Comparative Analysis of Topology-Dependent Chaos Compression in Lorenz-Hala and Sprott A-Hala

This technical note documents the contrasting discretization behaviors of two butterflyshaped chaotic attractors when subjected to the Hala operator’s successive controlled collapse protocol. We demonstrate that the presence or absence of equilibrium points fundamentally determines whether the Hala operator compresses a 3D chaotic attractor into discrete 0D fixed points (Lorenz case) or a continuous 1D limit cycle (Sprott A case). This topology-dependent behavior establishes the Hala operator as a universal yet adaptive chaos control mechanism.