holomorphic-dynamics

Holomorphic Dynamics Educational Pack

Overview

This educational pack teaches holomorphic dynamics from first principles through ten progressive modules. Starting with iteration on the complex plane (HD-01) and building through fixed points, fractal sets, topology, and deep learning connections, the pack culminates in data-driven dynamics with Dynamic Mode Decomposition (HD-09) and Koopman operator theory (HD-10).

The pack is self-contained: all complex arithmetic, iteration engines, fractal renderers, and DMD algorithms are implemented from scratch with no external math libraries. Every algorithm is transparent and educational rather than optimized for production performance.

What you will learn:

• How iteration of simple functions on the complex plane produces fractal

structures of extraordinary complexity

• The classification of fixed points by their multiplier and what each

classification means dynamically

• Why the Mandelbrot set is the "atlas" of all quadratic Julia sets
• The Fatou-Julia dichotomy: stable domains versus chaotic boundaries
• How period doubling cascades lead to chaos through Feigenbaum universality
• The topological properties that constrain what dynamical systems can do
• The surprising bridge between holomorphic dynamics and deep learning
• How skill-creator itself can be modeled as a dynamical system on the

complex plane, with skills as orbits converging to fixed points

• Data-driven dynamics: extracting coherent spatiotemporal modes from

snapshot data using DMD and its variants

• The Koopman operator: lifting nonlinear dynamics into an infinite-

dimensional linear framework where spectral analysis applies

Who this is for:

Developers, mathematicians, and curious minds who want to understand the mathematics behind fractals, chaos, and data-driven modeling. No prior knowledge of complex analysis is assumed; each module builds on the previous one.

Quick Start

Import any function directly from the holomorphic barrel:

import {
  computeOrbit, renderMandelbrot, classifyFixedPoint,
  dmd, classifyDMDEigenvalue, bridgeDMDToSkillDynamics,
} from '../src/holomorphic';

Each module in src/holomorphic/modules/HD-XX/ contains:

• content.md -- Educational text explaining the mathematics
• try-session.ts -- Interactive TypeScript code to experiment with

(some modules also have try-session.py for Python/PyDMD examples)

To run a try-session, import and call its exported function:

import { runTrySession } from '../src/holomorphic/modules/HD-01/try-session';
runTrySession();

Module Guide

HD-01: Iteration on the Complex Plane

The foundation. Defines orbits of the quadratic map f(z) = z^2 + c, escape radius, escape time, and the four fundamental orbit behaviors (converging, periodic, chaotic, escaping). Introduces the computational engine behind all of holomorphic dynamics.

Path: src/holomorphic/modules/HD-01/

HD-02: Fixed Points and Stability

Analyzes what happens when orbits converge. Classifies fixed points by their multiplier lambda: superattracting (|lambda| = 0), attracting (|lambda| < 1), indifferent (|lambda| = 1), and repelling (|lambda| > 1). Covers the linearization theorem and basins of attraction.

Path: src/holomorphic/modules/HD-02/

HD-03: The Mandelbrot Set

The parameter space of the quadratic family. Defines the Mandelbrot set M as the set of c-values for which the critical orbit remains bounded. Covers the cardioid and period bulbs, the relationship between M and Julia sets, and escape-time rendering algorithms.

Path: src/holomorphic/modules/HD-03/

HD-04: Julia Sets and Fatou Sets

The dynamical plane partition. For each c, the Julia set J(f) is the boundary between chaos and stability, while the Fatou set F(f) is the complement of stable, predictable behavior. Covers the Fatou-Julia dichotomy, connected versus Cantor dust Julia sets, and the relationship between the Mandelbrot set and Julia set topology.

Path: src/holomorphic/modules/HD-04/

HD-05: Cycles and Period Doubling

Periodic orbits and the route to chaos. Covers period-n cycles, the period-doubling cascade, Feigenbaum's universal constant (delta = 4.669...), and bifurcation diagrams. Shows how simple parameter changes drive a system from order through period doubling into chaos.

Path: src/holomorphic/modules/HD-05/

HD-06: Topology of the Complex Plane

The geometric and topological properties that constrain dynamics. Covers connectedness, simple connectedness, the Riemann sphere, conformal maps, and how topology determines the possible behaviors of holomorphic maps. Includes references to Meyerson, Greene-Lobb, and the MAT327 course.

Path: src/holomorphic/modules/HD-06/

HD-07: From Dynamics to Deep Learning

The bridge between holomorphic dynamics and neural networks. Shows how deep learning can be viewed as iterated function composition, how activation functions relate to holomorphic maps, and how concepts like fixed points, stability, and bifurcation appear in the training dynamics of neural networks.

Path: src/holomorphic/modules/HD-07/

HD-08: Skill-Creator as a Dynamical System

Maps the skill-creator system onto the complex plane. Skills are modeled as points z = r * e^(i*theta) where r is distance from mastery and theta encodes the skill domain. Skill iteration is a contractive affine map f(z) = alpha*z + beta. Classifies skill dynamics using the same fixed-point taxonomy from HD-02: superattracting (compiled skills), attracting (converging skills), indifferent (stalled skills), and repelling (degrading skills). The Fatou-Julia classification separates stable skill domains from chaotic boundary regions.

Path: src/holomorphic/modules/HD-08/

HD-09: Dynamic Mode Decomposition

Data-driven dynamics extraction. DMD takes snapshot matrices of observed system states and extracts spatiotemporal modes with associated eigenvalues (growth/decay rates and frequencies). Covers the standard DMD algorithm with SVD, eigenvalue classification on the unit circle, and four variants: DMDc (control inputs), mrDMD (multi-resolution time-scale separation), piDMD (physics-informed constraints), and BOP-DMD (robust bootstrap optimization). Includes Python try-session using PyDMD.

Path: src/holomorphic/modules/HD-09/

HD-10: Koopman Operator Theory

The theoretical foundation for data-driven dynamics. The Koopman operator K acts on observable functions rather than state vectors, lifting nonlinear dynamics into an infinite-dimensional linear space where spectral decomposition applies. Covers Extended DMD (EDMD) as a finite-dimensional approximation to K, dictionary-based lifting with polynomial and RBF observables, and the bridge from DMD eigenvalues to skill dynamics classifications. Includes Python try-session.

Path: src/holomorphic/modules/HD-10/

Core API

Complex Arithmetic (complex/arithmetic.ts)

| Function | Signature | Description | |----------|-----------|-------------| | add | (a, b) => ComplexNumber | Complex addition | | sub | (a, b) => ComplexNumber | Complex subtraction | | mul | (a, b) => ComplexNumber | Complex multiplication | | div | (a, b) => ComplexNumber | Complex division | | magnitude | (z) => number | Absolute value \|z\| | | argument | (z) => number | Phase angle arg(z) | | conjugate | (z) => ComplexNumber | Complex conjugate | | cexp | (z) => ComplexNumber | Complex exponential e^z | | cpow | (z, n) => ComplexNumber | Complex power z^n |

Constants: ZERO, ONE, I

Iteration Engine (complex/iterate.ts)

| Function | Description | |----------|-------------| | computeOrbit | Iterate f from z_0, collecting orbit points until escape or max iterations | | detectPeriod | Detect periodicity in an orbit using Floyd's algorithm | | computeMultiplier | Compute the derivative multiplier at a fixed point