<b><i>The Mathematics of Thick and Thin Chaos: When Chaos Connects</i></b>

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>