In the late 1800's Poincaré came up with geometric ways to understand dynamical systems. This is a way that you can understand more about the general behaviours of a dynamical system even without solving the equations. This allows you to ask questions about the stability of solutions (eg. will three planets remain locked together, or will one of them fly away?) .
Poincaré also glimpsed chaos: The sensitivity of a system to initial conditions.
This is the paths of three planets around each other and is an example of ‘chaotic’ system. We’ll discuss more of this later. This image is take from the video at https://vimeo.com/11993047.
Hang on - but if three planets are chaotic, and our solar system has a lot more than three, why doesn’t our solar system seem to be chaotic?
When you have a chaotic system, there is always a time-scale associated with it. That timescale roughly says that for times shorter than this, you can still make reasonable predictions, but over longer times, you no longer can. In the case of the solar system, that timescale is around 200 million years. This means that given the positions of the planets now, we can be very sure of where they are over the next hundred million years, but over the further hundreds of millions of years, we would be very quickly less and less sure of where they would be.
The reason that this timescale is so long in the solar system is because to a good approximation the dynamics of the solar system can be thought of as a lot of independent two-body problems. The sun is the dominant force in the solar system, and though the planets do feel each other’s pull, that pull is a lot less than that of the sun. If the mass of the sun and the planets were more similar, then it would be much harder to predict the movements of planets on much shorter timescales.
It wasn't until the 1950's that the next big breakthrough came. Although Poincaré had given us these graphical methods for studying systems, everything was still done by hand and that meant that we couldn't really simulate these chaotic and complex systems. In the 1950's computers allowed Edward Lorenz to study simplified weather systems. He found that these were inherently unpredictable and this was really the start of the formal study of chaos. He showed that there was structure in this chaos.
With the advancement of computers over the coming decades, and many more ingenious mathematical tools, we made huge progress in the understanding of dynamical systems. We can now model the firing of neurons in the brain, we have good models of the weather that can predict several days into the future (even though this system is chaotic), we can model populations of predators and prey, the motion of planets, stars and black holes, the motion of liquids and gasses, the interactions of elementary particles and more.