Nonlinear Dynamics & Chaos
Chaos theory is the study of nonlinear systems dynamics whose development is marked by iteration and feedback loops, making them sensitive to initial conditions (what is called the butterfly effect). Even though these nonlinear systems may be deterministic, such as a double pendulum, their long-term trajectory is typically not possible to predict without running the system.1
Linear & Nonlinear
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 do not 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 may be the product of multiple different interacting forces, have multiple attractor states and be able to change between different attractors over time. A classical example given of this is a double pendulum. A simple pendulum without a joint will follow the periodic and deterministic motion characteristic of linear systems with a single equilibrium. 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, the dynamics of a planet orbiting another is an example of a linear system with a single equilibrium and attractor, but when we add another planet into this equation, 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.
Sensitivity to Initial Conditions
Whereas with 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: “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 moreover it is primarily focused 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 nondeterministic. In these complex systems, we would, of course, expect all sorts of random, complex and chaotic behavior, but it is not something we would expect in simple deterministic systems. This chaotic and unpredictable behavior happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable, which is deeply counter-intuitive to us. A double pendulum essentially consists of only two interacting components – that is each limb – and these limbs are both strictly deterministic when taken in isolation. But when we join them, this very simple system can and does exhibit nonlinear and chaotic behavior. In these chaotic systems, their unpredictable nature emerges out of the interaction between their components.
Complex Chaotic Systems
Although chaos theory deals with simple nonlinear systems, the phenomenon of sensitivity to initial conditions is also a part of complex systems as we might expect. For example, say I am walking to the subway station on my way to work but as I pass the bus stop I notice a bus just pulling in that I recognize as one that will take me near to where I want to go. So I jump on the bus and it takes me off into a different basin of attraction than if I had arrived just 30 seconds later. In complex systems this sensitivity to initial conditions can be very acute during particular stages in their development, what are called phase transitions. When they are far-from-equilibrium, small fluctuations can push them into new basins of attraction.