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oai:figshare.com:article/34029486 2026-09-30T04:37:06Z category_29803 item_type_13 month_year_09_2026

<b><i>The Mathematics of Thick and Thin Chaos: When Chaos Connects</i></b> Doug Doucette (23673087) Mathematical physics not elsewhere classified thin chaos; thick chaos; transport connectivity; Lyapunov exponents; operational stability. <p dir="ltr"><i>The Mathematics of Thick and Thin Chaos</i> by Doug Doucette addresses a diagnostic ambiguity in nonlinear dynamics: a system can be chaotic yet remain functionally stable, while another can fail without a dramatic increase in local instability. The book argues that classical chaos diagnostics, especially positive Lyapunov exponents, establish only local sensitivity: nearby trajectories separate. They do not reveal whether those trajectories can escape a coherent region, cross barriers, diffuse through action space, or reach a failure boundary. To correct this, Doucette distinguishes thin chaos from thick chaos. Thin chaos is genuine local instability whose transport consequences remain confined; point prediction fails, but operational coherence survives. Thick chaos occurs when local instability becomes transport-effective, connecting formerly isolated chaotic regions to pathways that reach failure. The transition is framed as a connectivity transition, not merely a rise in chaos strength. Resonance overlap, barrier leakage, spectral-gap collapse, percolation, and transport graphs all describe how thin chaos can thicken. The thinness ratio compares chaotic transport over a chosen timescale with the displacement required for failure, making classification relative to function, timescale, and transport variable. The book develops definitions, propositions, measurement protocols, and a classification matrix. It explores implications for Hamiltonian systems, celestial mechanics, dissipative attractors, turbulence, quantum information spreading, prediction, and control. A final applied chapter operationalizes the framework for a MEMS resonator, showing how to measure local instability, define the coherent region, choose an amplitude transport variable, estimate chaotic transport, set the failure scale, compute the ratio, and decide whether the device is robustly thin, transitional, or thick. The conclusion captures the central image: thin chaos is a wrinkle, thick chaos is a tear. Chaos becomes dangerous when it connects. Stability can mean containment of chaos rather than its absence, and control may aim to keep chaos thin.</p> 2026-09-30T04:37:06Z Text Book 10.6084/m9.figshare.34029486.v1 https://figshare.com/articles/book/_b_i_The_Mathematics_of_Thick_and_Thin_Chaos_When_Chaos_Connects_i_b_/34029486 CC BY 4.0