Consider the logistic map \( x_{n+1} = r x_n (1 - x_n) \) for \(0 \leq x_n \leq 1\) and \(0 \leq r \leq 4\). Find all the fixed points and determine their stability
[Sol] The fixed points satisfy \( x^* = f(x^*) = r x^* (1 - x^*) \) Hence \(x^* = 0\) or \( 1 = r(1 - x^*), \text{ i.e., } x^* = 1 - \frac{1}{r}. \) The origin is a fixed point for all \(r\), whereas \(x^* = 1 - \frac{1}{r}\) is in the range of allowable \(x\) only if \(r \geq 1\)
Stability depends on the multiplier \( f'(x^*) = r - 2rx^*. \) Since \(f'(0) = r\), the origin is stable for \(r < 1\) and unstable for \(r > 1\). At the other fixed point, \( f'(x^*) = r - 2r(1 - \frac{1}{r}) = 2 - r. \) Hence \(x^* = 1 - \frac{1}{r}\) is stable for \(-1 < (2 - r) < 1\), i.e., for \(1 < r < 3\). It is unstable for \(r > 3\)