 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 of motion everywhere. The world is a bit more complex than that. 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 these different attractors over time. Before we get into the theory of this, let's take a few examples to try and illustrate the nature of nonlinear dynamic systems. A classical example given of this is the double pendulum. A single pendulum without 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 now has two limbs instead of one. Now the dynamical state of the system will be a product of these two parts interacting over time. And we will get a nonlinear dynamical system. Let's take a second example. In the previous section, we looked at the dynamics of a planet orbiting another with a single equilibrium and attractor. But what would happen if we added another planet into this equation? Physicists puzzled over this one for a long time. We now have two equilibria points creating a nonlinear dynamical system as our planet would be under the influence of two different gravitational fields of attraction. 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 we call chaos. Wikipedia has a good definition for chaos theory so let's 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 divergent outcomes for such dynamical systems, rending long term predictions impossible in general. Now we should note that chaos theory really deals with deterministic systems and moreover it is primarily focused on simple systems in that it often only deals with systems with 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 course expect all sorts of random, complex and chaotic behavior but it is not something we would naturally expect within these simple deterministic systems. This chaotic and unpredictable behavior happens even though these systems are deterministic meaning 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 non-linear and chaotic behavior. This once again reveals to us the source of non-linearity as a product of the interactions between components within the system. In these chaotic systems unpredictable behavior emerges out of the interactions between components. Although chaos theory deals with simple non-linear systems the phenomena of sensitivity to initial conditions is also a part of complex systems as we might expect. For example say I'm walking to the subway station on my way to work but as I pass the bus stop I notice a bus 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 into a different basin of attraction than if I'd arrived just 30 seconds earlier. In complex systems this sensitivity to initial conditions can be very acute during particular stages in their development. During what are called phase transitions when they are far from equilibrium small fluctuations can push them into new basins of attraction. We'll be covering phase transitions in a later lecture but in the next module we will continue our discussion on chaos by talking further about the butterfly effect.