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

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.

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.

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.

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.

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.

This section is a quick slide to explaine the long tail distirbution

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.

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.

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.

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.

We have encountered many extraordinary phenomena during this 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 and all the product of very simple iterative functions that we discussed in an earlier section.

In this last lecture to the course we present the real world phenomena of fractals in pictures to give you a sense for both there prevalence, extraordinary detail and visual attractiveness.