Another one of the fundamental properties of a chaotic system is dense periodic orbits. It’s a bit of an odd one: a chaotic system doesn’t have to have periodic orbits at all. But if it does, then they have to be dense.

The dense periodic orbit rule is, in many ways, very similar to the sensitivity to initial conditions. But personally, I find it rather more interesting a way of describing key concept. The idea is, when you’ve got a dense periodic orbit, it’s an odd thing. It’s a repeating system, which will cycle through the same behavior, over and over again. But when you look at a state of the system, you can’t tell which fixed path it’s on. In fact, minuscule differences in the position, differences so small that you can’t measure them, can put you onto dramatically different paths. There’s the similarity with the initial conditions rule: you’ve got the same basic idea of tiny changes producing dramatic results.

In order to understand this, we need to step back, and look at the some basics: what’s an orbit? What’s a periodic orbit? And what are dense orbits?

To begin with, what’s an orbit?

If you’ve got a dynamical system, you can usually identify certain patterns in it. In fact, you can (at least in theory) take its phase space and partition it into a collection of sub-spaces which have the property that if at any point in time, the system is in a state in one partition, it will never enter a state in any other partition. Those partitions are called orbits.

Looking at that naively, with the background that most of us have associated with the word “orbit”, you’re probably thinking of orbits as being something very much like planetary orbits. And that’s not entirely a bad connection to make: planetary orbits are orbits in the dynamical system sense. But an orbit in a dynamical system is more like the real orbits that the planets follow than like the idealized ellipses that we usually think of. Planets don’t really travel around the sun in smooth elliptical paths – they wobble. They’re pulled a little bit this way, a little bit that way by their own moons, and by other bodies also orbiting the sun. In a complex gravitational system like the solar system, the orbits are complex paths. They might never repeat – but they’re still orbits: a state where where Jupiter was orbiting 25% closer to the sun that it is now would never be on an orbital path that intersects with the current state of the solar system. he intuitive notion of “orbit” is closer to what dynamical systems call a periodic orbit: that is, an orbit that repeats its path.

A periodic orbit is an orbit that repeats over time. That is, if the system is described as a function f(t), then a periodic orbit is a set of points Q where ∃Δt : ∀q∈Q: if f(t)=q, then f(t+Δt)=q.

Lots of non-chaotic things have periodic orbits. A really simple dynamical system with a periodic orbit is a pendulum. It’s got a period, and it loops round and round through a fixed cycle of states from its phase space. You can see it as something very much like a planetary orbit, as shown in the figure to the right.

On the other hand, in general, the real orbits of the planets in the solar system are not periodic. The solar system never passes through exactly the same state twice. There’s no point in time at which everything will be exactly the same.

But the solar system (and, I think, most chaotic systems) are, if not periodic, then nearly periodic. The exact same state will never occur twice – but it will come arbitrarily close. You have a system of orbits that look almost periodic.

But then you get to the density issues. A dynamical system with dense orbits is one where you have lots of different orbits which are all closely tangled up. Making even the tiniest change in the state of the system will shift the system into an entirely different orbit, one which may be dramatically different.

Again, think of a pendulum. In a typical pendulum, if you give the pendulum a little nudge, you’ve changed its swing: you either increased or decreased the amplitude of its swing. If it were an ideal pendulum, your tiny nudge will permanently change the orbit. Even the tiniest perturbation of it will create a permanently change. But it’s not a particularly dramatic change.

On the other hand, think of a system of planetary orbits. Give one of the planets a nudge. It might do almost nothing. Or it might result in a total breakdown of the stability of the system. There’s a very small difference between a path where a satellite is captured into gravitational orbit around a large body, and a path where the satellite is ejected in a slingshot.

Or for another example, think of a damped driven pendulum. That’s one of the classic examples of a chaotic system. It’s a pendulum that has some force that acts to reduce the swing when it gets too high; and it’s got another force that ensures that it keeps swinging. Under the right conditions, you can get very unpredictable behavior. The damped driven pendulum produces a set of orbits that really demonstrate this, as shown to the right. Tiny changes in the state of the pendulum put you in different parts of the phase space very quickly.

In terms of Chaos, you can think of the orbits in terms of an attractor. Remember, an attractor is a black hole in the phase space of a system, which is surrounded by a basin. Within the basin, you’re basically trapped in a system of periodic orbits. You’ll circle around the attractor forever, unable to escape, inevitably trapped in a system of periodic or nearly orbits. But even the tiniest change can push you into an entirely different orbit, because the orbits are densely tangled up around the attractor.

One thing that I wanted to do when writing about Chaos is take a bit of time to really home in on each of the basic properties of chaos, and take a more detailed look at what they mean.

To refresh your memory, for a dynamical system to be chaotic, it needs to have three basic properties:

1. Sensitivity to initial conditions,
2. Dense periodic orbits, and
3. topological mixing

The phrase “sensitivity to initial conditions” is actually a fairly poor description of what we really want to say about chaotic systems. Lots of things are sensitive to initial conditions, but are definitely not chaotic.

Before I get into it, I want to explain why I’m obsessing over this condition. It is, in many ways, the least important condition of chaos! But here I am obsessing over it.

As I said in the first post in the series, it’s the most widely known property of chaos. But I hate the way that it’s usually described. It’s just wrong. What chaos means by sensitivity to initial conditions is really quite different from the more general concept of sensitivity to initial conditions.

To illustrate, I need to get a bit formal, and really define “sensitivity to initial conditions”.

To start, we’ve got a dynamical system, which we’ll call f. To give us a way of talking about “differences”, we’ll establish a measure on f. Without going into full detail, a measure is a function which maps each point x in the phase space of f to a real number, and which has the property that points that are close together in f have measure values which are close together.

Given two points x and y in the phase space of f, the distance between those points is the absolute value of the difference of their measures, .

So, we’ve got our dynamical system, with a measure over it for defining distances. One more bit of notation, and we’ll be ready to get to the important part. When we start our system at an initial point , we’ll write it .

What sensitivity to initial conditions means is that no matter how close together two initial points and are, if you run the system for long enough starting at each point, the results will be separated by as large a value as you want. Phrased informally, that’s actually confusing; but when you formalize it, it actually gets simpler to understand:

Take the system with measure . Then f is sensitive to initial conditions if and only if:

• Select any two points and such that:
• Let diff(t) = . (Let diff(t) be the distance between and at time .)
• (No matter what value you chose for G, at some point in time T, diff(T) will be larger than G.)

Now – reading that, a naive understanding would be that the diff(T) increases monotonically as T increases – that is, that for any two values and , if with measure . And for our non-chaotic system, we’ll use , with .

Think about arbitrarily small differences starting values. In the quadratic equation, even if you start off with a miniscule difference – starting at v₀=1.00001 and v₁=1.00002 – you’ll get a divergence. They’ll start off very close together – after 10 steps, they only differ by 0.1. But they rapidly start to diverge. After 15 steps, they differ by about 0.5. By 16 steps, they differ by about 1.8; by 20 steps, they differ by about 1.2×10⁹! That’s clearly a huge sensitivity to initial conditions – an initial difference of 1×10^-5, and in just 20 steps, their difference is measured in billions. Pick any arbitrarily large number that you want, and if you scan far enough out, you’ll get a difference bigger than it. But there’s nothing chaotic about it – it’s just an incredibly rapidly growing curve!

In contrast, they logistic curve is amazing. Look far enough out, and you can find a point in time where the difference in measure between starting at 0.00001 and 0.00002 is as large as you could possibly want; but also, look far enough out past that divergence point, and you’ll find a point in time where the difference is as small as you could possible want! The measure values of systems starting at x and y will sometimes be close together, and sometimes far apart. They’ll continually vary – sometimes getting closer together, sometimes getting farther apart. At some point in time, they’ll be arbitrarily far apart. At other times, they’ll be arbitrarily close together.

That’s a major hallmark of chaos. It’s not just that given arbitrarily close together starting points, they’ll eventually be far apart. That’s not chaotic. It’s that they’ll be far apart at some times, and close together at other times.

Chaos encompasses the so-called butterfly effect: a butterfly flapping its wings in the amazon could cause an ice age a thousand years later. But it also encompasses the sterile elephant effect: a herd of a million rampaging giant elephants crushing a forest could end up having virtually no effect at all a thousand years later.

That’s the fascination of chaotic systems. They’re completely deterministic, and yet completely unpredictable. What makes them so amazing is how they’re a combination of incredibly simplicity and incredible complexity. How many systems can you think of that are really much simpler to define that the logistic map? But how many have outcomes that are harder to predict?