Question: 3 (20 points) Consider the following system. 1= (1-r - x) (r -e) . (a) Draw a bifurcation diagram (r, r*), labeling the stability of

3 (20 points) Consider the following system. 1= (1-r - x) (r -e) . (a) Draw a bifurcation diagram (r, r*), labeling the stability of each fixed point. Ignore the borderline case(s): r = rc. Hint: This problem requires a linear stability analysis. When calculating f'(x), don't distribute, it will simplify when you substitute in fixed points. Check when f'(x) 0; taking log preserves inequality for 1 set of the fixed pts; the other set of fixed points only erists when r > 0, so that simplifies the inequality. (b) What type of bifurcation do you observe, and for what value(s) of r, does this occur? (c) For r = 0, completely describe what happens as lim r(t) for all z(0) ER. (d) Can hysteresis occur in this system? Explain why or why not

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