1

1. Determine the equilibria for each of the following and classify them as stable/unstable and oscillating/non-oscillating. If stability or oscillation can't be determined, say so. 
a) xn+1 = xn+ 10x„ + 20. b) xr,±1 = x2n — 5x„ + 8. 
2. For xn+1 = 10xn -I- 20, a cobweb diagram is given below with xo = —6.75. Does the solution converge to one of the equilbria you found in 1(a)? If it does converge, explain why. If it doesn't, explain how you could change xo to have a better chance to converge to an x* 

3. Draw two separate cobweb diagrams for xn±i = —1xn + 5 with the fol-lowing initial conditions. What does this seem to imply for the stability of the equilibrium? a) xo = 0, and b) xo = 8. 
4. For xn+1 = 4 — xn, sketch a cobweb diagram until you get to x3, with xo = 1. Where does the solution look like it's tending towards as n —> oo? 
5. Consider the system of difference equations modeling Romeo and Juliet's love for each other: 
xn-F1 = xn — ayn Yn+1 = yn + bxn 
Here, xn is Romeo's love for Juliet after n days, and yn is Juliet's love for Romeo. a) Write an Excel (Sheets) code that numerically solve the system of dif-ference equations up to n = 70 with xo = 1, yo = 0, a = .5, and b = .05. Note, you should have 3 columns, n, xn, and yn, with parameters xo, Yo, a, and b. 
b) Plot the results with n on the x-axis and xn and yn on the y-axis. c) What type of functions' graphs do the solution curves remind you of? d) Does the solution seem to represent love from lyrics below? 
Cause you're hot then you're cold You're yes then you're no You're in then you're out You're up then you're down