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.