The Thin-to-Thick Transition in Resonant Systems: A Transport-Geometric Framework for Operational Chaos Beyond Lyapunov Exponents

This paper introduces a transport-geometric framework that refines how we diagnose chaos in resonant Hamiltonian and periodically driven systems. Classical indicators—positive Lyapunov exponents or satisfaction of the Chirikov overlap criterion—flag the onset of local instability, but the paper argues they do not reliably signal operational failure.

The central innovation is the distinction between thin chaos and thick chaos:

• Thin chaos: Local exponential stretching occurs inside isolated stochastic layers or chaotic bands, yet surviving KAM tori, cantori, or spectral gaps confine transport. Action drift remains bounded by the narrow stochastic-layer width, so the system stays functionally coherent on operationally relevant timescales.
• Thick chaos: Resonance overlap, cantori breakdown, or attractor crisis connects formerly isolated chaotic regions into a transport-effective network. Trajectories can now reach or cross the failure scale, producing loss of confinement, mode-lock failure, or statistical (RRKM-like) behavior.

Key theoretical result: The Resonant Thinness Theorem proves that, in nearly integrable systems with finitely many isolated resonances whose Chirikov overlap parameter remains subcritical, any chaotic motion (positive finite-time Lyapunov exponents) is thin. Action drift stays parametrically smaller than any macroscopic failure scale up to Nekhoroshev timescales; the thinness ratio remains ≪ 1.

Applications demonstrate the framework’s sharpness:

• Trojan asteroids and Kordylewski dust clouds exhibit chaotic libration yet remain confined for astronomical times because their chaos is thin.
• Mode-locked lasers tolerate timing jitter (thin chaos) until modal overlap or detuning drift connects the locking basin to unlock or multi-pulsing regimes (thick chaos).
• Isolated Fermi resonances produce restricted IVR (thin); overlap enables facile, statistical energy flow (thick).

Practical payoff: Geometry-based diagnostics—resonance proximity, spectral-gap integrity, and connectivity of the resonance transport graph—detect the approach to the thin-to-thick transition earlier and more reliably than variance, autocorrelation, or Lyapunov magnitude alone. Control strategies shift from suppressing all chaos to thinness-preserving interventions that tolerate bounded local irregularity while preventing connectivity to failure boundaries.

In short, the paper’s strongest claim is that operational instability in resonant systems is a connectivity event, not an instability event. The thin-to-thick transition supplies a falsifiable, geometry-driven criterion that resolves long-standing paradoxes of “chaotic yet stable” resonant dynamics and yields sharper early-warning and design principles across astrodynamics, ultrafast optics, molecular control, and resonator engineering.