Topological Compression of One-Winged Manifolds: Achieving Dimension Collapse in the Roessler-Hala System via Nonlinear Feedback

This research investigates the mechanism of controlled dimensionality reduction in chaotic manifolds through the implementation of the Hala operator, a nonlinear feedback framework designed to suppress the vertical instabilities of the R¨ossler system. By introducing a statedependent damping term, −γ(x2+y2)z, we demonstrate a definitive topological transition where the characteristic “one-winged” 3D strange attractor collapses into lower-dimensional manifolds. Numerical analysis of the Lyapunov Spectrum confirms that as the control intensity γ reaches a critical threshold (γ ≈ 0.035), the largest Lyapunov exponent (λ1) undergoes a sign reversal, signaling a phase transition from DL ≈ 2.01 chaos to a 0D fixed-point singularity. Comparative studies with the Lorenz and Sprott A systems reveal that while the former mirrors this abrupt collapse, the latter plateaus at a 1D limit cycle due to a lack of equilibrium points, establishing a fundamental “Topological Limit” for nonlinear feedback. Robustness analysis further indicates that the temporal stability of the R¨ossler-Hala system is highly energy-sensitive, with recovery times (T) scaling from < 0.3 s to < 1.2 s depending on the internal system drive. These results, validated through TL082-based analog circuitry, provide a rigorous methodology for the discretization of chaotic flows into predictable “landing zones,” offering significant implications for secure signal processing and high-precision rhythmic oscillators.