Nonlinear Feedback Regulation of Hidden Attractors: The Hala Operator as a Deterministic Chaos Compressor Sprott A Dynamics

This paper investigates the application of the Hala operator—a state-dependent nonlinear feedback mechanism (z˙ → z˙ − γ(x2 + y2)z)—to the Sprott Case A attractor system, a foundational volume-preserving chaotic flow. While the original Sprott-A system is characterized by a maximal Lyapunov dimension (DL = 3.0) and a lack of equilibrium points, we demonstrate that the introduction of the Hala operator induces a tunable transition from conservative chaos to dissipative order. Computational analysis reveals that the operator acts as a discrete “deterministic chaos compressor,” collapsing the system’s dimensionality from a volume-filling state to a stable one-dimensional limit cycle. This operator can function as a physical thermostat for the Nos´e-Hoover oscillator, for example, so we explored its utility as a cooling mechanism in statistical mechanics. Further, we implement a “successive controlled collapse” protocol, showing that by periodically switching the control parameterγ, the continuous hidden attractor can be discretized into a programmable constellation of coordinates. This discretization bridges the gap between Hamiltonian-like chaotic flows and predictable dissipative systems. The findings suggest that applying the Hala operator to a minimally chaotic system, such as Sprott-A, provides a robust framework for physics applications requiring high-precision entropy reduction, such as the stabilization of physical plasma turbulence and the engineering of self-regulating thermal oscillators. Keywords: Sprott-A, Hala Operator, Chaos Control, Deterministic Chaos Compression, Nos´e-Hoover Thermostat, Hidden Attractors, Successive Controlled Collapse, Discretization.