In this segment, we're going to circle back around
to the notion of a fixed point, a point in the
state space of a dynamical system that's
stationary under the influence of the dynamics like,
for example, that point of the pendulum.
This is a stable fixed point because perturbations
to it will shrink over time. This is also a
fixed-point, in the sense that if you put the
system there, it stays there, but it's an
unstable fixed point because if you perturb it,
that perturbation grows. That's the right set
of ideas and terminology by the way for what
are called dissipative dynamical systems. Those
are the kinds that have attractors. In conservative
dynamical systems where there is no friction,
there are no attractors. There are still fixed points
in conservative dynamical systems and
there is still chaos, it's just that there are
not attracting fixed points and chaotic attractors.
A bit of this came up when we drew the state space
portrait of this beast, pretending that it had
no friction. Again, conservative systems can be chaotic
even though they don't have attractors.
The solar system, for instance, is a conservative
system, at least on human time scales. But,
remember that Pluto's orbit is chaotic.
We'll touch on so-called Hamiltonian chaos
at the very end of this course, but it is not
in our critical path yet. Okay, back to fixed points.
Recall the definition of a fixed point in a map. It's
a point that doesn't move under the influence of
the dynamics. The definition is the same in flows.
A point that doesn't move under the influence
of the dynamics. So how to think about that
properly? That's going to lead us to our first
encounter with the notion of a dynamical landscape;
a topography layered over the state space that is
defined by the dynamics of the system. We'll come
back to that idea again in unit 5 and formalize
it mathematically. In the mean time, we'll just draw.
Recall that the state variables of the pendulum are
theta and omega. Theta, the angle measured in radians.
And omega, the angular velocity measured in
radians per second. And the state space has those
variables as axes. Now, think about
the fixed point at the origin and what initial
conditions converge to it. You could imagine
starting the pendulum at a small positive angle
or a small negative angle, or a small velocity
of either sign. And the trajectories from all of
those initial conditions would converge to the
fixed point at the origin. So really, that fixed point
is in the bottom of some kind of bowl. It may not
be a hemispherical bowl, it may have some sort
of elliptical shape or something stranger,
but it's the kind of shape where a marble
will roll around the bowl and end up at the
bottom unless of course you push the
marble so hard that it hops out of the bowl.
That's an initial condition that's outside
the basin of attraction of that fixed point.
Now, the shape of the bowl is dictated by the
dynamics. Indeed, that shape
actually is the dynamics as we will see in
unit 5 when we do the math. That landscape
tells you for every value of the state variables
which way is downhill. A trajectory of a
dynamical system is a ball rolling on that
landscape defined by the dynamics. This
metaphor is useful, but really only in 2-D systems
that have 2 state variables, since you need
another axis to think about the landscape and
its slope. So an unstable fixed point is like
an upside-down bowl, but that's not the
only possible shape of the dynamical landscape
for an unstable fixed point. There's another kind of
topography of a landscape on which a marble
can balance, but will roll away from that point
if it's perturbed; and that's a saddle.
In the next segment, we are going to talk about more
about stability and saddle points.