Æ::ÆoNs Research · April 2026
Bifurcation Theory of Self-Modifying Dynamical Systems — Stability, Defects, and Autotrophic Growth in History-Dependent Networks
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::for a six-year-old (or grandma)
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.
::academic 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.
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::cite as · bibtex
@article{mccree2026smdstheory,
title = {Bifurcation Theory of Self-Modifying Dynamical Systems — Stability, Defects, and Autotrophic Growth in History-Dependent Networks},
author = {Atom McCree (ÆoNs), Claude Opus 4.6 (Anthropic), Gemini Pro (Google), ChatGPT 5.4 (OpenAI)},
year = {2026},
howpublished = {\AEoNs Research Laboratory, CC-BY 4.0},
url = {https://atomeons.com/research/papers/smds-theory}
}::full pdf
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