So let's summarize Unit Six.
This unit was on the phenomenon of universality.
I began with an observation; different functions have similar bifurcation diagrams.
We looked at the logistic equation in previous units, and explored its bifurcation diagram in quite some detail.
But in this unit, we looked at two other functions; the cubic function and the sine function.
These functions are all similar in that they have a single maximum and that maximum is not pointy, it's quadratic-like.
But they have very similar looking bifurcation diagrams.
They're not absolutely identical, but they're very similar and in particular we see this period doubling structure occur again and again
across different bifurcation diagrams and also within the same bifurcation diagram we see period doubling at different points.
So that led us to look geometrically at period doubling. So here's just a sketch of period two doubling to four doubling to eight.
And we said alright - what can we say geometrically about this?
So define delta one to be the length of this branch, delta two to be the length of that branch, delta three to be the length of that branch,
so that's what this equation says here.
The 'r' values are one or two or three are the values at which these bifurcations occur.
And then lowercase delta n is just the ratio of one length to the next.
So delta n tells us how many times larger this is than this - that would be delta one.
Delta two would tell us how many times larger this is than this, and so on.
So that's delta, and we calculated it for a few different functions.
And the main result is that delta n, as n approaches infinity and gets large, approaches this number delta and the value of this delta is universal.
So delta is a value of delta n for large n as we go looking at larger and larger periods.
So delta is universal, and what that means is it has the same value for all functions f(x) that map an interval to itself and have a single quadratic maximum.
So that's a very broad family of functions, pretty much any function that has a single maximum on some interval.
It will undergo period doubling and it will do so in a way such that the ratio of one bifurcation to the next to the next approaches the same value 4.669201.
The value of this number is often known as Feigenbaum's constant.
So that's the phenomenon of universality in maps, one dimensional functions that obey this criteria will have the same delta, the same ratio.
Then I said it turns out that we can also see universality in physical systems. In particular, the period doubling route is observed in physical systems.
There the parameter might be something like flow rate or the difference between temperature in a convection roll experiment,
the difference between temperature in the top and the bottom.
So instead of having this r, that we adjust on the computer, we have something else that we adjust in a real physical experiment.
But we can define delta and measure it the same way for these systems,
and the results that one gets are consistent with this universal value.
They don't agree exactly because of experimental error, and also this says that we don't expect this exact value until large n,
and usually we can only observe a few period doublings.
But what all this says is that some how these simple one-dimensional systems capture some feature of complicated physical systems
like dripping faucets, certain electric circuits and convection rolls in certain geometries.
Somehow, in a certain sense, there's only one way to have this period doubling transition into chaos.
That large, large numbers of systems, both mathematical systems and physical systems undergo period doubling in the same way,
governed by this ratio.
So there's something common about all of these different systems, and not just in a rough, "Oh they're kind of alike",
but we get this same number, something you can go out and measure.
So lastly, I tried to explain how universality is possible. There are two pieces to this.
One is how is it possible in one-dimensional maps, and the second piece is what do one-dimensional maps have to do with actual physical systems.
That latter question, I'll try to address in the next couple of units.
For the first question, how is universality possible in one-dimensional maps, that can be explained very nicely by a mathematical theory,
or set of techniques, called renormalization or the renormalization group.
Renormalization is a technique in math and physics, where one changes a length scale and then observes how other properties of the system change.
So for some types of transitions there is some sort of a fractal structure, properties are independent of length.
That was the case for the bifurcation diagram.
We see a fractal structure in the bifurcation diagram for the logistic equation and other equations.
We see those pitchforks or trees repeating again and again even as we zoom in deeper and deeper.
So this fact, or realization, that some properties are independent of length, together with this mathematical machinery for changing lengths,
leads to some techniques that can be used to derive critical exponents.
That's a term from statistical physics and delta is an example of a critical exponent.
Renormalization also explains why some model details don't matter.
So in dynamical systems, we can have many different initial conditions get pulled to the same attracting fix point, say.
When one applies renormalization, one has many different functions, that all go to the same attracting universal function.
So the details of those initial conditions, or the details of those functions, logistic, cubic, and so on, don't matter for this universal behaviour.
So I should say it one more time: universality is an amazing result.
It's an incredible mathematical fact that these different functions have the same or very similar bifurcation diagrams and this same ratio of 4.669201,
but what's really amazing is that this number, 4.669, can be found in physical experiments as well.
That real physical systems, much more complicated than one-dimensional functions, show period doubling transitions to chaos,
and that certain features of these transitions are universal - the same across a very, very broad range of systems.
I think this is a really nice and exciting result very much in the spirit of physics, that says there's some common mechanism or common structure
across a range of phenomena that initially might seem quite different and distant and unrelated.
The question then is, does the phenomena of universality, or what does the phenomenon of universality have to say about the study of complex systems.
Here I'm thinking of systems with many interacting parts, biological systems, economic systems, complicated ecologies with many interacting individuals.
Would we expect to see universal transitions in those, the way we do in the relatively simple physical systems, and very simple mathematical systems?
Well, I think the answer there, to be honest, is: nobody knows for sure.
My opinion, and just to be clear, this is not held by everybody, is that universality in the exact sense we've described here,
there being these critical exponents, things like delta, is likely not to appear in the study of complex systems.
I certainly think we'll see similarities between apparently different complex systems, that's sort of one of the central premises of the study of complex systems.
But I doubt that similarities will be as exact, as quantitative, as they are for universality in period doubling and related phenomena,
the way we've discussed in this unit.
Finally, I want to say a few words about fractals and the transition from periodic to chaotic behaviour.
In the bifurcation diagrams we've looked at in this unit, we see a certain type of transition from periodic to aperiodic chaotic behaviour,
an order disorder transition. Period doublings - they accumulate - and the system becomes chaotic.
This transition is accompanied by fractal behaviour and we see that in the bifurcation diagram.
Those period doublings bifurcations, those forks, those tree branches in the diagram,
we see them again and again at all scales, so that's an example of a fractal phenomenon.
And the number 4.669 is kind of like a fractal dimension, it tells us how much bigger the bigger branches are than the branches at a given scale.
In this case, we see an order disorder transition and that's associated with some sort of fractal or scale free behaviour.
In the natural and physical world and even in the social world, we see lots and lots of examples of scale free behaviour.
In distributions of earthquakes and mountain ranges, in populations of cities, there are all sorts of fractals all throughout the natural and physical world.
In fact lots of systems that we might think of as being complex systems have some sort of a fractal character to them.
However, that does not mean that these systems are necessarily poised at the edge of some order disorder transition, about to go into chaos.
The reason I say that is that there are many, many different ways to make these fractal distributions, these so-called power laws.
When one observes power laws in a system, there are many possible explanations.
Many different mechanisms give rise to power laws and fractals in physical and natural and social systems.
It doesn't have to be the result of an order disorder transition or something poised at the edge of chaos.
A cautionary note because I'm not sure that the point I just made about there being different mechanisms for fractals is widely appreciated.
I think sometimes folks, particularly physicists, become so excited by the phenomena of universality, and it is amazing,
as you've heard me say, that maybe we get a little bit carried away and extend it beyond the range that we should.
There are lots of ways to make fractals an order disorder transition or in general a phase transition of the universal type I've been talking about is just one possible way.
This brings us to the end of Unit Six on universality.
In the next unit, Unit Seven, we'll go back to differential equations, and we'll look at differential equations in two and three dimensions
and this will lead us to the notion of phase space, one of the really important abstractions from the study of dynamical systems.
It's a really useful geometric and analytic tool.
And then in the unit after that, we'll look at how strange attractors unfold in phase space.
So , see you next week in Unit Seven.
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