The Hala Operator as a State-Dependent Thermostat for a Nos´e-Hoover (Sprott – A Attractor) Oscillator

This paper investigates the application of the Hala operator—a state-dependent nonlinear feedback mechanism (z˙ → z˙ − γ(x2 + y2)z)—to the Sprott A attractor, 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 “deterministic chaos compressor,” monotonically suppressing the largest Lyapunov exponent and collapsing the system’s dimensionality from a volume-filling state to a stable one-dimensional limit cycle. Following the observation by J.C. Sprott that this operator functions as a physical thermostat for the Nos´e-Hoover oscillator, we explore its utility as a cooling mechanism in statistical mechanics. Furthermore, we implement a “successive controlled collapse” protocol, showing that by periodically switching the control parameter γ, 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 the Hala-Sprott A system provides a robust framework for physics applications requiring high-precision entropy reduction, such as the stabilization of plasma turbulence and the engineering of self-regulating thermal oscillators.Keywords: Sprott – A Attractor, Hala Operator, Chaos Control, Deterministic Compression,Nos´e-Hoover Thermostat, Hidden Attractors, Successive Controlled Collapse.