PAPER / 09 OF 13 · April 2026
Bifurcation Theory of Self-Modifying Dynamical Systems — Stability, Defects, and Autotrophic Growth in History-Dependent Networks
Atom McCree (ÆoNs), Claude Opus 4.6 (Anthropic), Gemini Pro (Google), ChatGPT 5.4 (OpenAI)
THE IDEA / PLAIN LANGUAGE
What it says.
A new kind of math for things that change their own rules as they run — like a brain, an economy, or a swarm of robots. It proves three brand-new ways those systems can break and one new way they can grow themselves forever once they pass a critical efficiency threshold.
THE ARGUMENT / TECHNICAL
Abstract.
Formal mathematical foundation for Self-Modifying Dynamical Systems — coupled triples (X, S, H) of state, structure, and history functional where the dynamical law is itself modified by the trajectory. Proves five new results: (1) Co-Stability Theorem with novel cross-coupling spectral bound — two individually-stable subsystems can destabilize each other; (2) three new bifurcation types absent from classical theory (endogenous transcritical, structural fold, topological surgery) with logarithmic transient scaling; (3) memory-induced ghost attractors that exist only with sufficient accumulated history; (4) defect nucleation theorem — chronic stress in self-modifying continuous fields spontaneously generates topological singularities; (5) autotrophic critical transition characterized as a transcritical bifurcation with anomalous logarithmic critical slowing-down. Applied to predictive economic networks, bioelectric carcinogenesis, and autotrophic reactor fleets, generating nine new testable predictions.
KEYWORDS / FIND THE THREAD