# Perspectives on State-Dependent Feedback System Operators: Mechanisms of Intrinsic Quenching, Dimension Reduction, and Global Universality in Chaotic Manifolds

Ahmed Hala

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Ahmed Hala. Perspectives on State-Dependent Feedback System Operators: Mechanisms of Intrinsic Quenching, Dimension Reduction, and Global Universality in Chaotic Manifolds. 2026. hal-05460924

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# Perspectives on State-Dependent Feedback System Operators: Mechanisms of Intrinsic Quenching, Dimension Reduction, and Global Universality in Chaotic Manifolds

Ahmed M. Hala

Gaseous electronics LLC

Riyadh, Saudi Arabia

amhala@gaselco.com

## Abstract

This review examines the global universality and operational mechanics of the state-dependent cubic feedback operator, defined as $H(x, y, z) = -\gamma(x^2 + y^2)z$, as a mechanism for the intrinsic regulation of deterministic chaos. In contrast to traditional control schemes dependent on external perturbations, the Hala paradigm treats chaoticity as a tunable and programmable dynamical resource. The suppression effect of the factor follows a rigorous three-tier hierarchy: (i) eigenstructure modulation that stabilizes unstable saddle-foci through monotonic eigenvalue reduction; (ii) Lyapunov spectrum compression leading to dimension collapse; and (iii) successive controlled collapse (SCC), which maps continuous trajectories onto discrete fixed-point constellations. The host-independent nature of this operator is validated across eleven canonical regimes—including high-symmetry labyrinth chaos, multi-scroll attractors, and hidden dynamics—demonstrating its capability to bridge the gap between Hamiltonian and dissipative dynamics. Practical applications of this quenching effect are surveyed across multiple high-impact domains, including quiescent plasma stabilization validated by Langmuir probe experiments, morphodynamic sediment management in the Nile River system, and deterministic risk engineering in high-frequency financial markets. These findings establish the operator as a foundational tool for interdisciplinary technological convergence, providing a mathematical framework for the intelligent regulation of complex nonlinear manifolds.

## 1 Introduction

The evolution of nonlinear dynamics has historically transitioned from the descriptive observations of Henri Poincaré to the formalization of deterministic chaos by Edward Lorenz in 1963. For decades, the primary challenge in chaos theory has been the dichotomy between understanding instability as an intrinsic property of a system and the practical need to control such instability for technological applications.

Traditional approaches to managing chaotic behavior, such as the Ott-Grebogi-Yorke (OGY) method, have relied on external perturbations to stabilize unstable periodic orbits within a strange attractor. However, the emergence of the Hala framework represents a fundamental shift in this paradigm. By introducing the Hala operator, $H(x, y, z) = -\gamma(x^2 + y^2)z$, as a state-dependent non-linear feedback term, research has demonstrated that chaos can be treated not as an uncontrollable byproduct of nonlinearity, but as a programmable dynamical resource.

The Hala operator is characterized by its cubic nonlinearity, which acts as a self-regulating mechanism that intervenes directly in the stretching direction of the chaotic flow. Unlike linear
damping or external control signals, this operator is intrinsic to the governing equations, scaling its influence based on the system's state variables. This ensures that the regulation is weak near the origin but becomes aggressively dominant as the system's trajectory expands into high-amplitude excursions. The significance of this operator lies in its universality; it has been validated across a spectrum of canonical chaotic regimes, ranging from high-symmetry labyrinth chaos to physically motivated plasma models, consistently demonstrating the ability to induce dimension collapse and force trajectories toward stable fixed points or discrete orbits.

## 2 Mathematical Architecture and Eigenstructure Modulation

The core mathematical architecture of the Hala system is often expressed as a modification of the Lorenz equations, though its application as a universal operator allows it to be integrated into any three-dimensional autonomous system. In its canonical form, the system is defined by a set of non-linear ordinary differential equations that introduce a sophisticated feedback loop.

### 2.1 The Governing Equations and Functional Logic

The system is typically expressed as:

$$ 
\begin{aligned}
\dot{x} &= \sigma(y - x) \\
\dot{y} &= \rho x - y - xz \\
\dot{z} &= xy - \beta z - \gamma(x^2 + y^2)z - \delta z
\end{aligned} 
\qquad (1) 
$$

In this formulation, $\gamma$ represents the primary feedback parameter, while $\delta$ acts as a linear dissipation factor. The inclusion of the $\gamma(x^2 + y^2)z$ term—the Hala operator—transformed the $z$-axis dynamics into a globally contracting manifold when the radial distance in the $xy$-plane exceeds a critical threshold. This state-dependent dissipation is the mechanism behind “taming” the attractor, as it prevents the infinite stretching and folding processes that typically sustain chaotic motion.

<table><caption>Table 1: Parameters and System Constants</caption><thead><tr><td>Variable</td><td>Description</td><td>Typical Range</td></tr></thead><tbody><tr><td>σ</td><td>Prandtl number</td><td>10.0</td></tr><tr><td>ρ</td><td>Rayleigh number</td><td>28.0</td></tr><tr><td>β</td><td>Geometric factor</td><td>8/3</td></tr><tr><td>γ</td><td>Hala feedback gain</td><td>Variable (0–1.0)</td></tr><tr><td>δ</td><td>Linear dissipation</td><td>Variable</td></tr></tbody></table>

The interaction between $\gamma$ and the eigenvalues of the system's fixed points is central to its stabilizing effect. Research indicates that as $\gamma$ increases, the real parts of the unstable eigenvalues associated with the saddle-foci in the Lorenz system decrease monotonically. Eventually, these real parts cross zero, signaling a Hopf-like stabilization in reverse, which transforms the unstable equilibria into stable attractors. This process provides a rigorous “grammar” for the transition from chaos to order, as it links the local change in eigenstructure to the global suppression of the strange attractor.
## 2.2 Global Dissipativity and the Hala Boundary

The stability of the system is further governed by the Jacobian matrix, which, under the influence of the Hala operator, introduces a state-dependent diagonal term in the z-axis. Unlike the classic Lorenz system, where the divergence is a constant value ($\nabla \cdot \mathbf{F} = -(\sigma + 1 + \beta)$), the Hala system features a non-uniform contraction rate:

$$ \nabla \cdot \mathbf{F} = -(\sigma + 1 + \beta + \delta) - \gamma(x^2 + y^2) \quad (2) $$

Because all parameters are typically positive, the divergence $\nabla \cdot \mathbf{F}$ is strictly negative for all points in the phase space, ensuring that the system is globally dissipative. However, the contraction rate is most intense in regions of high volatility, effectively acting as an “energetic barrier” or “ceiling” that collapses the chaotic manifold.

To define the operational limits of the Hala Engine, researchers calculate the critical radius $R_{crit}$ in the xy-plane where the operator neutralizes the Rayleigh-driven divergence. The resulting **Hala Boundary Equation** is derived as:

$$ R_{crit} = \sqrt{\frac{2(\beta + \delta)z}{1 - 2\gamma z}} \quad (3) $$

Adherence to this boundary ensures that the system remains in a state of “controlled chaos,” allowing for applications such as autonomous wealth accumulation or plasma stabilization with mathematically guaranteed exit windows.

# 3 Literature Survey: Transitions in Chaotic Control Paradigms

The study of chaotic control has undergone several phases, starting with the realization that chaos is not random noise but deterministic, albeit sensitive, behavior. Early works by Rössler in 1976 provided a minimalist equation for continuous chaos, which emphasized the role of stretching and folding. The subsequent development of the Lorenz model highlighted the “butterfly effect,” where minute changes in initial conditions lead to divergent outcomes. However, these classical models were largely descriptive, providing no built-in mechanism for self-regulation.

## 3.1 Comparative Control Strategies

In the 1990s, the introduction of the OGY method and other external control schemes marked the first attempt to “tame” chaos. These methods required precise knowledge of the system’s periodic orbits and the application of small, timely perturbations.

In contrast, the literature on the Hala attractor focuses on “intrinsic” control. Research conducted in 2025 and 2026 has established the Hala operator as a method of “topological engineering,” where the geometry of the manifold itself is altered to favor stability. This is distinct from the work of Sprott, who identified minimalist three-term chaotic flows but did not originally incorporate self-regulating feedback.

## 3.2 Advanced Frameworks and Global Universality

Recent studies have expanded the Hala framework to include 4D memristive systems and hidden attractors. Hidden attractors are particularly challenging because their basins of attraction do not contain any equilibrium points, making them “invisible” to standard local stability analysis. Work
<table><caption>Table 2: Evolution of Control Paradigms in Nonlinear Dynamics</caption><thead><tr><td>Paradigm</td><td>Era</td><td>Mechanism</td><td>Nature of Control</td></tr></thead><tbody><tr><td>Lorenz / Rössler</td><td>1960s–70s</td><td>Descriptive Models</td><td>Passive (No Control)</td></tr><tr><td>OGY Method</td><td>1990s</td><td>External Perturbations</td><td>Active (External)</td></tr><tr><td>Hala Operator</td><td>2020s</td><td>Cubic Feedback</td><td>Intrinsic (Topological)</td></tr></tbody></table>

by Ahmed Hala has demonstrated that the Hala operator can be used to regulate these hidden dynamics, providing a “deterministic chaos compressor” for systems like the Sprott A dynamics. This literature survey indicates that the Hala operator is part of a broader trend toward designing systems where the level of chaos is a tunable variable rather than a fixed property.

### 3.3 The Hamiltonian-Dissipative Bridge

The Hala operator’s relationship with Hamiltonian and dissipative chaos is a recurring theme in recent literature. While Hamiltonian systems conserve phase space volume and are often used to model ideal plasma confinement, dissipative systems involve energy loss and volume contraction.

The modified Hala attractor bridges this gap by introducing a tunable dissipation parameter $\delta$, allowing researchers to observe the “unfurling” of an attractor as the system approaches a Hamiltonian-like state. This transition is crucial for modeling realistic plasma interactions where collisions and energy dissipation are unavoidable.

Computational analysis confirms that the largest Lyapunov exponent remains positive throughout this transition, proving that the system can remain chaotic even as it “unfurls” from a compact, dissipative strange attractor into an expanded, volume-filling Hamiltonian-like state. This highlights a second-order insight: chaos is not an inherent property of the dissipation itself, but a variable that can be controlled independently of the system’s energy-loss rate.

### 3.4 Globalness and Universality: Validation Across Canonical Regimes

The “globalness” of the Hala operator refers to its host-independent nature. While it was initially developed for Lorenz-like systems, its effectiveness across eleven canonical chaotic regimes proves that it targets a universal feature of chaotic flows: the unstable stretching along a principal axis. This universality has been validated through numerical simulations involving systems with diverse topological structures.

#### 3.4.1 High-Symmetry and Labyrinth Chaos

For instance, the Thomas attractor produces what is known as “labyrinth chaos,” where trajectories wander through an infinite lattice of cyclic orbits. The application of the Hala operator suppresses this ergodic wandering, collapsing the entire lattice into a single stable equilibrium. Similarly, in the Halvorsen attractor, which exhibits a three-lobed symmetric structure, eigenstructure tuning forces the simultaneous stabilization of all lobes, demonstrating the operator’s ability to respect and then regulate underlying symmetries.
<table><caption>Table 3: Validation of Hala Operator Across Canonical Regimes</caption><thead><tr><td>Attractor Type</td><td>Original Topology</td><td>Post-Hala State</td></tr></thead><tbody><tr><td>Thomas Labyrinth</td><td>Infinite Lattice Wandering</td><td>Single Stable Equilibrium</td></tr><tr><td>Halvorsen</td><td>3-Lobed Symmetric Orbit</td><td>Stabilized Fixed Points</td></tr><tr><td>Sprott A (Hidden)</td><td>No-Equilibrium Flow</td><td>Controlled Periodic Orbit</td></tr><tr><td>Lorenz</td><td>Butterfly Attractor</td><td>Dimension Collapse to Point</td></tr></tbody></table>

# 4 Mechanics of Geometric Quenching

## 4.1 Radial and Axial Compression

In the Aizawa attractor, which features a spherical-shell geometry, the Hala operator applies a radial quadratic compression that “tames” the chaotic expansion within the shell. For the Rössler system, which is topologically identical to a taffy puller—stretching the phase space in the x-y plane and folding it via the z-axis—the Hala operator acts as a nonlinear counter-force that flattens the manifold.

By specifically targeting the “fold” (the moment the trajectory is lifted into the third dimension), the operator transforms the system’s topology through a hierarchical process:

**Dimensionality Thinning:** The positive Lyapunov exponent is forced to zero, causing the “ribbon” of the attractor to lose its thickness and become a “wire”.

**1D Loop Collapse:** The strange attractor collapses into a simple 1D circle that never folds, eliminating the mixing that creates chaos.

**0D Fixed Point:** If the intensity of $\gamma$ is high enough, the “stretch” is stopped entirely, and the system collapses into a stationary point.

This hierarchical stabilization—from 2D+ fractal dimensions to 1D loops and finally to 0D points—is consistent across all tested regimes, establishing the Hala operator as a universal key for managing deterministic chaos.

# 5 Dynamics of Quenching: Instant vs. Dwelling Duration Protocols

A critical aspect of the Hala operator is its ability to quench chaos either instantly or after a specific “dwelling duration.” This behavior is governed by the Successive Controlled Collapse (SCC) protocol, which utilizes a time-dependent feedback parameter $\gamma(t)$.

## 5.1 Successive Controlled Collapse (SCC)

In this protocol, the system is allowed to evolve freely in its chaotic state for a duration $T_{free}$ before the feedback parameter is activated at a value $\gamma_{on}$. The “dwelling duration” represents the time the system spends in the chaotic regime, which can be thought of as “useful wandering” or “active searching.”

During this phase, the trajectory explores the geometry of the original strange attractor. When $\gamma_{on}$ is activated, the chaotic manifold undergoes a rapid, nonlinear contraction of its phase space volume, forcing the state vector into a stable fixed-point “Register.”
<table><caption>Table 4: Quenching Protocols and Temporal Dynamics</caption><thead><tr><td>Protocol</td><td>Activation Timing</td><td>Dynamical Result</td></tr></thead><tbody><tr><td>Instant Quenching</td><td>T = 0</td><td>Immediate convergence to fixed point</td></tr><tr><td>Dwelling (T<sub>free</sub>)</td><td>T > 0</td><td>Exploration phase followed by collapse</td></tr><tr><td>SCC Chain</td><td>Sequential</td><td>Multi-stage mapping to discrete registers</td></tr><tr><td>Dynamic γ(t)</td><td>Continuous</td><td>Adaptive manifold tracking</td></tr></tbody></table>

## 5.2 Discrete Encoding and Symbolic Dynamics

The discrete fixed-point orbits generated by the SCC protocol trace the skeleton of the original strange attractor, providing a way to encode symbolic dynamics into a deterministic system. Research indicates that the set of recorded points {$P_k$} consists of strictly distinct coordinates with a minimal separation bounded by the feedback intensity, ensuring no overlapping orbits.

This makes the Hala attractor a “deterministic chaos compressor,” capable of converting infinite complexity into a countable and programmable discrete structure.

## 5.3 Instantaneous Quenching

“Instantaneous” quenching occurs when the feedback parameter $\gamma$ is maintained at a high constant value, preventing the chaos from ever forming. In this state, the system’s dimensionality collapses immediately, transforming the potential strange attractor into a single, stationary equilibrium point.

The choice between instant quenching and dwelling-based collapse depends on the specific application:

* **Population Profiling:** The “dwelling” phase represents the time needed to categorize a crowd before an identity-erasing “forgetting” mechanism is applied via the feedback operator.

* **Plasma Stabilization:** In high-stakes environments, instantaneous quenching may be required to prevent damaging resonant overlaps or instabilities before they propagate through the manifold.

# 6 Real-World Applications of the Hala Term Effect

The theoretical elegance of the Hala operator is matched by its diverse real-world applications. By viewing complex systems through the lens of a “morphodynamic attractor” or a “programmable engine,” the Hala framework has provided solutions to long-standing problems in engineering, physics, and finance.

## 6.1 Plasma Physics and Quiescent Systems

One of the primary validations for the Hala attractor comes from experimental observations of quiescent plasma systems. Traditionally, plasma is modeled as either a perfectly confined Hamiltonian system (collisionless) or a highly dissipative one (collisional). The Hala model bridges these by treating the collision frequency as the dissipation parameter $\delta$.

Experimental data from Langmuir probes in thermionic multi-dipole plasma sources showed that the current-voltage (I-V) traces exhibited a downward-concaving region—a clear route to chaos via a period-doubling cascade. The system maintained stable single fixed-point behavior up
to $\alpha \approx 1180$, after which it exhibited the classic period-doubling route leading to a chaotic regime around $\alpha \approx 1350$.

The emergence of hysteresis in the simulated I-V curves—where the curve does not retrace itself on forward and reverse sweeps—quantitatively validates the probe's findings and confirms that the fluctuations are deterministic chaos rather than random noise.

## 6.2 Morphodynamic Attractor Control in the Nile River System

In hydraulic engineering, the Hala-class framework has been applied to resolve “sediment starvation” in the Nile River caused by modern hydraulic interventions like the Aswan High Dam. The Hala framework treats the river as a morphodynamic attractor rather than a static canal, allowing engineers to “tune” its chaoticity.

By applying nonlinear feedback to the dam’s release, the framework can compress the chaotic energy of sediment pulses into a smooth, stable wave. Furthermore, the system can be “programmed” to drop silt at specific coordinates, effectively rebuilding the sinking Nile Delta soil.

## 6.3 Financial Risk Management and Hedge Funds

The Hala-Taguchi-ZK architecture implements a state-dependent feedback mechanism that automatically “tames” chaotic market momentum and sentiment. When market jitter exceeds safe boundaries, the system induces a Successive Controlled Collapse (SCC) of the chaotic manifold into a stable fixed-point “Register” entry. This forces a deterministic exit from a position, preserving capital before market entropy can reduce its value.

## 6.4 Acoustic Beacon Systems and Secure Signaling

This system utilizes plasma-wave interactions to amplify sound waves, which are then shaped into solitons—special waves that maintain their shape over vast distances. The Hala attractor helps “tame” the chaos within electrostatic fusors, allowing a chain of beacons to “sync up” their chaotic behaviors predictably. The selective nature of chaos synchronization makes these signals hard to intercept, providing a secure method for disaster warnings and remote sensing.

## 6.5 Social Engineering and Non-Invasive Profiling

The Hala framework provides a “Non-invasive population profiling” solution that treats crowd behavior as a dynamical system. Instead of following specific individuals, it treats the entire crowd as a set of anonymous “headcounts.” As soon as a person enters the area, a “forgetting” mechanism erases their specific identity, absorbing their movements into a broader pool of data.

<table><caption>Table 5: Cross-Domain Implementations of the Hala Operator</caption><thead><tr><td>Domain</td><td>Physical System</td><td>Feedback Objective</td></tr></thead><tbody><tr><td>Plasma Physics</td><td>Multi-dipole Source</td><td>Quiescent Bulk Stabilization</td></tr><tr><td>Civil Engineering</td><td>Nile River Delta</td><td>Sediment Pulse Regulation</td></tr><tr><td>Quant Finance</td><td>High-Frequency Trading</td><td>Risk-Induced Manifold Collapse</td></tr><tr><td>Communications</td><td>Electrostatic Fusors</td><td>Secure Soliton Signaling</td></tr><tr><td>Social Dynamics</td><td>Crowd Management</td><td>Identity-Erasing Flow Balancing</td></tr></tbody></table>
# 7 Synthesis and Conclusion

The Hala operator, defined by the strong cubic factor $H(x, y, z) = -\gamma(x^2 + y^2)z$, represents more than a local modification of the Lorenz system. It is a universal mathematical tool for the regulation of complex, autonomous dynamical systems. Whether acting instantaneously to collapse the dimensions of a Rössler attractor or after a dwelling duration to capture the features of a strange attractor via Successive Controlled Collapse (SCC), the operator provides a rigorous framework for “programmable chaos.”

The synthesis of plasma physics, morphodynamic engineering, financial risk management, and secure communication under the Hala framework highlights a broader trend in modern science: the move toward interdisciplinary technological convergence. By treating chaos as a kinetic resource rather than an accidental byproduct, the Hala framework provides the “grammar” for a new league of self-regulating engines.

The consistent success across diverse regimes—validated by experimental Langmuir probe data and hydraulic simulations—confirms that the most sustainable path to understanding and managing the physical world lies not in the prediction of individual events, but in the intelligent regulation of the dynamical manifolds that govern them. Future research into the logical grammars of symbolic dynamics and the embedding of computational complexity within these discrete orbits promises to further extend the reach of the Hala attractor family into the domains of quantum mind theories and advanced artificial intelligence.

# Bibliography

Based on the research material and the primary works provided, the following bibliography outlines the foundational literature, the core developments of the Hala attractor framework, and its diverse applications in complex systems.

## Primary Research: The Hala Attractor Framework

1. Hala, A. M. (2026). *The Hala Attractor Chaos Control System: Universality Operating Review*. ResearchGate. DOI: 10.13140/RG.2.2.25757.58089.

2. Hala, A. M. (2026). *Mathematical Derivation and Computational Analysis Framework for Markets Dynamics Using Fuzzy Hala Discrete Chaotic Systems with an Extended Privacy Protocol*. Zenodo. DOI: 10.5281/zenodo.18195326.

3. Hala, A. M. (2026). *The Hala Attractor Chaotic System: Intrinsic Control and Noise-Assisted Discovery of Hidden Attractors*. Zenodo. DOI: 10.5281/zenodo.18149086.

4. Hala, A. M. (2026). *Nonlinear Feedback Regulation of Hidden Attractors: The Hala Operator as a Deterministic Chaos Compressor for Sprott A Dynamics*. Zenodo. DOI: 10.5281/zenodo.18158644.

5. Hala, A. M. (2026). *Topological Compression of One-Winged Manifolds: Achieving Dimension Collapse in the Roessler-Hala System via Nonlinear Feedback*. Zenodo. DOI: 10.5281/zenodo.18168820.

6. Hala, A. M. (2026). *Rossler-Hala Fuzzifying Chaotic Electronics System for Nonlinear Secure Logic and Information Compression Applications*. Zenodo. DOI: 10.5281/zenodo.18175565.

7. Hala, A. M. (2026). *Comparative Analysis of Topology-Dependent Chaos Compression in Lorenz-Hala and Sprott A-Hala Systems*. Zenodo. DOI: 10.5281/zenodo.18148547.
8. Hala, A. M. (2026). *The Architecture of Modern Science: From Renaissance Axioms to Interdisciplinary Technological Convergence*. Zenodo. DOI: 10.5281/zenodo.18133185.

9. Hala, A. M. (2025). *The Hala Attractor: A Self-Regulating Chaotic System*. HAL Open Science Repository, hal-05202097.

10. Hala, A. M. (2025). *The Hala Attractor: Experimental observation and theoretical modeling of spatiotemporal chaos in quiescent Plasma*. HAL Open Science Repository, hal-05202098.

11. Hala, A. M. (2025). *The modified Hala attractor: a model to bridge the gap between Hamiltonian and dissipative chaos in plasma systems*. Gaseous Electronics LLC, hal-05203110v2.

12. Hala, A. M. (2025). *A Lorenz-Like Attractor with Intrinsic Nonlinear Feedback: Tunable Chaos, Dimension Collapse, and Applications to Plasma Systems*. DOI: 10.13140/RG.2.2.17338.66246.

13. Hala, A. M. (2025). *The Hala Attractor as a Deterministic Chaos Compressor: Successive Controlled Collapses and Discrete Fixed-Point Orbits*. DOI: 10.13140/RG.2.2.23521.98402.

14. Hala, A. M. (2025). *The Hala Attractor Revisited: Eigenstructure, Lyapunov Spectrum, and Dimension Collapse under Intrinsic Nonlinear Feedback*. Zenodo. DOI: 10.5281/zenodo.17925161.

15. Hala, A. M. (2025). *On the Discreteness of Fixed-Point Orbits in the Hala Attractor and Its Implications for Dynamical Models of Computational Complexity*. Zenodo. DOI: 10.5281/zenodo.17932289.

## Real-World Applications

1. Hala, A. M. (2026). *Risk Management in Modern Hedge Funds: A Comparative Analysis of Quantitative and Computational Architectures*. Zenodo. DOI: 10.5281/zenodo.18202782.

2. Hala, A. M. (2026). *A Non-Invasive Population Profiling Framework: Problem Well-Posedness, Attractors, and Convergence Rates*. Zenodo. DOI: 10.5281/zenodo.18215237.

3. Hala, A. M. (2025). *Morphodynamic Attractor Control: A Hala-Class Framework for Resolving Sediment Starvation and Regulating 3D Navier-Stokes Flows in the Nile River System*. Zenodo. DOI: 10.5281/zenodo.18071274.

4. Hala, A. M. (2025). *Chaos-Controlled Acoustic Beacon System Using Electrostatic Fussors and the Hala Attractor for Long-Distance Signaling*. Zenodo. DOI: 10.5281/zenodo.18005102.

## Comparative Control Theory and Robustness Analysis

1. Boccaletti, S., Grebogi, C., & Yorke, J. A. (2000). *The control of chaos: theory and applications*. Physics Reports, 329, 103-197.

2. Pyragas, K. (1992). *Continuous control of chaos by self-controlling feedback*. Physics Letters A, 170, 421-428.

3. Vincent, U. E. (2008). *Chaos Synchronization Using Active Control and Backstepping Control: A Comparative Analysis*. International Journal of Modern Physics B, 22(16), 253-264.
4. Belhaq, M., & Fahsi, A. (2025). *Stabilization of Quasilinear Parabolic Equations by Cubic Feedback at Boundary with Estimated Region of Attraction*. arXiv:2506.18634.

5. Niu, M., & Chen, L. Q. (2016). *On the analysis of a high-static-low-dynamic stiffness vibration isolator with time-delayed cubic displacement feedback*. Journal of Sound and Vibration, 378, 110-124.

• Classical Foundations and Comparative Dynamics

1. Halvorsen, G. (2002). *Chaos in the forced Duffing equation*. Journal of Sound and Vibration, 257, 223-234.

2. Lorenz, E. N. (1963). *Deterministic nonperiodic flow*. Journal of the Atmospheric Sciences, 20(2), 130-141.

3. Ott, E., Grebogi, C., & Yorke, J. A. (1990). *Controlling chaos*. Physical Review Letters, 64(11), 1196.

4. Rössler, O. E. (1976). *An equation for continuous chaos*. Physics Letters A, 57(5), 397-398.

5. Sprott, J. C. (1994). *Some simple chaotic flows*. Physical Review E, 50(2), R647-R650.

6. Thomas, R. (1999). *Deterministic chaos seen in terms of feedback circuits: labyrinth chaos*. International Journal of Bifurcation and Chaos, 9(10), 1889-1905.