Diffeomorphic Connectivity of Sprott A-Hala and Rössler-Hala Systems: A Framework for Chaos Compression and Switching Logic

This paper provides a rigorous proof for the existence of a C1 diffeomorphism connecting the Sprott A-Hala and Rössler-Hala dynamical systems. By utilizing the Hala Operator (γ) as a universal chaos compressor, we demonstrate a "Successive Controlled Collapse" (SCC) that maps high-dimensional chaotic flows onto lower-dimensional manifolds. We establish that while the Sprott A system acts as a non-equilibrium "Clock" (1D limit cycle) and the Rössler system as a "Digital Switch" (0D logic sink), they are topologically conjugate under specific feedback intensities. This framework provides the mathematical foundation for secure digital communication and plasma diagnostics by treating chaos as a programmable medium.