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,

miniscule 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 pertubation 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.