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