The contribution of degenerate tori: The study of strong Kolmogorov–Arnold–Moser stability and barrier development

We examine the degenerate tori and their stability analysis in this study. These tori are crucial to the development of robust transport barriers in Hamiltonian systems. Applications in fields including atmospheric dynamics (such as the ozone hole problem), plasma physics, and celestial mechanics—where Hamiltonian systems with degenerate tori [violating the twist requirement ω(P)≠0] are frequent—what drives this research. This work presents an original technique to investigate the identification of transport barriers and chaotic structures in periodic and quasiperiodic flows. Unlike transient recurring flows, which possess exact forward and backward time reversibility, real-world flows often lack this ideal property. By leveraging the precise characterization of transport barriers in periodic flows as material surfaces generated by trajectories from invariant sets in Poincaré maps, we explore fluid transport and chaotic structures in two-dimensional incompressible flow using Kolmogorov–Arnold–Moser (KAM) theory. Furthermore, we analyze invariant tori within the flow and examine the bounded domain of the flow in finite time intervals with multi-periodic time dependence. Additionally, we illustrate strong KAM stability phenomena near degenerate tori in Hamiltonian systems. To validate our findings, we conduct an in-depth analysis of the Arnold–Beltrami–Childress (ABC) flow, a canonical example of transport and mixing phenomena.