mfe-emergencelisted

# Emergence Part IX: Growing — Chapters 28, 29, 30, 31 — Plane Position: (0.5, 0) radius 0.4 — 36 Primitives ## Workflow 1. **Identify the dynamical system** — determine whether it is discrete (logistic map, cellular automaton) or continuous (ODE system) 2. **Compute the Lyapunov exponent** to classify behavior: λ > 0 indicates chaos, λ < 0 indicates convergence to periodic orbit 3. **Analyze bifurcations** by varying parameters — classify as saddle-node, pitchfork, Hopf, or period-doubling 4. **Measure fractal dimension** for strange attractors using d = log(N)/log(1/r) for self-similar structures 5. **Estimate prediction horizon** using t_predict ≈ (1/λ) × ln(Δ/δ₀) to quantify how far ahead the system remains predictable ## Key Concepts **Neural Network** (definition): An artificial neural network is a computational graph: y = f_L(W_L * f_{L-1}(... f_1(W_1 * x + b_1) ...+ b_L)), where W_i are weight matrices, b_i are bias vectors, and f_i are nonlinear activation functions. A perceptron is the single-layer case: y = sigma(w^T x + b). - Approximating complex input-output mappings from data - Pattern recognition in images, text, and audio - Building flexible function approximators for regression and classification **Logistic Map** (definition): The logistic map is the discrete dynamical system x_{n+1} = r * x_n * (1 - x_n), where x_n in [0,1] and r in [0,4]. It exhibits period doubling, bifurcations, and chaos as r increases, serving as the canonical example of de