Nonlinear Dynamics & Chaos
Channel: Systems Innovation
Date: 2015-07-19

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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.

Isolated systems tend to evolve towards a single equilibrium, a special state that has been the focus of many-body research for centuries. But when we look around us we don't see simple periodic patterns everywhere, the world is a bit more complex than this and behind this complexity is the fact that the dynamics of a system maybe the product of multiple different interacting forces, have multiple attractor states and be able to change between different attractors over time. Before we get into the theory lets take a few examples to try and illustrate the nature of nonlinear dynamic systems.
A classical example given of this is a double pendulum; a simple pendulum with out a joint will follow the periodic and deterministic motion characteristic of linear systems with a single equilibrium that we discussed in the previous section. Now if we take this pendulum and put a joint in the middle of its arm so that it has two limbs instead of one, now the dynamical state of the system will be a product of these two parts interaction over time and we will get a nonlinear dynamic system.
To take a second example; in the previous section we looked at the dynamics of a planet orbiting another in a state of single equilibrium and attractor, but what would happen if we added another planet into this equation, physicists puzzled over this for a long time, we now have two equilibrium points creating a nonlinear dynamic system as our planet would be under the influence of two different gravitational fields of attraction.
Where as with our simple periodic motion it was not important where the system started out, there was only one basin of attraction and it would simply gravitate towards this equilibrium point and then continue in a periodic fashion. But when we have multiple interacting parts and basins of attraction, small changes in the initial state to the system can lead to very different long-term trajectories and this is what is called chaos.
Wikipedia has a good definition for chaos theory so lets take a quote from it. “Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Small differences in initial conditions yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general”.
We should note that chaos theory really deals with deterministic systems
and more over it is primarily focuses on simple systems, in that it often deals with systems that have only a very few elements, as opposed to complex systems where we have very many components that are non deterministic, in these complex systems we would of cause expect all sorts of random, complex and chaotic behavior, but it is not something we would expect in simple deterministic systems.


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