In the last segment, I talked a bit more generally
about the state variables of a dynamical system.
The values of these variables,
like the angles and angular velocities of the double pendulum,
evolved continuously under the influence
of the dynamics of the system.
For example, gravity is acting on the double pendulum,
but the constraints of the ball bearings,
which only allow rotational motion, effect that dynamics as well.
So, what do you think the state variables would be for an elbow?
An elbow... unless you're damaged, can only move in this direction
So, you could fully describe the state of an elbow
thinking about this angle and the rate of change of that angle.
So, the state space there would have two axes.
What about a wrist?
A wrist can move this way
or it can move that way.
So, we need two angles to describe its position
and two angular velocities to describe how it's moving.
So, the state space for a wrist would have four axes.
What about a shoulder?
A shoulder can move this way or that way
or this way.
So, a shoulder has three state variables that describe the position
and three that describe the velocity,
and its state space would have six axes.
The state space is arguably the most powerful of the many representations
we use in the field of nonlinear dynamics.
It suppresses time and brings out the patterns
that emerge as the system evolves.
So, it's kind of like mounting a camera
over that eddy in the stream that I used as an example before,
opening the shutter, putting a wood chip in the eddy,
letting it go around with the camera taking pictures the whole time,
and then turning off the shutter of the camera.
What the camera would have recorded
is the path of the wood chip through the eddy.
If you wanted to explore a wider expanse of the water
of the dynamical system in the metaphor,
you would want to repeat that experiment,
dropping the wood chip in lots of different parts of the state space
and seeing what happens.
And, that's the idea of a state space portrait of a dynamical system...
you're picking initial conditions to explore
that are interesting and representative of the dynamics of the whole system.
And, just like in a portrait that you would draw of a person's face,
you have to focus your exploration.
An artist will draw an eye very carefully,
but he or she will not devote much work to the cheek.
And, we do that also with state space portraits.
Let's draw one for the pendulum.
As we established, the axis of the state space are:
theta (θ) - the angle, and omega (ω) - the angular velocity.
The state space of the double pendulum would have four axes.
That gets hard to draw.
So, this state of the system is zero, zero (0, 0) on those axes.
This state of the system...
is pi, zero (π, 0) on those axes
if I define angle positively in this direction.
This state of the system...
would be negative pi, zero (-π, 0) on those axes.
So, if this is zero, zero (0, 0)...
that state is two pi, zero (2π, 0)
and that state is negative two pi, zero (-2π, 0)
Here's the pendulum against a better backdrop
so that you can really see what's going on.
And, I want to circle back around to what I just said.
If this is theta equals zero (θ=0)
this looks just like that to us.
But, if somebody is keeping track of
how many times it's wound around the axis back here,
that's actually a different angle.
It's actually different by two pi (2π),
and, if I have defined theta positive (θ) going this way,
that's actually two pi (2π)
So... here is...
theta equals zero (θ=0)
theta equals two pi (θ=2π)
theta equals four pi (θ=4π)
two pi (2π)
zero (0) again
and now, negative two pi (-2π)
There's a way of measuring angle
that kind of discounts all of the times it's wound itself around the axis -
you can use the mathematical modulo operator.
You can divide the angle by two pi (2π)
and only keep the fractional part.
That is equivalent to...
measuring the angle the way we actually see it
without keeping track of the winding up.
This point here would be pi over two, zero (π/2, 0)
but if I started it with a little bit of a push
that would be pi over two (π/2) and then a small amount of velocity.
The next question is: what is the sine of that velocity?
Well, I defined angle positive going this way,
so a velocity that is making the angle go that way is a negative velocity.
So, you get the idea.
Now, those were all thought experiments about states at specific points in time.
Now, let's think about how the state evolves.
First of all, most of the examples that we just went through
involve fixed points of the dynamics
So, this point right here is a fixed point of the dynamics.
You put the system there - it doesn't move.
It's a stable fixed point because, if I deliver a small perturbation,
that perturbation shrinks over time.
The point...
negative two pi, zero (-2π, 0) is also a stable fixed point.
The point...
two pi, zero (2π, 0) is a stable fixed point.
So, the points at even multiples of pi (π) are stable fixed points.
The points at odd multiples of pi (π) are unstable fixed points
This one I think I can get to balance.
There is a fixed point of this dynamics.
It is unstable.
So, that's one.
There's another one here.
So, how many unstable fixed points do you think this device has?
An infinite number - same thing for stable fixed points.
Here are some of the stable fixed points in blue
and here are some of the unstable fixed points in red
at the odd multiples of pi (π).
So, that means we're done with some of the portrait -
all those points on the theta (θ) axis that I just drew.
Now, we need to explore some of the other points.
So, let's look at this one.
What does that trajectory look like in the state space representation?
Like an ellipse, actually, if the angle is small enough.
We'll get back to the exact form of the curve
in the next two segments of this unit.
Alright, let's try a slightly larger initial condition.
That's a bigger ellipse in the state space.
For even larger theta (θ)
those ellipses deform into rugby balls.
We'll get back to the mathematics of that
when we study ordinary differential equations later in this unit.
What about this one?
What does that look like on the state space portrait?
To understand this, think about the fact that
the pendulum slows down every time it goes over the top
and speeds up every time it goes past the bottom.
Or, this one.
In one of these trajectories, theta (θ) is winding itself up.
It keeps getting more and more positive because the velocity is positive.
In the other, theta (θ) keeps getting smaller because the velocity is negative.
Something interesting happens in between these closed curves
and these so-called "running solutions."
And, that interesting thing involves the unstable fixed points.
We'll talk about that at the beginning of unit four.
In the meantime - here's our portrait.
I've added a few things.
The behavior is the same around the two pi (2π) fixed point
as the zero pi (0π) fixed point as the four pi (4π) fixed point.
I've also added some faster running solutions.
If I pushed the pendulum faster over the top it would spin faster,
which is why those curves are further out the omega (ω) axis,
and it would slow down less over the top.
So, the wiggles are supposed to be drawn a little bit less wiggly.
And, this portrait was our goal for today.
Two important points here:
first of all, as some of you have probably noticed,
the portrait that we just drew...
is actually not representative of this particular device,
because this particular device...
has friction.
It's going through a damped oscillation
to a fixed point.
The portrait I drew assumes no friction.
With no friction, there's no attractors
because there's nothing causing a transient to die out,
nothing causing the trajectory to converge to anything.
So, the fixed points in the portrait that we drew
are actually not technically attracting fixed points.
A system without friction
is called a "conservative system" or a "Hamiltonian system."
That's a synonym for:
"it doesn't have any dissipation or friction."
And, the points at zero (0), two pi (2π), four pi (4π) negative two pi (-2π)
are called "elliptic fixed points,"
and that's because of those ellipses around them.
All that is a little bit beyond our scope in this course.
Another important point: uniqueness.
Notice that the trajectories in our picture don't cross.
In fact, that's a mathematical requirement.
As you'll see in the next segments, you can think of a trajectory
as a ball rolling downhill on a landscape defined by a dynamics.
And, in the kind of systems that we''ll play with,
there's only one direction - that's dynamically downhill...
unless someone is moving the landscape underneath you.
More on that a little bit later.