The Hala Attractor Chaotic System: Intrinsic Control and Noise-Assisted Discovery of Hidden Attractors

This paper presents the Hala Chaotic System, a novel three-dimensional autonomous dynamical system engineered to generate and stabilize intrinsic self-regulated non-equilibrium steady states (NESS). In contrast to conservative chaotic flows with hidden attractors—exemplified by the volume-preserving Sprott A system, which exhibits no natural basins of attraction and resists conventional stabilization techniques due to phase-space incompressibility and the absence of connected equilibria—we introduce a dual-mechanism control strategy leveraging the synergy between additive stochastic forcing (intensity σ) and state-dependent nonlinear feedback (strength γ). Stochastic perturbations serve as an ergodic exploration mechanism, akin to a probe illuminating inaccessible regions of phase space and facilitating the localization of hidden invariant manifolds. Upon intersection with these manifolds, the Hala Operator—a directional nonlinear feedback term—acts as a meta-stable attractor inducer, effecting a deterministic phase-space volume contraction that successively collapses the extended chaotic sea into discrete, stable fixed-point orbits or low-dimensional limiting sets. Numerical integrations, complemented by Lyapunov spectrum analysis and bifurcation diagrams, reveal a noise-induced order-to-chaos transition analogous to a critical point, wherein excessive stochastic intensity disrupts the feedback-induced ordering, leading to a reversal of the dimension collapse. We furnish analytical insight into the mechanism of this successive Kaplan-Yorke dimension reduction and quantify the effective “dissipative cost”—manifest as transient energy dissipation required to sustain the stabilized discrete orbits against perturbations. These findings establish the Hala framework as a robust paradigm for noise-facilitated localization and deterministic compression of hidden chaotic dynamics, reconfiguring conservative ergodic flows into structured, information-rich manifolds amenable to programmable steady states. The approach holds particular promise for applications in plasma confinement and turbulence suppression, high-dimensional chaotic computing, and the design of self-organizing autonomous structures far from thermodynamic equilibrium. Keywords: Hala Attractor, Hala Operator, Hidden Attractors, Deterministic Chaos Compression,Stochastic Resonance, Non-Equilibrium Steady States, Dimension Collapse, Sprott A System.