You can make artistic images using mathematics. That doesn't make these particular images \"art\".
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Let me try my hand at explaining what these images actually are, mathematically. It takes a few conceptual steps to get there, but I'll try not to include more actual math than is necessary.
In algebra you might be given an equation like x^2+2x-3=0 and asked to find solutions (they happen to be x=1 and -3.)
What if, instead of just solving for x^2+2x-3=0, I asked you about solutions for (the same stuff)=1 and then (stuff)=5 and (stuff)=-8 and so on? That'd be tedious; you'd be doing the same work over and over again with minor changes. But if you graphed y=(stuff) (try it with an
online graphing program; don't put the y= part in), then you could look at any given y value on your graph and come up with approximate solutions. In other words, your graph shows the solutions for (stuff)=number, for any number I pick.
One thing you would find is that, for y<-4, there are no real solutions -- the graph doesn't go below that point. So you can think of y=-4 as a point at which the equation y=x^2+2x-3 changes its behavior. We'll come back to the idea of looking for changes in behavior in a moment.
Now, for a definition: a
dynamical system is a system that changes over time according to a mathematical rule (In calculus terms, it might be written as dx/dt = f(x,t).) The system can be one, two, three, or more dimensions, but for now, let's think of it as two-dimensional. Now we're no longer looking for a single solution like x=1 or (x,y)=(3,2); we're looking at the way the whole system changes over time -- the way everything in the whole plane moves. With such a system, there are a few things we might care about:
- are there any points that the system tends to move towards over time? (\"stable fixed point\")
- are there points the system moves away from? (\"unstable fixed point\")
- does it settle into cycles? (\"limit cycles\")
- if there's more than one stable fixed point, how can you determine which of those points you'll go towards? (\"basins of attraction\")
You can characterize all of these by drawing what's called a \"
phase portrait\". It's basically just arrows that show where you go from where you are. If all the arrows point in at a single spot, that tells you that the system will move toward that spot over time no matter where you started. If they all spiral inward to a circle, that tells you the system will eventually tend to lead to cycles (it'll move around that circle.) So a phase portrait tells you how the system behaves. Some of the stuff in the original video is basically artistically-rendered phase portraits. But some of it goes a step further.
What if I took that system dx/dt=f(x,t) and I made some little tweak to it? Would the system behave the same way, or would it be radically different? If it had just one stable fixed point, could my tweak give it two, or none?
With our original equation, it always had two solutions as long as y>-4; the exact values were different but there were always two of them. But if we went to y<-4 it had no solutions. We call y=-4 a \"bifurcation value\" for that system.
For our more complicated system, let's say it depends on a \"parameter\" called m that we're going to tweak. We'd write the equation as dx/dt=f(x,t;m) to denote this. For different values of the parameter m, the system might behave differently -- sometimes it might lead to limit cycles, other times to fixed points, and other times it might be
chaotic (which basically means that it settles down into a sort of pattern, but not into a simple and predictable cycle or fixed point.) Since our system changes its behavior depending on m, we can create a \"bifurcation diagram\" that shows exactly how changes in m effect the system.
Most of the pictures in the original video are, essentially, artistically colored bifurcation diagrams based on a complex-valued parameter of some system. The famous Mandelbrot Set is similar, showing parameter values for which a particular equation behaves in a particular way near zero. In all cases,
they're just diagrams showing how a system changes its behavior as you tweak some little piece of it, with the colors showing different types or speeds of behavior.
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