Find the equilibria of the difference equation and classify them as stable or unstable. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) xt + 1 = 1 − xt2

Question

Find the equilibria of the difference equation and classify them as stable or unstable. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)

x_{t + 1} = 1 − x_t²

Please show detailed steps and how to classify stable and unstable.

So far I have

Stable: DNE

Answer 1

Hi Joie T.,

If you're still following this query:

Stable means that successive x points, as t increases, approach a constant value. (there are some other kinds of stable, such as a set of repeated values ("Julia set"), but that's for a higher-level course than yours, I think, and more common on the complex number set). So if there IS a stable value(s), it means that essentially x(t+1) = x(t).

so plug that into the equation for propagation:

x(t+1) = x(t) = 1-x(t)^2 and solve, just use x:

x = 1 - x^2

x^2+x-1=0

x=(-1+-(1+4))/2 (the quadratic formula!)

That's two separate values: -(1/2)+5^0.5/2 and -(1/2)-5^0.5/2

Now, just b/c you solved the equation, doesn't mean you're done! You still NEED to check for stability -- these values might repeat for the exact value, but drift away for values NEAR the exact value (i.e. be unstable!). So calculate the exact value(s), then plug in values close to but both less than and greater than each exact value, and see if they converge towards the exact value.

you might also want to try plugging in some t=0 values for x that are far away from those possible stable points -- do they still converge towards one or the other of the values?

This is a neat problem to mine for mathematical insights!

One question you might explore is, are there separate regions for x that "feed" specific stable point(s)? If there are separate regions, where is the "dividing point" between the regions? Where is the "dividing point" between regions that "feed" a stable point vs. those that diverge away unstably (if such exist)?

What kinds of functions in a difference equation create stable points? What kinds of functions destroy stable points? Can you rank functions based on their power for creating or destroying a stable point (i.e. if a creating function is battling a destroying function, which is stronger)? Is that ranking of functions absolute across the entire number line for x, or can it be limited to only certain regions?

These are the kinds of questions that you might explore to make you really good at math. Suggestion, though: if you want to check for regions of stability, etc. set up an Excel spreadsheet on a PC. Don't sit and punch up calculations on a calculator, what a waste of time! If you don't yet know how to set up formulas or otherwise use Excel, find someone to show you, or get something online to do the equivalent tutorial. With Excel, it's easy to set up the difference equation calculations and observe results.

Hope this answer finds and helps you,

-- Cheers, -- Mr. d.