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arXiv · 2606.22322

Singular barriers and quartic integrability breaking in the TTW system

Adrian M. Escobar Ruiz·Miguel E. Gómez Quintanar·Lidia Jiménez-Lara·Jaume Llibre
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Abstract

We study the competition between resonance-induced integrability breaking and centrifugal confinement in a symmetric quartic deformation of the classical $k=1$ Tremblay--Turbiner--Winternitz (TTW) system, $H=\frac{1}{2}\left(P_X^2+P_Y^2\right)+X^2+Y^2 +\frac{\gamma}{X^2}+\frac{\gamma}{Y^2}+\kappa X^2Y^2.$ The inverse-square terms are the centrifugal remnants of an $SO(2)\times SO(2)$ reduction of a four-dimensional oscillator, while the $X^2Y^2$ interaction couples the reduced radial modes and connects the Smorodinsky--Winternitz and Contopoulos limits. For $\gamma>0$, the axes are impenetrable and split configuration space into invariant sectors. For sufficiently small nonzero $\kappa$, resonant averaging constructs a phase-locked periodic orbit on each energy surface $H=h>4\sqrt{\gamma}$ whose fixed-energy reduced Poincar\'e map has no unit characteristic multiplier. Poincar\'e's criterion therefore excludes a second independent $C^1$ first integral in any invariant neighbourhood of this orbit. At finite coupling, Poincar\'e sections and finite-time Lyapunov maps show the breakup of invariant curves and the growth of chaotic layers. Comparison with the barrier-free limit separates two effects: the quartic interaction breaks integrability, whereas the singular barriers reduce phase-space connectivity and chaotic transport without restoring it.

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Adrian M. Escobar Ruiz, Miguel E. Gómez Quintanar, Lidia Jiménez-Lara, Jaume Llibre. 2026-06-21. Singular barriers and quartic integrability breaking in the TTW system. https://arxiv.org/abs/2606.22322

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