3. Consider the logistic difference equation, a. we constrain x to the interval [0, 1]. well-defined...

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3. Consider the logistic difference equation, a. we constrain x to the interval [0, 1]. well-defined (i.e. x = [0, 1] for all i > 0). xn+1 = axn (1xn) Determine amin and amax for which the difference equation is b. For a difference equation xn+1 = f(x), an equilibrium point x* satisfies x* = equilibria of the logistic difference equation. f(x). Determine all c. For a difference equation xn+1 = f(xn), an equilibrium point x* is asymptotically stable if | f'(x*)| < 1. Furthermore, if 0 < f'(x*) < 1, the system approaches the equilibrium monotonically; if -1 < f'(x*) < 0, the system approaches the equilibrium oscillatory. Analyze in detail teh stability of each equilibrium for our difference equation. d. Find the value a2 [amin, amax] after which 2-period oscillations begin. Plot the position of 2 period oscillations for a [a2, amax]. Hint: f exhibits n period oscillations at x if x = f(x) and x f(x) for 0 3. Consider the logistic difference equation, a. we constrain x to the interval [0, 1]. well-defined (i.e. x = [0, 1] for all i > 0). xn+1 = axn (1xn) Determine amin and amax for which the difference equation is b. For a difference equation xn+1 = f(x), an equilibrium point x* satisfies x* = equilibria of the logistic difference equation. f(x). Determine all c. For a difference equation xn+1 = f(xn), an equilibrium point x* is asymptotically stable if | f'(x*)| < 1. Furthermore, if 0 < f'(x*) < 1, the system approaches the equilibrium monotonically; if -1 < f'(x*) < 0, the system approaches the equilibrium oscillatory. Analyze in detail teh stability of each equilibrium for our difference equation. d. Find the value a2 [amin, amax] after which 2-period oscillations begin. Plot the position of 2 period oscillations for a [a2, amax]. Hint: f exhibits n period oscillations at x if x = f(x) and x f(x) for 0