What Are Phase Portaits and Phase Space in Chaos Theory?
When you were a student, you may remember different methods for graphing information in physics. We would assign the x-axis and the y-axis with certain units and plot data to gather insight into an experiment we were running. Typically, we like to look at how position, velocity, acceleration, and time in high school physics. But are there other possible methods for graphing, and one you may not have heard of is phase portraits of phase space. What is it, and how does it help scientists?
Phase space is a way to visualize dynamic systems that have complex movements to them. We like to have the x-axis be position and the y-axis be either momentum or velocity, for many physics applications. It gives us a way to extrapolate and predict future behavior of the changes in the system, typically represented as some differential equations. But by utilizing a phase diagram, or a graph in phase space, we can observe the motion and perhaps see a potential solution by mapping out all possible paths on a single diagram (Parker 59-60, Millis).
To see phase space in action, a great example to examine is a pendulum. When you plot the time versus position, you get a sinusoidal graph, showing the back and forth motion as amplitude goes up and down. But in phase space, the story is different. So long as we are dealing with a simple harmonic oscillator (our angle of displacement is rather small) pendulum, aka idealized, we can get a cool pattern. With position as the x-axis and velocity as the y-axis, we start as a point on the positive x-axis, for the velocity is zero and position is a maximum. But once we let the pendulum down, it eventually makes to max velocity in the negative direction, so we have a point on the negative y-axis. If we keep proceeding in this fashion, we eventually arrive back where we started. We made a trip around a circle in a clockwise direction! Now that is an interesting pattern, and we call that line a trajectory and the direction it goes the flow. If our trajectory is closed, like with our idealized pendulum, we call it an orbit (Parker 61-5, Millis).
Now, this was an idealized pendulum. What if I increase the amplitude? We would get an orbit with a bigger radius. And if we graph many different trajectories of a system, we end up with a phase portrait. And if we are getting real technical, we know the amplitude decreases with each successive swing because of energy loss. This would be a dissipative system, and its trajectory would be a spiral going towards the origin. But even all of this is still too clean, for many factors impact the amplitude of a pendulum (Parker 65-7).
If we kept increasing the amplitude of the pendulum, we would eventually reveal some nonlinear behavior. That is what phase diagrams were designed to help with, because they are a doozy to solve analytically. And more nonlinear systems were being uncovered as science progressed, until their presence demanded attention. So, let’s go back to the pendulum. How does it really work? (67-8)
As the pendulum’s amplitude grows, our trajectory goes from a circle to an ellipse. And if the amplitude gets large enough, the bob goes completely around and our trajectory does something odd – the ellipses seem to grow in size and then break and form horizontal asymptotes. Our trajectories are no longer orbits, for they are open at the ends. On top of that, we can start to change the flow, going clockwise or counterclockwise. On top of that, trajectories start to cross over each other are called separatrices and they indicate where we change from types of motion, in this case the change between a simple harmonic oscillator and the continuous motion (69-71).
But wait, there’s more! Turns out, this was all for a forced pendulum, where we offset any energy losses. We haven’t even begun to talk about the dampened case, which has many tough aspects to it. But the message is the same: our example was a good starting point for getting familiar with phase portraits. But something remains to be pointed out. If you took that phase portrait and wrapped it as a cylinder, the edges line up so that the separatrices line up, showing how the position is actually the same and the oscillatory behavior is maintained (71-2).
Like other mathematical constructs, phase space has dimensionality to it. That dimension required to visualize the behavior of the object is given by the equation D=2σs, where σ is the number of objects and s is the space they exist in our reality. So, for a pendulum, we have one object moving along a line of one dimension (from its viewpoint), so we need 2D phase space to see this (73).
When we have a trajectory that flows to the center no matter the starting position, we have a sink which demonstrates that as our amplitude decreases, so does our velocity and in many cases a sink shows the system returning to its rest state. If instead we always flow away from the center, we have a source. While sinks are a sign of stability in our system, sources are definitely not because any change in our position changes how we are moving from the center. Anytime we have a sink and a source cross over each other, we have a saddle point, an equilibrium position, and the trajectories that did the crossing over are known as saddles or as separatrix (Parker 74-76, Cerfon).
Another important topic for trajectories is any bifurcation that may occur. This is a matter of when a system goes from stable motion to unstable, much like the difference between balancing on the top of a hill versus the valley below. One can cause a big problem if we fall, but the other doesn’t. That transition between the two states is known as the bifurcation point (Parker 80).
An attractor, however, looks like a sink but doesn’t have to converge to the center but instead can have many different locations. The main types are fixed point attractors aka sinks of any location, limit cycles, and torus’s. In a limit cycle, we have a trajectory that falls into an orbit after a portion of flow has passed by, therefore closing off the trajectory. It might not start off well but it will eventually settle down. A torus is a superposition of limit cycles, giving two different period values. One is for the larger orbit while the other is for the smaller one. We call this quasiperiodic motion when the ratio of the orbits is not an integer. One shouldn’t get back to their original position but the motions are repetitive (77-9).
Not all attractors result in chaos, but strange ones do. Strange attractors are a “simple set of differential equations” in which the trajectory converges towards it. They also depend on initial conditions and have fractal patterns. But the strangest thing about them is their “contradictory effects.” Attractors are meant to have trajectories converge, but in this case a different set of initial conditions can lead to a different trajectory. As for the dimension of strange attractors, that can be tough because trajectories don’t cross over, despite how the portrait appears. If they did then we would have choices and the initial conditions would not be so particular to the portrait. We need a dimension larger than 2 if we want to prevent this. But with these dissipative systems and initial conditions, we cannot have a dimension larger than 3. Therefore, strange attractors have a dimension between 2 and 3, therefore not an integer. Its fractal! (96-8)
Now, with all that established, read the next article on my profile to see how phase space plays its role in chaos theory.
Cerfon, Antoine. “Lecture 7.” Math.nyu. New York University. Web. 07 Jun. 2018.
Miler, Andrew. “Physics W3003: Phase Space.” Phys.columbia.edu. Columbia University. Web. 07 Jun. 2018.
Parker, Barry. Chaos in the Cosmos. Plenum Press, New York. 1996. Print. 59-80, 96-8.
© 2018 Leonard Kelley