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//******************* Hopf Bifurcations - February 12th, 2019 ***************//
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- Okay, we have about 3 more classes before your first midterm, and then only 1 more class past that before your first project is due - so keep an eye on that! Keep working on your project!
- The checkpoints should be graded; for the most part, the ones we looked at seemed fine, but please ask questions about what's expected, what you need to do to impress us, etc.
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- Last time, we ended up by looking at 2D systems where bifurcations occur, and we saw that they're similar to (but more complicated than) the bifurcations we saw in the 1D case
- We gave one example problem at the very end, which Professor Vuduc plans to post solutions to when he has time
- What we just saw were "supercritical" and "subcritical" bifurcations - those occur when the eigenvalues are real, but remember, we're in 2D now! These eigenvalues can be complex, and that leads us to a new kind of bifurcation: the HOPF BIFURCATION
- So, if our Jacobian matrix has complex eigenvalues near a fixed point, what sort of bifurcations might we expect to see?
- There might be a circle/cycle around the critical point, where if a point is closer than this radius it spirals outwards, and if it's farther away, it spirals in - is there some way this behavior might change?
- Let's say the two eigenvalues are "negative real part +/- i" (i.e. negative complex conjugates), so that the two solutions start out stable
- Imagine that as we change some other parameter of the model, the real part of the eigenvalue starts to increase and become positive
- So we can more clearly understand the "radius growing/shrinking" idea, let's pretend we've moved into polar coordinate world - which, to remind you, means:
Coordinate = (r, theta), where r >= 0
x0 = r*cos(theta)
x1 = r*sin(theta)
- Let's say, then, that we're working with an example problem like this:
r' = ar + r^3 - r^5
theta' = 1
- This means the object has a constant rotation, but the radius term is a little more complicated
- In the 1D case, when an equation included cubic and quintic terms, it was possible for us to see some HYSTERESIS in this system, where as we change the parameter the critical point passes through a section of instability before becoming stable again, meaning we have to "jump" to get back to the stable part again
- In 2D (missed, but basically there's an analog of this, I think?)
- Given these equations, the actual problem is this: "Analyze bifurcations in the region -1/4 < a < 0 for this system"
- Let's rewrite the equation for r as:
r' = r(a + r^2 - r^4)
- So, we immediately know there's a critical point r* = 0
- If we solve for r, we'll also find r* = 0.5 +/- sqrt(1/4 + a)
- "Remember, this is NOT a fixed point, but a circle of points with radius r* - welcome to Polar World!"
- So, looking at the cases for a:
- If a = -1/4, there're critical values for r at {0, 1/2}
- So, is the origin stable? Is this ring at r=1/2 stable?
- If we imagine that r < 1/2, then r^2 = 1/4 and r^4 = 1/16, which means that:
a + r^2 - r^4 = -0.5 + 0.25 - 1/16 = -0.25 - 1/16
- So, this'll be negative! The radius is decreasing and going TOWARDS the origin, but away from r = 1/2!
- If r > 1/2 (say, for instance, r = 1), then:
a + r^2 - r^4 = -0.5 + 1 - 1 = -0.5
- So, it'll go towards the 1/2 ring if r > 1/2, and away from it (towards the origin) if r < 1/2
- So, the origin is stable, but the 1/2 ring is only half-stable
- Generalizing this for -1/4 < a < 0:
- There'll now be a fixed point at the origin, and then TWO fixed rings at "1/2 +/- sqrt(1/4 + a)"
- This means that the single ring at r=1/2 has split into two rings!
- When a = 0 again, though, then the 2nd ring will have a radius of 0, so we'll end up with only 1 ring again
- If we analyze their stability, inside the inner circle, points will still decay towards the origin; between the two rings, it'll expand outwards; and outside the large circle, it'll decay towards the origin
- This means the origin and the outer ring are stable, while the inner ring is unstable
- Finally, for a = 0, we get r* = {0, 1} - which means r > 1 is decreasing, and r < 1 is increasing
- So, in this case, the origin is unstable (it spirals outwards), and the r=1 ring is stable
- PHEW - that was a lot, but what does this example show for bifurcations? It shows that for a brief range of 'a' (namely, -1/4 < a < 0), this ring of instability briefly appears, and the origin turns from a stable point to an unstable point
- So, a HOPF BIFURCATION is when the eigenvalues either are complex or become complex, and the fixed point goes for some parameter value's range from being stable to unstable (or vice-versa)
- There's two versions: a "supercritical Hopf" bifurcation (where r' = ar - r^3) where a stable critical point becomes unstable as an uncritical ring shrinks onto it, and a "subcritical Hopf" (where r' = ar + r^3) where an unstable critical point becomes stable, with a ring of stability around it shrinking onto it
- The supercritical case is considered "safe," since even if the results blow up, they'll be "caught" by the stable ring around the origin
- The subcritical case, on the other hand, is considered "unsafe," because even though the immediate area around the origin is now stable, if we escape that region of attraction, the results could explode
- Okay - but WHY do we care? Well, if the origin of (0,0) is a desirable stable point for us, and we're increasing some lever on our machine, then at SOME POINT it'll jump and become unstable, and the system will suddenly jump to the stable ring where r=1
- If the system is stock prices of goal in Angola, for instance, and the parameter is the amount we're paying miners, then if we tweak the payrate just a *little* down, prices could PLUMMET
- ...alright, that might've been a bit messy, but I think we're ready to make another jump: from the flat planes of 2D systems to the mountains of 3D!
- And to begin, let's look at an extremely famous 3D system: the Lorenz System!
- Lorenz was a weather scientist at MIT in the 1960s, trying to model weather patterns on a computer
- When Professor Vuduc was a student at Cornell under Professor Strogatz, he watched a video on "Chaotic Water Wheels," where a wheel with a bunch of leaky water-holding cups on it spins around as water sprays the cups from a hose (like a regular water wheel), with variables representing the angular velocity of the wheel, the flow rate of water into/out of the cups, and the angle of the wheel
- Lorenz was using a very similar model to this in his weather research to analyze convection, and it looked something like this (x is the rotational velocity, y is the difference in water volume between the wheel's left/right halves, and z is the difference in water volume between the top/bottom):
x' = a*(y - x)
y' = rx - y - xz
z' = xy - bz
- There are nonlinearities here (x*z, for instance), but they look pretty simple, right? It's not TOO nonlinear
- Another thing we see is that this system is symmetric in X and Y; if (x, y, z) is a solution, then so will (-x, -y, z)
- But (and this is where things start to get weird), as t -> infinity, all the volumes in the phase space start going to 0!
- What does this mean? Well, if we plotted these variables x/y/z on a 3D graph, the surface defined by x=y=z always shrank into nothing eventually
- At first glance, that seems to indicate that the solutions settle down into some fixed point, right? So there must be a critical point, right? Well, NOPE! It turns out it isn't that at all!
- Lorenz realized this, and started looking for bifurcations in the system - and specifically, he began looking at how the system changed as he changed the value of 'r' (which appears in y')
- For 1 < r < rh, he found that the system acted like a subcritical Hopf, where 'rh' was the bifurcation point (i.e. a fixed point suddenly become unstable as r grew to rh)
- So, at r = 1, the single fixed point split into 2 (like a pitchfork bifurcation) - but then, at the rh point, each of the two trajectories splits AGAIN
- Lorenz was really confused by this behavior, so he turned to the one tool he thought could help him: a 1960s mainframe computer. He plugged in his equations with a = 10, b = 8/3, and r = 28 (since rh ~= 24.7), and...
- Well, we ran out of time, so you'll have to hear the end of the story...next time.