce the key sources of nonlinearity as the type of relations between components within a system where these relations add or subtract some value to the overall system. We will talk about synergies and interference that make the system either greater or less than the simple sum of its components. We will then cover the second source of nonlinearity, what are call feedback loops that allow for both exponential growth and decay.
Exponentials, Power laws & Long tail distributions
In this module we will discuss the dynamics of exponentials and their counterparts power laws that represent an exponential or power relation between two entities, we talk about long tail distributions, sometimes called the fat tail, so called because it results in there being an extraordinary large amount of small occurrences to an event and a very few very large occurrences with there being no real average or normal to the distribution.
Systems dynamics & Chaos
For many centuries the idea prevailed that if a system was governed by simple rules that were deterministic then with sufficient information and computation power we would be able to fully describe and predict its future trajectory, the revolution of chaos theory in the latter half of the 20th century put an end to this assumption showing how simple rules could in fact lead to complex behavior. In this module we will describe how this is possible when we have the phenomena of what is called sensitivity to initial conditions.
Fractals
We will have encountered many extraordinary phenomena by this stage in the course but fractals may top them all, self-similar geometric forms that repeat themselves on various scales, they can both contain infinite detail, as we zoom in and the very counter intuitive phenomena of infinite length within a finite form with this all being the product of very simple iterative rules.
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1
What Is A System?In this module we give an overview to the model of a system that will form the foundations for our discussion on nonlinear systems, we will quickly present the basic concepts from systems theory such as elements, system's boundary, environment etc. and give some real world example of systems.
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2
Linear Systems Theory
In this lecture we will discuss linear systems theory that is based upon what is called the superposition principals of additivety and homogeneity, we will explore both of these principal separately to get a clear understanding of what they mean and the basic assumptions behind each
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3
Nonlinear Systems Overview
Having now laid our foundations this is where our discussion on nonlinearity really starts. We will talk about why and how linear systems theory breaks down as soon as we have some set of relations within a system that are non-addative, which appears to be often the case in the real world, we also look at how feedback loops over time work to defy the homogeneity principle with the net result being nonlinear behavior.
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4
Synergies & Interference
In this section we introduce the key sources of nonlinearity as the type of relations between components within a systems where these relations add or subtract some value to the overall system. We will talk about synergies that represent a positive interaction between components resulting in the system being more than the simple sum of its parts. Next we talk about interference which works in the inverse direction to synergies making the system less than the sum of its part.
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5
Feedback Loops & Equilibrium
In this lecture we will discuss the second source of nonlinearity, what are call feedback loops, where the previous output to the system has some effect on its environment and this then in turn feeds back to effect the current or future input to the system making exponential growth and decay possible.
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6
Exponentials & Power Laws
Exponentials are a signature key of nonlinear systems, unlike linear growth exponential grow represents a phenomenon where the actual rate of growth is growing itself to generate an asynchronous development with respect to time and some very counter intuitive events. In this module we will discuss the dynamics of exponentials and their counterparts power laws that represent an exponential or power relation between two entities.
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7
Long Tail Distributions
One result of the power laws that we discovered in the previous section are long tail distributions which is a type of graph we get when we plot a power law relation between two things. The long tail distribution, sometimes called the fat tail, is so called because it results in there being an extraordinary large amount of small occurrences to an event and a very few very large occurrences with there being no real average or normal to the distribution.
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8
Long Tail Distributions ExplainedThis section is a quick slide to explaine the long tail distirbution
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9
Dynamical Systems Introduction
Dynamical systems is a area of mathematics and science that studies how the state of systems change over time, in this module we will lay down the foundations to understanding dynamical systems as we talk about phase space and the simplest types of motion, transients and periodic motion, setting us up to approach the topic of nonlinear dynamical systems in the next module.
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10
Nonlinear Dynamics & Chaos
For many centuries the idea prevailed that if a system was governed by simple rules that were deterministic then with sufficient information and computation power we would be able to fully describe and predict its future trajectory, the revolution of chaos theory in the latter half of the 20th century put an end to this assumption showing how simple rules could in fact lead to complex behavior. In this module we will describe how this is possible when we have what is called sensitivity to initial conditions.
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11
The Butterfly Effect
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. In this module we give an overview to the concept and its origins in meteorology with the famous Edward Lorenz computer experiment.
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12
Phase Transitions & Bifurcation Theory
A phase transition is the transformation of a system from one state to another through a period of rapid change. The classical example of this is the transition between solid, liquid and gaseous states that water passes through given some change in temperature, phase transitions are another hallmark of nonlinear systems. In this module we discuss the concept in tandem with its counterpart bifurcation theory.
