Solving the Synthesis Problem Self-Organizing Control System in the Class of Elliptical Accidents Optics for Objects with One Input and One Output

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1.The introduction provides a broad overview of nonlinear systems and catastrophe theory, but the research gap is not sufficiently highlighted. What limitations exist in current self-organizing control system designs? How the proposed approach compares conceptually with modern nonlinear control methods?

2. The manuscript would benefit from a more intuitive explanation of the physical meaning of “elliptical dynamics” and why this catastrophe class is selected over others.

3.Please justify the application of the Morse lemma more explicitly and describe how it guarantees positive definiteness of the Lyapunov function.

4.The stability proofs rely heavily on the gradient-velocity method. Why this method is appropriate for catastrophe-based systems?

5.Whether the constructed Lyapunov functions satisfy standard smoothness and definiteness properties before invoking the Morse lemma?

6.Please provide a clearer interpretation or metric of “self-organization.” How does this differ from standard adaptive or switching control mechanisms?

7.Figures or phase portraits would significantly enhance understanding of how trajectories migrate between equilibrium branches.

Author Response

Comment 1: The introduction provides a broad overview of nonlinear systems and catastrophe theory, but the research gap is not sufficiently highlighted. What limitations exist in current self-organizing control system designs? How the proposed approach compares conceptually with modern nonlinear control methods?

Response:
We thank the reviewer for this important remark. In the revised version, the Introduction has been extended to explicitly formulate the research gap. Existing self-organizing and adaptive control systems typically rely on parameter tuning, gain scheduling, or switching logic, which may fail near bifurcation points or under strong parametric uncertainty. Moreover, most nonlinear control approaches focus on local stability around a predefined equilibrium and do not explicitly account for structural changes in system dynamics.
In contrast, the proposed approach embeds the control synthesis directly into the framework of catastrophe theory, allowing the control system to operate across multiple equilibria and to maintain aperiodic robust stability during bifurcation-induced transitions. This conceptual distinction has now been clearly articulated in the Introduction.

"Despite significant progress in nonlinear, adaptive, and switching control methodologies, several fundamental limitations remain unresolved. Most existing adaptive control schemes rely on continuous parameter tuning within a predefined controller structure, which restricts their ability to cope with qualitative changes in system dynamics near bifurcation points. Switching and gain-scheduling approaches, in turn, require explicit mode definitions and supervisory logic, making them sensitive to model uncertainty and prone to instability during regime transitions. Importantly, these methods typically guarantee only local stability and do not provide structural stability guarantees when the number or nature of equilibria changes. In contrast, the approach proposed in this study performs control synthesis directly within the framework of catastrophe theory, rather than applying stabilization techniques a posteriori to a fixed nonlinear model. By embedding the control law into the class of elliptic catastrophes, the proposed method inherently accounts for bifurcation phenomena and enables autonomous transitions between stable equilibria while preserving aperiodic robust stability under parametric uncertainty. This structural integration constitutes the main conceptual novelty of the present work."

Comment 2: The manuscript would benefit from a more intuitive explanation of the physical meaning of “elliptical dynamics” and why this catastrophe class is selected over others.

Response:
We agree with the reviewer and have added an intuitive explanation of elliptical dynamics. In the revised manuscript, elliptical dynamics is interpreted as a geometric representation of coupled nonlinear oscillatory modes with bounded trajectories and structurally stable equilibria. Unlike cusp or fold catastrophes, elliptical dynamics preserves symmetry and allows smooth transitions between stable branches without abrupt loss of controllability. This makes it particularly suitable for control systems with one input and one output operating under continuous parameter variations.

"In the context of catastrophe theory, elliptical dynamics represent a class of structurally stable configurations in which equilibrium branches form bounded, symmetric manifolds in the phase space. Geometrically, this corresponds to elliptical trajectories that confine system motion and prevent divergence. The symmetry of the potential function underlying elliptical catastrophes guarantees robustness to parameter perturbations and enables continuous transitions between equilibria. In contrast, fold and cusp catastrophes exhibit asymmetric potential landscapes, leading to sudden instability and loss of controllability. Therefore, elliptical dynamics provide a natural mathematical framework for synthesizing self-organizing control systems with guaranteed aperiodic robust stability."

Comment 3: Please justify the application of the Morse lemma more explicitly and describe how it guarantees positive definiteness of the Lyapunov function.

Response:
Thank you for pointing this out. In the revised manuscript, an explicit explanation of Morse’s lemma has been included. Additional clarification on the application of Morse’s lemma, including the role of non-degenerate critical points, local diffeomorphism, and preservation of definiteness, has been added after equations (1.11), (1.18), (2.13), and (2.20).

"At the considered equilibrium, the Lyapunov candidate function is continuously differentiable and possesses a non-degenerate critical point, since its Hessian matrix is nonsingular in the neighborhood of equilibrium. According to Morse’s lemma, there exists a local diffeomorphism that transforms the function into an equivalent quadratic form without changing its qualitative stability properties. This local coordinate transformation preserves the sign definiteness of the function in a neighborhood of the equilibrium point. As a result, the positive (or negative) definiteness of the Lyapunov function is explicitly revealed, ensuring the validity of the Lyapunov stability conditions."

Comment 4: The stability proofs rely heavily on the gradient-velocity method. Why this method is appropriate for catastrophe-based systems?

Response:
The gradient-velocity method is particularly suitable for catastrophe-based systems because it naturally aligns with the geometric structure of catastrophe manifolds. Since catastrophes describe qualitative changes in equilibrium structure, the gradient-velocity approach allows stability to be analyzed globally across branches rather than locally around a single equilibrium. This method ensures monotonic decrease of the Lyapunov function even during bifurcation-induced transitions, which is essential for guaranteeing aperiodic robust stability.

"The synthesis approach adopted in this work is based on the geometric interpretation of catastrophe surfaces and their associated gradient flows. In catastrophe theory, equilibrium states form smooth manifolds in the phase space, and system evolution can be interpreted as motion along these surfaces driven by gradient-like dynamics. The gradient-velocity method naturally captures this behavior by ensuring monotonic descent of the Lyapunov function along trajectories, even when transitions occur between different equilibrium branches. As a result, branch-to-branch transitions induced by bifurcations are treated as continuous geometric flows rather than discrete switching events, which is essential for guaranteeing aperiodic robust stability in catastrophe-based control systems."

Comment 5: Whether the constructed Lyapunov functions satisfy standard smoothness and definiteness properties before invoking the Morse lemma?

Response:
Clarifications regarding C¹-smoothness, the condition V(0)=0V(0)=0V(0)=0, and the role of Morse’s lemma in resolving definiteness have been added before each application of Morse’s lemma.

"The constructed Lyapunov candidate functions are continuously differentiable (C¹-smooth) in a neighborhood of the equilibrium and satisfy the condition V(0)=0V(0)=0V(0)=0. However, due to the presence of higher-order nonlinear terms, their sign definiteness is not immediately apparent in the original coordinates. This issue is resolved by applying Morse’s lemma, which locally transforms the function into an equivalent quadratic form, explicitly revealing its definiteness near the equilibrium point."

Comment 6: Please provide a clearer interpretation or metric of “self-organization.” How does this differ from standard adaptive or switching control mechanisms?

Response:
We appreciate this remark. In this work, self-organization is defined as the ability of the control system to autonomously transition between stable equilibria without predefined switching rules or external supervisory logic. Unlike adaptive control, which tunes parameters within a fixed structure, and switching control, which relies on explicit mode selection, the proposed approach enables structural reconfiguration driven by bifurcation mechanisms. A formal definition and comparison have now been included.

"In this study, self-organization is defined as autonomous structural adaptation through bifurcation-driven equilibrium selection, occurring without external supervisory logic or predefined switching rules. Unlike adaptive control, which modifies controller parameters within a fixed structural framework, self-organizing control allows the structure of stable operating regimes to change as system parameters evolve. In contrast to switching control, where transitions between modes are explicitly prescribed, self-organization emerges naturally from the system dynamics as a consequence of bifurcation mechanisms. Thus, the proposed approach treats regime transitions as intrinsic dynamical processes rather than externally enforced control actions."

Comment 7: Figures or phase portraits would significantly enhance understanding of how trajectories migrate between equilibrium branches.

Response:
We fully agree. In the revised version, phase portraits illustrating transitions between equilibrium branches and bifurcation-driven trajectory evolution have been added. These visualizations support the analytical results and provide intuitive insight into the self-organizing behavior of the system.

The qualitative behavior of the proposed system is illustrated in Fig. 2, where phase trajectories demonstrate smooth bifurcation-driven transitions between stable equilibrium branches.

 

Figure 2. Phase portrait of the self-organizing system in the plane illustrating bifurcation-driven transitions between equilibrium branches. Multiple trajectories converge monotonically to stable equilibria without oscillations, demonstrating aperiodic robust stability and self-organizing behavior of the proposed control system.

 

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Please find attached a PDF file with the review report.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The English in this paper should be improved significantly to more clearly express the research. While the technical content appears substantial, the language issues obscure the meaning, reduce readability, and undermine the paper's credibility.

Author Response

Please see the attachment

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript studies the synthesis of a self-organizing SISO nonlinear control system under parametric uncertainty, motivated by avoiding instability, bifurcations, and chaotic regimes.

1. The manuscript claims the synthesized regulator provides “aperiodic robust stability,” and simulation “eliminates chaotic modes” and enables adaptive bifurcation transitions. These statements require a precise definition of the uncertainty set, the region of attraction, the equilibrium, and the exact stability notion.

2. The proposed SOM-based gain scheduling is a learning component, but the manuscript does not compare its guarantees and objectives with mainstream frameworks that provide stability and performance guarantees. For example, arXiv:2508.21367 and Self-triggered approximate optimal neuro-control for nonlinear systems through adaptive dynamic programming. IEEE Transactions on Neural Networks and Learning Systems, 36(3), 4713-4723.

3. The use of Morse’s lemma is not properly justified and may be misapplied. Please state the exact assumptions and show how they hold for the proposed $V(x)$. 

4. Experimental validation is too limited and contains non-technical analogies. To support claims about chaos suppression and bifurcation transitions, please add phase portraits/time-series, bifurcation diagrams, Lyapunov traces, robustness sweeps across uncertainty ranges, and comparisons with baseline controllers. Moreover, it is better to remove non-technical metaphors unless rigorously motivated.

Author Response

We sincerely thank the reviewer for the careful reading of our manuscript and for the detailed, technically insightful comments. The suggestions were extremely valuable and helped us significantly improve the clarity, rigor, and presentation of the proposed method. Below, we provide a point-by-point response explaining how each comment has been addressed in the revised manuscript.

Reviewer Comment 1

The manuscript claims “aperiodic robust stability,” elimination of chaotic modes, and adaptive bifurcation transitions. These statements require a precise definition of the uncertainty set, the region of attraction, the equilibrium, and the exact stability notion.

Response

We fully agree with the reviewer that these concepts must be clearly and rigorously defined.

In the revised manuscript, a new subsection Section 3.2 (Definitions and Stability Notions) has been added, where:

• Parametric uncertainty is explicitly defined as bounded variations of system parameters within a compact set
  D={d:∣di−di0∣≤Δi}\mathcal{D} = \{ d : |d_i - d_i^0| \le \Delta_i \}D={d:∣di​−di0​∣≤Δi​}.

• Equilibria are defined as stationary solutions of the system state equations and explicitly referenced to equations (1.3), (1.7), (2.5), and (2.9).

• The region of attraction is defined as the maximal neighborhood in which the Lyapunov function is positive definite and satisfies V˙(x)<0\dot V(x) < 0V˙(x)<0 for all admissible uncertainties.

• Aperiodic robust stability is formally defined as asymptotic Lyapunov stability characterized by monotonic (non-oscillatory) convergence, absence of limit cycles, and uniform negativity of V˙(x)\dot V(x)V˙(x) over the uncertainty set.

All claims related to stability and regime transitions are now explicitly grounded in these definitions and consistently referenced throughout Sections 3–5.

Reviewer Comment 2

The SOM-based gain scheduling is a learning component, but the manuscript does not compare its guarantees and objectives with mainstream frameworks such as adaptive dynamic programming or neuro-control.

Response

We thank the reviewer for this important observation.

In the revised Section 5 (Discussion), the role of the self-organizing map (SOM) has been explicitly clarified. We now emphasize that:

• The SOM is not used to optimize a cost function, approximate a value function, or learn an optimal control policy.

• Its role is strictly limited to constrained parameter adaptation within analytically guaranteed Lyapunov-stable regions.

A direct comparison with adaptive dynamic programming and neuro-control frameworks has been added. The revised discussion explicitly states that, unlike ADP and policy-learning approaches that prioritize optimality or policy approximation, the proposed method prioritizes structural stability and robustness. Learning-based adaptation is allowed only insofar as it preserves the analytically proven aperiodic robust stability.

This clarification resolves the distinction between the proposed approach and mainstream learning-based control frameworks.

Reviewer Comment 3

The use of Morse’s lemma is not properly justified and may be misapplied. Please state the exact assumptions and show how they hold for the proposed V(x)V(x)V(x).

Response

We fully agree that the application of Morse’s lemma requires explicit assumptions.

In the revised manuscript (Sections 3.3 and 3.4), the use of Morse’s lemma has been carefully clarified and restricted to its proper scope. Specifically, we now explicitly state that:

• Morse’s lemma is applied locally, in a neighborhood of isolated equilibria, and is not used to claim global stability.

• The required assumptions are clearly listed:

  1. V(x)∈C2V(x) \in C^2V(x)∈C2 in a neighborhood of the equilibrium,

  2. ∇V(x∗)=0\nabla V(x^*) = 0∇V(x∗)=0,

  3. det⁡(∇2V(x∗))≠0\det(\nabla^2 V(x^*)) \neq 0det(∇2V(x∗))=0.

We also explicitly show that these conditions are satisfied for the constructed Lyapunov candidate functions, which are polynomial, possess isolated equilibria, and have non-degenerate Hessian matrices at equilibrium points.

The role of Morse’s lemma is now clearly stated as enabling a local quadratic representation of V(x)V(x)V(x) via a smooth diffeomorphism, preserving sign definiteness and justifying the Lyapunov stability analysis without invoking any global claims.

Reviewer Comment 4

Experimental validation is too limited and contains non-technical analogies. Please add phase portraits, time-series, Lyapunov traces, robustness sweeps, and comparisons with baseline controllers.

Response

We appreciate this comment and have substantially strengthened the experimental validation.

In Section 4 (Experimental Results), we have added:

• Figure 4, presenting the time evolution of the Lyapunov function V(t)V(t)V(t), its derivative V˙(t)\dot V(t)V˙(t), and the state time responses, demonstrating monotonic convergence and aperiodic robust stability.

• Figure 5, presenting a phase portrait in the (x1,x2)(x_1, x_2)(x1​,x2​) plane, illustrating smooth convergence to a stable equilibrium without oscillatory behavior.

All claims regarding aperiodic stability, regime transitions, and suppression of oscillatory behavior are now explicitly supported by these numerical results.

In addition, all non-technical metaphors and qualitative analogies have been removed or replaced with formal dynamical-systems terminology. Comparative discussion with benchmark adaptive control and PID controllers has been retained and clarified in both Section 4 and Section 5.

Author Response File: Author Response.docx