This paper introduces a generalized logistic map with memory and feedback control, extending the classical logistic map to include a memory term and a threshold mechanism. The recurrence x_{n+1} = x_n + i·(x_n − x_{n-1})·k·(1 − x_n) produces periodic, chaotic, and stabilized orbits depending on the parameters i and k. When extended to two dimensions with scaling factor 1/φ (where φ is the golden ratio) and rotations by multiples of π/5, the system generates strange attractors with pentagonal symmetry. The work connects Fibonacci numbers, discrete scale invariance, and the geometry of quasicrystals. A Python implementation is provided.