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//************* Stability and Bifurcations - January 17th, 2019 *************//
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- ...welp, it appears only 3 people joined me at the front, so I'm directly in the line of fire. This terrifies me.
- *squirms*
- "While the projector is warming up, someone mentioned I made a math mistake when modeling the logistic curve last lecture - I've posted some stuff correcting that on Piazza"
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- Okay, last time we talked about the logistic curve to wrap up the day, and classifying its stable/unstable points
- "If you're looking for ways to practice this stuff, look either in the Sayama book, or the questions I and the TAs hope to post shortly"
- As we mentioned, though, this logistic model seems to not work for organisms more complicated than bacteria...so what's this model missing?
- Well, animal populations tend to look a little more oscillatory - they go up and down over time
- But the logistic curve's derivative can't produce oscillations - we need additional state variables so that x can go back and forth over time! (I think?)
- Our analysis yesterday was largely based on the phase portrait, but there's another mode of analysis called LINEAR STABILITY ANALYSIS
- Basically, this is the quickie derivation for the "negative stable/positive unstable/zero idunno" thing from Differential Equations
- This is in the Sayama book (I believe in Chapter 7?), but we'll do a quick overview here
- Consider a system like this:
x' = f(x)
x(t) = x* + n(t), where n(t) is "small"
- Therefore (since derivative of a constant value x* is 0),
x'(t) = n'(t) = f(x* + n) (i.e., their derivatives are the same!)
- So, since we have the equation in terms of "f(x* + something)," let's choose n = some-really-small-away-from-x* something, and then we can do a Taylor expansion to try and understand it better:
= f(x*) + n*f'(x*) + O(n^2) + O(n^3) + ...
- "Since we know n is a super small value we chose away from our base point, let's pretend that all the squared terms are small enough to ignore"
~= f(x*) + nf'(x*)
- By definition of critical points,
~= 0 + nf'(x*) = nf'(x*)
- HOWEVER, we said this was equals to n'(t) AS WELL, so we have:
n'(t) = n*f'(x*)
- "What function is equal to its derivative? The exponential!" SO, we know
n(t) = e^(f'(x*) * t)
- SO, based on this interpretation, we can see how the derivative changes based on the value of f'(x*):
- If it's < 0, the function is decaying
- If it's > 0, the function is growing
- If it's 0, we're...uncertain
- ...basically, we're taking a *little* step to the left or right of our point and seeing if the stability of the function changes as a result
- As a more concrete example, let's say we have:
x' = sin(x) = f(x)
- So, we know sin(x) = 0 at x* = k*pi
- Furthermore, f'(x) = cos(x)
- (...might need to read the book for this, wasn't 100% sure where he was going unless he just meant that the value of the derivative)
- So, if we have a fixed point we've identified, and we've done this Taylor analysis, then we can hopefully determine the stability of the point!
- If this were a math class, we'd have to talk about if solutions exist/if those solutions are unique or not...but for the most part, we'll just assume that solutions do actually exist in this class
- i.e. just because we computed one solution, there could be others
- One useful theory thing to know: the "Sufficient Condition for Uniqueness" theorem:
- If we're given an IVP (i.e. x(0) = x0, x'=f(x)), and both f(x) and f'(x) are continuous in the interval (a,b), then there exists a time interval (-t, +t) about t=0 where x(t) has a unique solution
- So, we can figure out if a solution definitely exists, but the range for that solution might be very small
- e.g. x' = 1 + x^2, x0 = 0
- The solution is x(t) = tan(t), but it only exists in the range (-pi/2, pi/2) - past those asymptotes, the function blows up!
- "Again, we won't deal with this stuff very much in our class, but it is a concern for resarchers"
- Alright, let's now talk about bifurcations
- Here's a question we have: as the parameters of a given model change, how does its stability change?
- For instance, if we have a block that's perfectly balanced on a beam, and we place larger and larger blocks on it, what'll happen? Well, eventually it'll start to bend, and possibly even break!
- So, it was stable while it was straight, but changing the mass eventually made it (mathematically and literally) unstable
x = shape of the beam
r = mass of block
- In particular, we're interested in SADDLE-NODE BIFURCATIONS
- Consider the diff. eq. x' = r + x^2
- At r = 0, this equation is just a concave-up parabola with one critical point
- By our stability analysis stuff, this is "half stable;" the derivative is pointing towards the origin on the negative side and away from it on the positive
- If r < 0, the parabola will move DOWN, an we'll now have 2 critical points (one stable and one unstable)
- If r > 0, then the parabola will be above the axis, so there'll be NO critical points and therefore no stable ones!
- So, if we graph the critical points (x*) on the y-axis and the mass (r) on the x-axis, we'd end up with a sideways parabola open toward the negative side (w/ the negative points stable, and the positive ones unstable)
- This shows how the critical points change in response to changes in other parameters, and which of the points are stable/unstable
- This particular case, where there are 2 critical points, looks vaguely like a saddle, and is thus known as a saddle bifurcation
- "To be specific, a SADDLE NODE BIFURCATION is the point where 2 critical points merge into 1 for some value of r, as r changes"
- In general, "bowl-shaped" functions like parabolas tend to create these points
- As an example, try doing the x*/r plots for "x' = r - x^2"
- If you do an LSA for this, what'll you find? Well, you'll find that
x* = +- sqrt(r)
- Which'll mean that the critical points should only exist where r is positive - which makes sense!
- Another one to try: x' = r - x - e^(-x)
- "This looks complicated, since it's not easy to find the zeroes, but geometrically we could try drawing the equivalent r - x = e^(-x) and find roughly where they intersect"
- From there, we can conclude that there are 2 fixed/critical points when r > 1, exactly 1 point when r = 1 (when the "r-x" line is tangent to e^-x, i.e. their derivatives are equal), and none when r < 1
- In this example, r=1 is the "critical r" where the bifuraction occurs
- "If you do a Taylor expansion of e^-x, you should see some x^2-like 'bowlish' functions begin appearing, and that's partly why it has a saddle bifurcation"
- There's a whole system of theory called "normal forms" that defines what bifurcations can appear for different types of curves, and x^2 is the demesne of the saddle
- *20 minutes left*..."Hmmm, should I do the complicated one or the easy one..."
- Also, scary notification noise! "New assignment: for the next week of your life, just count how many times something beeps"
- (ended up doing more examples)
- So, we'll probably spend at least 1 more class going over analyzing this nonlinear ODE dynamics stuff before we start getting into discrete modeling
- "We're...um...some sports analogy away from getting the random teams out to you guys, so be on the lookout for those sometime this weekend."
- In the meantime, have a good weekend, see you Tuesday, and hopefully I didn't hurt your brain too much!