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

Abstract

Nonlinear single-input single-output (SISO) systems operating under parametric uncertainty often exhibit bifurcations, multistability, and deterministic chaos, which significantly limit the effectiveness of classical linear, adaptive, and switching control methods. This paper proposes a novel synthesis framework for self-organizing control systems based on catastrophe theory, specifically within the class of elliptic catastrophes. Unlike conventional approaches that stabilize a predefined system structure, the proposed method embeds the control law directly into a structurally stable catastrophe model, enabling autonomous bifurcation-driven transitions between stable equilibria. The synthesis procedure is formulated using a Lyapunov vector-function gradient–velocity method, which guarantees aperiodic robust stability under parametric uncertainty. The definiteness of the Lyapunov functions is established using Morse’s lemma, providing a rigorous stability foundation. To support practical implementation, a data-driven parameter tuning mechanism based on self-organizing maps (SOM) is integrated, allowing adaptive adjustment of controller coefficients while preserving Lyapunov stability conditions. Simulation results demonstrate suppression of chaotic regimes, smooth bifurcation-induced transitions between stable operating modes, and improved transient performance compared to benchmark adaptive control schemes. The proposed framework provides a structurally robust alternative for controlling nonlinear systems in uncertain and dynamically changing environments.

1. Introduction

The control of nonlinear single-input single-output (SISO) systems under parametric uncertainty remains a fundamental challenge in modern control engineering [1,2]. Such systems frequently exhibit bifurcations, multistability, and deterministic chaos when system parameters vary, which may lead to performance degradation or complete loss of stability [3,4,5]. These phenomena are especially relevant for autonomous systems, cyber–physical systems, and nonlinear actuators operating in uncertain and dynamically changing environments [6,7].

Classical linear control techniques guarantee stability only in a local neighborhood of an equilibrium point and become ineffective near bifurcation boundaries, where qualitative changes in system dynamics occur [8]. Adaptive and gain-scheduling approaches rely on continuous parameter adjustment within a fixed controller structure and therefore cannot adequately handle structural changes in the number or nature of equilibria [9,10]. Switching and hybrid control methods introduce supervisory logic to manage multiple operating modes, but they are often sensitive to model uncertainty, prone to instability during mode transitions, and lack guarantees of structural stability [11].

Catastrophe theory provides a mathematical framework for describing qualitative changes in system behavior caused by smooth variations in parameters [12]. Within this framework, certain classes of catastrophes exhibit structural stability, meaning that their qualitative behavior is preserved under small perturbations. In particular, elliptic catastrophes represent a class of structurally stable configurations in which equilibrium branches form bounded and symmetric manifolds in the phase space [13,14]. These properties make elliptic catastrophes especially attractive for control synthesis, as they naturally accommodate bifurcation-driven transitions between stable operating regimes while preserving robustness to parameter uncertainty.

In this work, self-organization is understood as autonomous structural adaptation of the control system through bifurcation-driven equilibrium selection, occurring without external supervisory logic or predefined switching rules [15,16]. Unlike adaptive control, which modifies controller parameters within a fixed structural framework, the proposed approach allows the structure of stable operating regimes itself to evolve as system parameters change. As a result, regime transitions emerge as intrinsic dynamical processes rather than externally imposed control actions.

The main contributions of this paper are as follows:

• a novel synthesis framework for self-organizing control of nonlinear SISO systems based on elliptic catastrophe theory;
• a Lyapunov vector-function gradient–velocity method that guarantees aperiodic robust stability under parametric uncertainty;
• a rigorous application of Morse’s lemma to establish the definiteness of Lyapunov functions in the control-theoretic context;
• a data-driven parameter tuning mechanism based on self-organizing maps (SOM) constrained by Lyapunov stability conditions;
• comparative numerical validation against benchmark adaptive control methods.

The remainder of the paper is organized as follows. Section 2 reviews related research on nonlinear, adaptive, and self-organizing control systems. Section 3 presents the mathematical foundations of the proposed synthesis method. Section 4 provides numerical experiments and comparative analysis. Section 5 discusses the implications and limitations of the approach, and Section 6 concludes the paper.

2. Related Research

The stages of rapid development in the field of automation, space technologies, and control of complex dynamic systems have led to many new challenges and discoveries in control theory. The problem of controlling nonlinear objects is particularly acute, as their dynamics differ significantly from linear systems both in terms of complexity of behavior and sensitivity to external influences and internal parametric disturbances.

One of the effective approaches to solving such problems is the use of self-organizing control systems. This approach is based on the ability of the system to rebuild its behavior based on internal feedback mechanisms, without explicit external programming. It is especially important to apply these principles in the context of managing objects with one input and output in conditions of parametric uncertainty, when traditional regulators lose stability and accuracy.

The theoretical basis for the synthesis of self—organizing systems is the theory of catastrophes, namely, the class “elliptical dynamics”, within which it is possible to formalize the processes of bifurcation, loss of stability and transition to a new stable state. Mathematical structures such as stable maps have an important property of structural stability—the ability to preserve the qualitative characteristics of the system’s behavior with small changes in parameters [6]. This is critically important for systems operating in unstable or poorly predictable external conditions.

Previously, similar approaches were used in the theory of dynamical systems, topology, and nonlinear differential equations [7]. However, the integration of the principles of structural stability and self-organization directly into the synthesis of regulators for nonlinear technical objects remains a relatively new and little-explored area.

Attempts have been made in the scientific literature to analyze the stability and suppression of chaotic regimes using Lyapunov functions, phase portrait methods, and numerical modeling [8]. However, most of these approaches are focused on evaluating already built systems, rather than on a formal synthesis with guaranteed quality of transients and robust stability [9]. Some works have raised the issue of constructing Lyapunov functions, but a universal method of their synthesis for self-organizing systems has not yet been proposed [10,11,12].

It is in this context that the contribution of this study is found: the proposed gradient–velocity method allows not only to construct Lyapunov vector functions based on equations of state, but also to determine the regions of aperiodic robust stability for both the model and the control system. Moreover, a procedure has been proposed for matching these areas in order to calculate the coefficients of a regulator that ensures the required control quality [13,14].

It is important to note that the importance of this approach is not limited to management theory. The principles of self-organization are widely used in related fields, from bioinspired algorithms to neural networks and autonomous transport systems [15]. Thus, the developed methodology can be used as the basis for a more general approach to managing complex adaptive systems operating under conditions of uncertainty.

In the following sections of the article, the mathematical foundations of the proposed approach, model construction, stability analysis and the results of calculations of the regulator coefficients will be considered [16,17]. This will allow us to form a holistic view of the possibilities of synthesizing self-organizing control systems based on disaster theory and their application in engineering practice.

3. Materials and Methods

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.

The nonlinear terms in the proposed state equations represent intrinsic energy exchange between system states and are responsible for bifurcation behavior. Cubic terms model saturation and nonlinear stiffness, while quadratic coupling terms capture cross-state interactions. The elliptic catastrophe structure ensures bounded trajectories and prevents divergence even under parameter variations.

3.1. Synthesis of a Self-Organizing Control System in the Class of Disasters “Elliptical Dynamics”

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.

Modern research in the field of nonlinear control systems emphasizes the importance of developing management strategies that can maintain stability in conditions of parametric uncertainty. Deterministic chaos and instability that occur when parameters exceed the boundaries of the robust stability domain represent a special class of structural uncertainty models. In this context, robust stability is defined as the ability of a system to maintain asymptotic behavior under the influence of unpredictable disturbances.

To eliminate the modes of chaos and instability, this paper considers the problem of synthesizing a self-organizing control system built in the class of disasters “elliptical dynamics”. It is assumed that there are several stationary states, each of which may lose or gain stability as a result of bifurcation processes when the parameters change [18]. When a bifurcation occurs, the control system is able to independently move from a state that is losing stability to a new one with robust stability.

3.2. Definitions and Stability Notions

In this study, parametric uncertainty is modeled as bounded variations in system parameters within a compact set:

$$D=\left\{d:\left|d_i-d_i^0\right|\le {∆}_i, i=1, \dots , n\right\},$$

where $d_i^0$ denotes nominal parameter values and ${∆}_i>0$ specifies admissible uncertainty bounds.

The equilibria of the considered nonlinear SISO system correspond to stationary solutions of the state equations and are explicitly given by (1), (7), (25), and (29). These equilibria may lose or gain stability as system parameters vary, giving rise to bifurcation-driven transitions.

The region of attraction is defined as the maximal neighborhood of an equilibrium point in which the constructed Lyapunov function V(x) is positive definite and its time derivative satisfies V(x) < 0 for all admissible uncertainties d ∈ D.

Throughout this paper, aperiodic robust stability is understood as asymptotic Lyapunov stability characterized by monotonic (non-oscillatory) convergence of system trajectories, absence of limit cycles and sustained oscillations, and uniform validity of the condition V(x) < 0 over the entire uncertainty set.

3.3. Methodology of System Synthesis

The problem of synthesizing a self-organizing control system with one input and one output in the class of elliptical dynamics disasters is solved in three stages:

-

Selection and construction of a model with desired transients, with aperiodic robust stability.

-

Analysis and determination of the conditions of aperiodic robust stability of a self-organizing control system with one input and one output in the class of disasters “elliptical dynamics”.

-

Calculation of the regulator coefficients that ensure the fulfillment of stability conditions for both the model and the real control system.

3.4. Mathematical Model of a Self-Organizing System

A system with one input and one output is considered, the behavior of which is described by the following system:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3\\ \dots \\ {\dot{x}}_{n-1}=x_n\\ {\dot{x}}_n=-x_2^3-3x_2{x}_1^2-d_{12}\left(x_1^2+x_2^2\right)+d_1^1{x}_1+d_1^2{x}_2,\dots , -x_n^3\\ -3x_n{x}_{n-1}^2-d_{n-1,n}\left(x_{n-1}^2+x_n^2\right)+d_{n-1}^1{x}_{n-1}+d_{n-1}^2{x}_n,\end{array}\right.$$

(1)

where x(t) $\in R^n$ is the state vector of the model with the desired transients.

$d_{i,i+1}, d_i^1, d_i^2, i=1, \dots ,$ n − 1, are known parameters of the model, which are determined by the results of a simulation experiment on the model for the quality of the desired transition process.

A self-organizing model (1) with the desired dynamics, constructed in the class of disasters “elliptical dynamics”, and with the specified parameters has a stationary state as a solution to a system of algebraic equations:

$$\left\{\begin{array}{c}{x}_2=0,\hspace{1em} x_3=0, \dots , x_n=0\\ -x_2^3-3x_2{x}_1^2-d_{21}\left(x_1^2+x_2^2\right)+d_1^1{x}_1+d_1^2{x}_2,\dots , -x_n^3-3x_n{x}_{n-1}^2\\ -d_{n-1,n}\left(x_{n-1}^2+x_n^2\right)+d_{n-1}^1{x}_{n-1}+d_{n-1}^2{x}_n=0\end{array}\right.$$

(2)

A trivial solution to the system of algebraic Equation (2) is

$$x_1=0,\hspace{1em}{x}_2=0, \dots , x_n=0$$

(3)

Other stationary states of the model are determined from the critical points of the disaster functions; the “elliptical harmonic” is denoted by the function:

$$\begin{gathered} F_k\left(x_i,x_{\left(i+1\right)},d_{i,i+1,}{d}_i^2,\right)=-x_{i+1}^3-{3x}_{i+1}{x}_i^2 \\ -d_{i,i+1}\left(x_{i+1}^2+x_i^2\right)+d_i^1{x}_i+d_i^2{x}_{i+1} \end{gathered}$$

The critical points $F_k$ are determined from the following equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial F}_k}{{\partial x}_i}}=-6x_i{x}_{i+1}-2d_{i,i+1}{x}_i+d_i^1=0,i=1, \dots , n-1, k=1,\dots ,n, \\ {\displaystyle \frac{{\partial F}_k}{{\partial x}_{i+1}}}-3x_{i+1}^2-2d_{i,i+1}{x}_{i+1}+d_i^2\ne 0, i=1,\dots , n-1, k=1, \dots , n, \end{array}\right.$$

(4)

$${\displaystyle \frac{{{\partial }^{2}F}_k}{{\partial x}_i^2}}=-3x_{i+1}-d_{i,i+1},{\displaystyle \frac{{{\partial }^{2}F}_k}{{\partial x}_i{\partial x}_{i+1}}}=-3x_i$$

$${\displaystyle \frac{{{\partial }^{2}F}_k}{{\partial x}_{i+1}{\partial x}_i}}=-3x_i,{\displaystyle \frac{{{\partial }^{2}F}_k}{{\partial x}_{i+1}^2}}=-3x_{i+1}-d_{i,i+1}$$

$$F_{ij}=‖\begin{array}{c}-\left(3x_{i+1}+d_{i,i+1}\right)-3x_i\\ -3x_i-\left(3x_{i+1}+d_{i,i+1}\right) \end{array}‖$$

When all the elements of the stability matrix [19] vanish corresponds to three times degenerate critical points. Doubly degenerate critical points can be found from the condition that at least one eigenvalue of the stability matrix vanishes. If this is the case, then the determinant of the stability matrix is zero, i.e.,

$$det{F}_{ij}^i=(3x_{i+1}+d_{i,i+1}{)}^2-9x_i^2=0$$

From here, we can write

$$3x_i=3x_{i+1}+d_{i,i+1}, d_{i,i+1}=3\left(x_i-x_{i+1}\right)$$

(5)

From (4), taking into account (5), we obtain

$$\left\{\begin{array}{c}{-6x}_i{x}_{i+1}-{6x}_i^2+6x_i{x}_{,i+1}+d_i^1=0\\ -3x_{i+1}^2-6x_i{x}_{i+1}{+6x}_{i+1}^2+d_i^2=0,\end{array}\right.$$

(6)

From (6), we obtain

$$\left\{\begin{array}{c}{x}_i^{2,3}=\pm \sqrt{{\displaystyle \frac{1}{6}}{d}_i^1}={\mathrm{A}}_i,\\ x_{i+1}^{2,3}=\sqrt{{\displaystyle \frac{1}{6}}{d}_i^1}\pm \sqrt{{\displaystyle \frac{1}{6}}{d}_i^1-{\displaystyle \frac{1}{6}}{d}_i^2}=B_i,\\ d_{i,i+1}=\pm 3\sqrt{{\displaystyle \frac{1}{6}}{d}_i^1-{\displaystyle \frac{1}{3}}{d}_i^2}=C_i\end{array} \right.$$

(7)

Here, the system of nonlinear algebraic Equation (2) has a trivial solution (3) and other solutions (7) for $d_i^1$ > 0 and ${\displaystyle \frac{1}{6}}{d}_i^1-{\displaystyle \frac{1}{3}}{d}_i^2>0$, i = 1, …, n. With negative values of $d_i^1$ < 0 and ${\displaystyle \frac{1}{6}}{d}_i^1-{\displaystyle \frac{1}{3}}{d}_i^2<0$, the system of Equation (4) has imaginary solutions that cannot correspond to any physically possible situation [20]. Solutions (7) merge with (3) when $d_i^1$ = 0 and ${\displaystyle \frac{1}{6}}{d}_i^1-{\displaystyle \frac{1}{3}}{d}_i^2=0$, i = 1, …, n, i.e., at the point $d_i^1>0$ and ${\displaystyle \frac{1}{6}}{d}_i^1-{\displaystyle \frac{1}{3}}{d}_i^2=0$, i = 1, …, n, a bifurcation occurs. In other words, the branches $x_i^{2,3}$ and $x_{i+1}^{2,3}$ appear as a result of a “bifurcation” at the moment when the state (3) loses stability, and these branches (7) themselves are also stable. These statements are verified by the gradient velocity method of Lyapunov vector functions.

• The apperiodic robust stability of the stationary state of the model is investigated (3). From the equation of state of the model (1), the components of the gradient vector from the vector of Lyapunov functions are determined $V\left(x\right)={(V}_1\left(x\right),\dots ,V_n(x))$:

  $$\left\{\begin{array}{c}{\displaystyle \frac{{\partial V}_1(x)}{{\partial x}_{i+1}}}=-x_{i+1},i=1,\dots , n-1;\\ {\displaystyle \frac{{\partial V}_n\left(x\right)}{{\partial x}_i}}={3x}_{i+1}{x}_1^2+d_{i,i+1}{x}_1^2-d_i^1{x}_i, \\ {\displaystyle \frac{{\partial V}_n(x)}{{\partial x}_{i+1}}}=x_{i+1}^3+d_{i,i+1}{x}_{i+1}^2-d_i^2{x}_{i+1}, i=1,3,\dots , n-1\end{array}\right.$$

  (8)

The components of the decomposition of the velocity vector along the coordinates of the model are determined from the equation of the state of the model (x₁,…, x_n):

$$\left\{\begin{array}{c}{\left({\displaystyle \frac{{dx}_1}{dt}}\right)}_{x_{i+1}}=x_{i+1},i=1,\dots ,n-1\\ {\left({\displaystyle \frac{{dx}_1}{dt}}\right)}_{x_n}={-3x}_{i+1}{x}_i^2-d_{i,i+1}{x}_1^2+d_i^1{x}_i;\\ {\left({\displaystyle \frac{{dx}_1}{dt}}\right)}_{x_{i+1}}=-x_{i+1}^3-d_{i,i+1}{x}_{i+1}^2+d_i^2{x}_{i+1}, i=1,3,\dots ,n-1\end{array}\right.$$

(9)

The total time derivative of the vector of Lyapunov functions V(x) is defined as the scalar product of the gradient vector from the vector of the Lyapunov function (6) by the velocity vector (7):

$$\begin{gathered} \left({\displaystyle \frac{dV(x)}{dt}}\right)=\sum _{i=1}^{n-1}-{(x_{i+1})}^2-\sum _{i=1,3,}^{n-1}{(3x_{i+1}{x}_i^2+d_{i,i+1}{x}_i^2-d_i^1{x}_i)}^2 \\ -{\sum }_{i=1,3,}^{n-1}{(x_{i+1}^3+d_{i,i+1}{x}_{i+1}^2-d_i^2{x}_{i+1})}^2, \end{gathered}$$

(10)

It follows from (10) that the total time derivative of the Lyapunov vector functions will always be a sign-negative function, i.e., a sufficient condition for aperiodic robust stability for the stationary state (3) will always be satisfied. Then, from (6), we can construct a vector of Lyapunov functions in scalar form:

$$\begin{array}{c}\hfill V(x) = x_2{x}_i^2+{\displaystyle \frac{1}{3}}{d}_{12}{x}_1^3-{\displaystyle \frac{1}{2}}{d}_1^1{x}_1^2+{\displaystyle \frac{1}{4}}{x}_2^4+{\displaystyle \frac{1}{3}}{d}_{12}{x}_2^3-{\displaystyle \frac{1}{2}}{(d}_1^2+1)x_2^2+,\dots ,+x_n{x}_{n-1}^3\\ \hfill +{\displaystyle \frac{1}{3}}{d}_{n-1,n}{x}_{n-1}^3-{\displaystyle \frac{1}{2}}{(d}_{n-1}^1+1)x_{n-1}^2+{\displaystyle \frac{1}{4}}{x}_n^4+{\displaystyle \frac{1}{3}}{d}_{n-1,n}{x}_{n-1}^3-{\displaystyle \frac{1}{2}}{(d}_{n-1}^1+1)x_n^2,\end{array}$$

(11)

At this stage, the definiteness of the Lyapunov candidate function (11) is not immediately apparent due to the presence of higher-order nonlinear terms. To rigorously justify its local definiteness, Morse’s lemma is applied.

Morse’s lemma is applied locally in a neighborhood of isolated equilibria and is not used to claim global stability. The application of the lemma requires the following conditions to be satisfied:

(i)

The Lyapunov function satisfies V(x) ∈ C² in a neighborhood of the equilibrium point x*;

(ii)

The equilibrium x* is a critical point of V(x), i.e., ∇V(x*) = 0;

(iii)

The Hessian matrix ∇²V(x*) is nonsingular, i.e., det(∇²V(x*)) ≠ 0.

All these conditions are satisfied for the Lyapunov candidate functions constructed in this work, since they are polynomial functions with isolated equilibria and non-degenerate Hessian matrices at the equilibrium points.

This allows a local quadratic representation of V(x) via a smooth coordinate diffeomorphism, preserving the sign definiteness of the Lyapunov function in a neighborhood of the equilibrium.

Morse’s lemma is employed to locally transform the Lyapunov candidate function into an equivalent quadratic form via a smooth coordinate diffeomorphism. This transformation preserves qualitative stability properties and explicitly reveals the definiteness of the Lyapunov function in a neighborhood of equilibrium, enabling rigorous verification of aperiodic robust stability. From (11), the condition of positive or negative certainty of functions is not obvious; therefore, the Morse Lemma from catastrophe theories is applied to functions (11). Then, functions (11) can be transformed into a square shape:

$$V(x) = -{\displaystyle \frac{1}{2}}{d}_1^1{x}_1^2-{\displaystyle \frac{1}{2}}{(d}_1^2+1)x_2^2+,\dots ,-{\displaystyle \frac{1}{2}}{(d}_{n-1}^1+1)x_{n-1}^2-{\displaystyle \frac{1}{2}}{(d}_{n-1}^1+1)x_n^2,$$

(12)

From (12), the condition of positive certainty is written

$${-d}_1^1>0,-{(d}_1^2+1)>0,\dots , -{(d}_{n-1}^1+1)>0,-{(d}_{n-1}^2+1)>0$$

(13)

The domain of aperiodic robust stability of the stationary state (3) is determined by the system of inequalities (13).

2.

The aperiodic robust stability of the stationary state (7) is investigated. For this equation of state, the model (1) is represented in deviations relative to the stationary state (7):

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3\\ \dots \\ {\dot{x}}_{n-1}=x_n\\ {\dot{x}}_n=-x_2^3-3x_2{x}_1^2-3A_1{x}_1^2-3B_1{x}_{1 }^2-{\displaystyle \frac{3}{2}}{C}_1{x}_1{x}_2+d_1^1{x}_1-d_1^2{x}_2,\dots , \\ -x_n^3-3x_n{x}_{n-1}^2-{3A_{n-1}{x}_{n-1}^2-3B_{n-1}{x}_{n }^2-{\displaystyle \frac{3}{2}}{C}_{n-1}{x}_n{x}_{n-1}-d_{n-1}^{1}x}_{n-1}-d_{n-1}^2{x}_n,\end{array}\right.$$

(14)

The components of the gradient vector from the vector of Lyapunov functions are determined from the equation of state in deviations (14), $V_1\left(x\right),\dots ,V_n\left(x\right)):$

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial V}_i(x)}{{\partial x}_{i+1}}}=-x_{i+1},i=1,\dots , n-1;\\ {\displaystyle \frac{{\partial V}_n\left(x\right)}{{\partial x}_i}}= {3x}_{i+1}{x}_1^2+3A_i{x}_i^2-{\displaystyle \frac{3}{2}}{C}_i{x}_i{x}_{i+1}+d_i^1{x}_i, \\ {\displaystyle \frac{{\partial V}_n(x)}{{\partial x}_{i+1}}}=x_{i+1}^3+3B_i{x}_{i+1}^2-d_i^2{x}_{i+1}, i=1,3,\dots , n-1\end{array}\right.$$

(15)

From the equation of state in deviations (14), the components of the decomposition of the velocity vector along the coordinates of the system are determined ($x_i,\dots ,x_n):$

$$\left\{\begin{array}{c}{\left({\displaystyle \frac{{dx}_i}{dt}}\right)}_{x_{i+1}}=x_{i+1},i=1,\dots ,n-1;\\ {\left({\displaystyle \frac{{dx}_i}{dt}}\right)}_{x_i}={-(3x}_{i+1}{x}_i^2+3A_i{x}_i^2+{\displaystyle \frac{3}{2}}{C}_i{x}_i{x}_{i+1}+d_i^1{x}_i);\\ {\left({\displaystyle \frac{{dx}_i}{dt}}\right)}_{x_{i+1}}=-{(x}_{i+1}^3+3B_i{x}_{i+1}^2+d_i^2{x}_{i+1}), i=1,3,\dots ,n-1\end{array}\right.$$

(16)

The total time derivative of the Lyapunov vector of functions is defined as the scalar product of the gradient vector (15) and the velocity vector (16):

$$\begin{array}{c}{\displaystyle \frac{dV(x)}{dt}}=-{\sum }_{i=1,3,}^{n-1}{\left(3x_{i+1}{x}_i^2+3A_i{x}_i^2+{\displaystyle \frac{3}{2}}{C}_i{x}_i{x}_{i+1}+d_i^1{x}_i\right)}^2\hfill \\ -{\sum }_{i=1,3,}^{n-1}{(x_{i+1}^3+3B_i{x}_{i+1}^2+d_i^2{x}_{i+1})}^2-{\sum }_{i=2}^{n-1}{(x_{i+1})}^2,\hfill \end{array}$$

(17)

It follows from (17) that the total time derivative of the Lyapunov vector functions will always be a sign-negative function, i.e., a sufficient condition for aperiodic robust stability for the stationary state (7) will always be satisfied.

Then, from (15), we can construct a vector of Lyapunov functions in scalar form:

$$\begin{array}{c}\hfill V(x) = x_2{x}_1^3+A_1{x}_1^3+{\displaystyle \frac{3}{4}}{C}_1{x}_1^2{x}_2+{\displaystyle \frac{1}{2}}{d}_1^1{x}_1^2+{\displaystyle \frac{1}{4}}{x}_2+B_1{x}_2^3\hfill \\ \hfill +{\displaystyle \frac{1}{2}}{(d}_1^2-1)x_2^2+,\dots ,+x_n{x}_{n-1}^3+A_{n-1,n}{x}_{n-1}^3+{\displaystyle \frac{3}{4}}{C}_{n-1}{x}_n{x}_{n-1}^2+{\displaystyle \frac{1}{2}}{(d}_{n-1}^1-1)x_n^2\hfill \\ +B_{n-1}{x}_n^3+{\displaystyle \frac{1}{2}}{(d}_{n-1}^2-1)x_n^2,\hfill \end{array}$$

(18)

From (18), the condition of positive definiteness of the vector of Lyapunov functions is not obvious. But all the conditions of Morse’s Lemma are fulfilled by one of the theories of catastrophes. The function (18) turns to zero at the origin and continuously differentiates. Therefore, the function (18) around the origin can be represented in a quadratic form:

$$V(x)={\displaystyle \frac{1}{2}}{d}_1^1{x}_1^2+{\displaystyle \frac{1}{2}}{(d}_1^2-1)x_2^2+,\dots ,+ {\displaystyle \frac{1}{2}}{(d}_{n-1}^1-1)x_{n-1}^2+ {\displaystyle \frac{1}{2}}{(d}_{n-1}^2-1)x_n^2,$$

(19)

The domain of aperiodic robust stability of the stationary state (7) is determined by a system of inequalities.

$$d_1^1>0, d_1^2-1>0,\dots ,d_{n-1}^1-1>0, d_{n-1}^2-1>0,$$

(20)

2.1.

Investigation of the Conditions of Aperiodic Robust Stability of a System with One Input and One Output

Let the state vector of the control object be fully measured and the control system have one input and one output. The dynamics of a stationary system with linear objects is described by the equation

$$\dot{x}=Ax+Bu,x\left(t\right)\in R^n,u\left(t\right)\in R^1$$

(21)

where

$$A=‖\begin{array}{ccccc}0\hfill & \hfill 1& \hfill 0& \hfill \dots \hfill & \hfill 0\\ 0\hfill & \hfill 0& \hfill 1& \hfill \dots \hfill & \hfill 0\\ \hfill \dots \hfill & & \hfill \dots \hfill & & \hfill \dots \hfill \\ 0\hfill & \hfill 0& \hfill 0& \hfill \dots \hfill & \hfill 1\\ a_n\hfill & \hfill a_{n-1}& \hfill a_{n-2}& \hfill \dots \hfill & \hfill a_1\end{array}‖, B=‖\begin{array}{c}0\\ 0\\ \dots \\ 0\\ b_n\end{array}‖$$

A self-organizing control system with one input and one output in the class of disasters elliptical optics is considered:

$$u_i\left(t\right)=-x_{i+1}^3-{3x}_{i+1}{x}_i^2-k_{i,i+1}\left(x_i^2+x_{i+1}^2\right)+k_i^1{x}_i+k_i^2{x}_{i+1},$$

(22)

The system (21) in its expanded form, taking into account the law of management (22), is written as

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3\\ \dots \\ {\dot{x}}_{n-1}=x_n\\ {\dot{x}}_n=b_n[-x_2^3-3x_2{x}_1^2-k_{12}\left(x_1^2+x_2^2\right)+{(k}_1^1+{\displaystyle \frac{a_n}{b_n}})x_1+{(k}_1^2+{\displaystyle \frac{a_{n-1}}{b_n}})x_2] \\ +,\dots ,+b_n[-x_n^3-3x_n{x}_{n-1}^2-k_{n-1,n}\left(x_{n-1}^2+x_n^2\right)+(k_{n-1}^1+{\displaystyle \frac{a_2}{b_n}})x_{n-1}\\ +{(k}_{n-1}^2+{\displaystyle \frac{a_1}{b_n}})x_n],\end{array}\right.$$

(23)

The stationary states of the system (23) will be determined by solving a system of algebraic equations:

$$\left\{\begin{array}{c}{x}_i=0,i=2,\dots ,n\\ -x_{i+1}^3-3x_{i+1}{x}_i^2-k_{i,i+1}\left(x_1^2+x_{i+1}^2\right)+{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_i\\ +{(k}_1^2+{\displaystyle \frac{a_{n-i+2}}{b_n}})x_{i+1}= 0,i=1,\dots ,n-1\end{array}\right.$$

(24)

A trivial solution to the system of algebraic Equation (24) is

$$x_i=0,i=2,\dots ,n$$

(25)

Other solutions are determined from the system of equations

$$\begin{gathered} -x_{i+1}^3-3x_{i+1}{x}_i^2-k_{i,i+1}\left(x_1^2+x_{i+1}^2\right)+{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_i \\ +{(k}_1^2+{\displaystyle \frac{a_{n-i+2}}{b_n}})x_{i+1}=0,i=1,\dots ,n-1 \end{gathered}$$

The function is denoted as

$$\begin{array}{c}\hfill F_k=-x_{i+1}^3-3x_{i+1}{x}_i^2-k_{i,i+1}\left(x_1^2+x_{i+1}^2\right)+{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_i\\ +{(k}_1^2+{\displaystyle \frac{a_{n-i+2}}{b_n}})x_{i+1}=0,i=1,\dots ,n-1, k=1,\dots ,n\hfill \end{array}$$

The critical points are determined from the equations

$${\displaystyle \frac{{\partial F}_k}{{\partial x}_i}}=-6x_{i+1}{x}_i-2k_{i,i+1}{x}_i+k_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}}=0,i=1, \dots , n-1, k=1,3,\dots ,n-1,$$

(26)

$${\displaystyle \frac{{\partial F}_k}{{\partial x}_i}}=-3x_{i+1}^2-3x_i^2-2k_{i,i+1}{x}_{i+1}+k_i^2+k_1^2+{\displaystyle \frac{a_{n-i+2}}{b_n}}=0$$

$$\left\{\begin{array}{c}{\displaystyle \frac{{{\partial }^{2}F}_k}{\partial x_i^2}}=-6x_{i+1}-2k_{i,i+1},{\displaystyle \frac{{{\partial }^{2}F}_k}{\partial x_i\partial x_{i+1}}}=-6x_i,{\displaystyle \frac{{{\partial }^{2}F}_k}{\partial x_{i+1}^2}}=6x_{i+1}-2k_{i,i+1}, \\ {\displaystyle \frac{{{\partial }^{2}F}_k}{\partial x_{i+1}\partial x_i}}=-6x_{i+1}, i=1,\dots , n-1, k=1, \dots , n, \end{array}\right.$$

(27)

The stability matrix is determined as

$$F_{ij}=‖\begin{array}{c}{-(3x}_{i+1}+k_{i,i+1})\\ {-3x}_i\end{array}\begin{array}{c}{-3x}_i\\ {-(3x}_{i+1}+k_{i,i+1})\end{array}‖$$

When all the elements of the stability matrix vanish, three times degenerate critical points appear. Doubly degenerate critical points can be found from the condition that at least one eigenvalue of the stability matrix vanishes. If this is the case, then the determinant of the stability matrix is zero.

$${detF}_{ij}={{(3x}_{i+1}-k_{i,i+1})}^2-{9x}_i^2=0, i=1,\dots ,n-1$$

From here, you can write

$$k_{i,i+1}=3{(x}_i-x_{i+1}), i=1,\dots , n-1$$

(28)

Other solutions of the system of algebraic Equation (24), taking into account (28), are calculated from (26) and (27):

$$\left\{\begin{array}{c}{x}_i^{2,3}=\pm \sqrt{{\displaystyle \frac{1}{6}}{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})}={\mathrm{A}}_i,\hspace{1em}\hspace{1em}i=1,3,\dots ,n-1\\ x_{i+1}^{2,3}=\pm \sqrt{{\displaystyle \frac{1}{6}}{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})}\pm \sqrt{{\displaystyle \frac{1}{3}}{[(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})-{(k}_i^2+{\displaystyle \frac{a_{n-i+2}}{b_n}})] }=B_i,\\ k_{i,i+1}=\pm \sqrt{{\displaystyle \frac{1}{3}}{[(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})-{(k}_i^2+{\displaystyle \frac{a_{n-i+2}}{b_n}})] }=C_i, i=1,3,\dots ,n-1\end{array} \right.$$

(29)

Here, the system of nonlinear algebraic Equation (24) has a trivial solution (25) and a nontrivial solution (29) for

$$k_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}}>0, {(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})-\left(k_i^2+{\displaystyle \frac{a_{n-i+2}}{b_n}}\right)>0,i=1,3,\dots ,n-1$$

Equation (24) has an imaginary solution, which cannot correspond to any physically possible situation. Solution (29) merges with (25) when $k_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}}$ = 0 и ${(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})-\left(k_i^2+{\displaystyle \frac{a_{n-i+2}}{b_n}}\right)=0,i=1,3,\dots ,n-1$ and branch off from it when $k_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}}$> 0 и ${(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})-\left(k_i^2+{\displaystyle \frac{a_{n-i+2}}{b_n}}\right)>0, i=1,3,\dots ,n-1$. At the point $k_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}}$> 0 и ${(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})-\left(k_i^2+{\displaystyle \frac{a_{n-i+2}}{b_n}}\right)>0,i=1,\dots ,n-1,$ a “bifurcation” occurs. In other words, branches $x_i^{2,3}$, $x_{i+1}^{2,3}$ appear as a result of “bifurcations” at the moment when the state (25) loses stability, and these branches themselves are stable. Consequently, self-organization occurs in the system (23). These statements are verified using the Lyapunov vector-function gradient–velocity method.

2.2

The aperiodic robust stability of the stationary state (25) is investigated. The components of the gradient vector from the vector of Lyapunov functions are determined from the equation of state of the control system (23) V(x) = ($V_1\left(x\right),\dots , V_n(x)$):

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial V}_i(x)}{{\partial x}_{i+1}}}=-x_{i+1}, i=1,\dots ,n-1 \\ {\displaystyle \frac{{\partial V}_n\left(x\right)}{{\partial x}_i}}=b_n(3x_{i+1}{x}_i^2+k_{i,i+1}{x}_1^2-{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_i), i=1,3,\dots ,n-1\\ {\displaystyle \frac{{\partial V}_n\left(x\right)}{{\partial x}_{i+1}}}=b_n(x_{i+1}^3+k_{i,i+1}{x}_{i+1}^2-{(k}_i^2+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_{i+1}), i=1,3,\dots ,n-1\end{array}\right.$$

(30)

From the equation of state of the control system (23), the components of the decomposition of the velocity vector along the coordinates of the system are determined ($x_i,\dots ,x_n$):

$$\left\{\begin{array}{c}{\left({\displaystyle \frac{{dx}_i}{dt}}\right)}_{x_{i+1}}=x_{i+1}, i=1,\dots ,n-1 \\ {\left({\displaystyle \frac{{dx}_n}{dt}}\right)}_{x_i}={-b}_n(3x_{i+1}{x}_i^2+k_{i,i+1}{x}_1^2-{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_i), i=1,3,\dots ,n-1\\ {\left({\displaystyle \frac{{dx}_n}{dt}}\right)}_{x_{i+1}}=-b_n(x_{i+1}^3+k_{i,i+1}{x}_{i+1}^2-{(k}_i^2+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_{i+1}), i=1,3,\dots ,n-1\end{array}\right.$$

(31)

The total time derivative of the vector of the Lyapunov function V(x) is defined as the scalar product of the gradient vector from the vector of the Lyapunov function (2.10) by the velocity vector (31):

$$\begin{array}{c}{\displaystyle \frac{dV\left(x\right)}{dt}}=-x_2^2-x_3^2-,\dots ,-x_n^2-b_n^2{\left(3x_2{x}_1^2+k_{12}{x}_1^2-(k_1^1+{\displaystyle \frac{a_n}{b_n}})x_1\right)}^2\hfill \\ -b_n^2{\left(x_2^3+k_{12}{x}_2^2-(k_1^2+{\displaystyle \frac{a_{n-1}}{b_n}})x_2\right)}^2-,\dots , -b_n^2{\left(3x_n{x}_{n-1}^2+k_{n-1,n}{x}_{n-1}^2-(k_{n-1}^1+{\displaystyle \frac{a_2}{b_n}})x_{n-1}\right)}^2-b_n^2{\left(x_n^3+k_{n-1,n}{x}_n^2-(k_{n-1}^2+{\displaystyle \frac{a_1}{b_n}})x_n\right)}^2,\hfill \end{array}$$

(32)

From (32), the total time derivative of the vector of Lyapunov functions is always a sign-negative function, i.e., a sufficient condition for aperiodic robust stability for the stationary state (25) is fulfilled. Then, from (31), it is possible to construct a vector of Lyapunov functions in scalar form:

$$\begin{gathered} V\left(x\right)=b_n\left(x_2{x}_1^3+{\displaystyle \frac{1}{3}}{k}_{1,2}{x}_1^3-{\displaystyle \frac{1}{2}}{(k}_1^1+{\displaystyle \frac{a_n}{b_n}})\right)x_1^2 \\ +b_n\left({\displaystyle \frac{1}{4}}{x}_2^4+{\displaystyle \frac{1}{3}}{k}_{12}{x}_2^3-{\displaystyle \frac{1}{2}}{(k}_1^2+{\displaystyle \frac{a_{n-1}+1}{b_n}})x_2^2\right)+,\dots , \\ +b_n\left(x_n{x}_{n-1}^3+{\displaystyle \frac{1}{3}}{k}_{n-1,n}{x}_{n-1}^3-{\displaystyle \frac{1}{2}}{(k}_{n-1}^1+{\displaystyle \frac{a_2+1}{b_n}})x_{n-1}^2\right) \\ +b_n\left({\displaystyle \frac{1}{4}}{x}_n^4+{\displaystyle \frac{1}{3}}{k}_{n-1,n}{x}_n^3-{\displaystyle \frac{1}{2}}{(k}_{n-1}^2+{\displaystyle \frac{a_1+1}{b_n}})x_n^2\right), \end{gathered}$$

(33)

From (33), the condition of positive or negative definiteness of functions is not obvious; therefore, the Morse Lemma of catastrophe theories is applied to functions (33). Then, functions (33) can be transformed into a quadratic form:

$$\begin{gathered} V\left(x\right)=-{\displaystyle \frac{1}{2}}{(b_{n}k}_1^1+a_n)x_1^2-{\displaystyle \frac{1}{2}}{(b_{n}k}_1^2+a_{n-1}+1)x_2^2-,\dots , \\ -{\displaystyle \frac{1}{2}}{(b_{n}k}_1^2+a_2+1)x_{n-1}^2-{\displaystyle \frac{1}{2}}{(b_{n}k}_{n-1}^2+a_1+1)x_n^2, \end{gathered}$$

(34)

From (34), the condition of positive certainty is written:

$$\begin{gathered} {-(b_{n}k}_1^1+a_n)>0, {-(b_{n}k}_1^2+a_{n-1}+1)>0,\dots , {-(b_{n}k}_{n-1}^1+a_2+1)>0 \\ {-(b_{n}k}_{n-1}^2+a_1+1)>0 \end{gathered}$$

(35)

The domain of aperiodic robust stability of the stationary state (25) is determined by the system of inequalities (35).

The aperiodic robust stability of the stationary state (29) is investigated. For this, the equations of state of the control system (23) are represented in deviations relative to the stationary state (29):

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3\\ \dots \\ {\dot{x}}_{n-1}=x_n\\ {\dot{x}}_n={-b}_n(x_2^3+3x_2{x}_1^2-3A_1{x}_i^2+3B_1{x}_2^2+{\displaystyle \frac{3}{2}}{C}_1{x}_1{x}_2+{(k}_1^1+{\displaystyle \frac{a_n}{b_n}})x_1\\ +{(k}_1^2+{\displaystyle \frac{a_{n-1}}{b_n}})x_2)-,\dots ,-b_n(x_n^3+3x_n{x}_{n-1}^2-3A_{n-1}{x}_{n-1}^2+3B_{n-1}{x}_n^2\\ +{\displaystyle \frac{3}{2}}{C}_{n-1}{x}_{n-1}{x}_n+{(k}_{n-1}^1+{\displaystyle \frac{a_2}{b_n}})x_{n-1}+{(k}_{n-1}^2+{\displaystyle \frac{a_1}{b_n}})x_n).\end{array}\right.$$

(36)

The components of the gradient vector from the vector of Lyapunov functions are determined from the equation of state of the control system in deviations (36):

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial V}_i(x)}{{\partial x}_{i+1}}}=-x_{i+1}, i=1,\dots ,n-1; \\ {\displaystyle \frac{{\partial V}_n\left(x\right)}{{\partial x}_i}}=b_n(3x_{i+1}{x}_i^2+3A_i{x}_i^2+{\displaystyle \frac{3}{2}}{C}_i{x}_i{x}_{i+1}+{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_i), \\ {\displaystyle \frac{{\partial V}_n\left(x\right)}{{\partial x}_{i+1}}}=b_n(x_{i+1}^3+3B_i{x}_{i+1}^2+{(k}_i^2+{\displaystyle \frac{a_{n-i+2}}{b_n}})x_{i+1}), i=1,3,\dots ,n-1.\end{array}\right.$$

(37)

From the equation of state of the control system, the components of the expansion of the velocity vector along the coordinates of the system are determined in deviations (36) ($x_i,\dots , x_n$):

$$\left\{\begin{array}{c}{\left({\displaystyle \frac{{dx}_i}{dt}}\right)}_{x_{i+1}}=x_{i+1}, i=1,\dots ,n-1,\\ {\left({\displaystyle \frac{{dx}_n}{dt}}\right)}_{x_i}={-b}_n(3x_{i+1}{x}_i^2+3A_i{x}_i^2+{\displaystyle \frac{3}{2}}{C}_i{x}_i{x}_{i+1}+{(k}_i^1+{\displaystyle \frac{a_{n-i+1}}{b_n}})x_i), \\ {\left({\displaystyle \frac{{dx}_n}{dt}}\right)}_{x_{i+1}}=-b_n(x_{i+1}^3+3B_i{x}_{i+1}^2+{(k}_i^2+{\displaystyle \frac{a_{n-i+2}}{b_n}})x_{i+1}), i=1,3,\dots ,n-1\end{array}\right.$$

(38)

The total time derivative of the vector of the Lyapunov function V(x) is calculated by the scalar product of the gradient vector from the vector of the Lyapunov function (37) to the velocity vector (38):

$$\begin{gathered} {\displaystyle \frac{dV\left(x\right)}{dt}}=-x_2^2-x_3^2-,\dots ,-x_n^2-b_n^2(3x_2{x}_1^2+3A_1{x}_1^2+{\displaystyle \frac{3}{2}}{C}_1{x}_1{x}_2+{(k}_1^1+{\displaystyle \frac{a_n}{b_n}})x_1{)}^2 \\ \hspace{1em}-b_n^2\left(x_2^3+3B_1{x}_2^2+{(k}_1^2+{\displaystyle \frac{a_{n-1}}{b_n}}\right)x_2{)}^2-,\dots , -b_n^2(3x_n{x}_{n-1}^2+3A_{n-1}{x}_{n-1}^2+ \\ {\displaystyle \frac{3}{2}}{C}_{n-1}{x}_{n-1}{x}_n-{(k}_{n-1}^1+{\displaystyle \frac{a_2}{b_n}}){x_{n-1})}^2-b_n^2{\left(x_n^3+3B_{n-1}{x}_n^2+{(k}_{n-1}^2+{\displaystyle \frac{a_1}{b_n}})x_n\right)}^2, \end{gathered}$$

(39)

From (39), the total time derivative of the vector of Lyapunov functions is always a sign-negative function, i.e., a sufficient condition for aperiodic robust stability for the stationary state (29) is guaranteed to be fulfilled. Then, from (37), it is possible to construct a vector of Lyapunov functions in scalar form:

$$\begin{array}{c}\hfill V\left(x\right)=b_n(x_2{x}_1^3+A_1{x}_1^3+{\displaystyle \frac{3}{4}}{C}_1{x}_1^2{x}_2+{\displaystyle \frac{1}{2}}\left(k_1^1+{\displaystyle \frac{a_n}{b_n}}\right)x_1^2\hfill \\ \hfill +b_n\left({\displaystyle \frac{1}{4}}{x}_2^4+3B_1{x}_2^3+{\displaystyle \frac{1}{2}}{(k}_1^2+{\displaystyle \frac{a_{n-1}-1}{b_n}})x_2^2\right)+,\dots , \hfill \\ \hfill +b_n(x_n{x}_{n-1}^3+A_{n-1}{x}_{n-1}^3+ {\displaystyle \frac{3}{4}}{C}_{n-1}{x}_{n-1}^2{x}_n+{\displaystyle \frac{1}{2}}{(k}_{n-1}^1+{\displaystyle \frac{a_n-2}{b_n}})x_{n-2}^2)\hfill \\ \hfill {+b}_n\left({\displaystyle \frac{1}{4}}{x}_n^4+3B_{n-1}{x}_n^3+{\displaystyle \frac{1}{2}}{(k}_{n-1}^2+{\displaystyle \frac{a_1-1}{b_n}})x_n^2\right),\end{array}$$

(40)

From (40), the condition of positive or negative certainty of functions is not obvious; therefore, the Morse lemma from catastrophe theories is applied to functions (40). Then, the function (40) can be transformed into a quadratic form:

$$\begin{gathered} V\left(x\right)={{\displaystyle \frac{1}{2}}(b}_n{k}_1^1-a_n)x_1^2+{{\displaystyle \frac{1}{2}}(b}_n{k}_1^2-a_{n-1}-1)x_2^2+,\dots , \\ +{{\displaystyle \frac{1}{2}}(b}_n{k}_{n-1}^1+a_2-1)x_{n-1}^2+{{\displaystyle \frac{1}{2}}(b}_n{k}_{n-1}^2+a_1-1)x_n^2, \end{gathered}$$

(41)

From (41), the condition of positive certainty is written:

$$\begin{gathered} b_n{k}_1^1+a_n>0,b_n{k}_1^2+a_{n-1}-1>0, b_n{k}_{n-1}^1+a_2-1>0, \\ b_n{k}_{n-1}^2+a_1-1>0, \end{gathered}$$

(42)

The domain of aperiodic robust stability of the stationary state (29) is determined by the system of inequalities (42).

3.5. Calculation of the Regulator Coefficients

A new approach to solving the problem of synthesizing control systems of a given quality is proposed. The problem of synthesizing a self-organizing control system with one input and one output in the class of elliptical harmonic disasters of a given quality is solved. The task of synthesizing control systems is solved on the basis of the Lyapunov vector-function gradient–velocity method. A gradient–velocity study of the Lyapunov vector of functions makes it possible to identify areas of aperiodic robust stability of stationary states of a self-organizing model with desired transients with specified parameters and a self-organizing control system in the class of elliptical dynamics disasters with one input and one output. At the same time, models and control systems with good indicators of the quality of the transient process are distinguished into the class of aperiodically stable, for which the mathematical norm of solutions to the equations of state of the model and control system decreases monotonously aperiodically, i.e., technically, transients are aperiodic in nature.

The aperiodic robust stability of the stationary states of the model and control system also ensures self-organization and guarantees protection from instability and deterministic chaos. The specified quality indicators of a self-organizing control system are provided by the results of a simulation experiment on a self-organizing model with desired transients, with specified parameters.

The coefficients of the regulator of a self-organizing control system are calculated from the conditions of aperiodic robust stability of the stationary states of the model and control system. For this, the conditions of aperiodic robust stability (23) and (35) or (40) and (42) are equated:

$$\left\{\begin{array}{c}-(b_n{k}_1^1+a_n)={-d}_1^1\\ -\left(b_n{k}_1^2+a_{n-1}+1\right)={-(d}_1^2+1)\\ \dots \dots \dots \\ -\left(b_n{k}_{n-1}^1+a_2+1\right)={-(d}_{n-1}^1+1)\\ -\left(b_n{k}_{n-1}^2+a_1+1\right)={-(d}_{n-1}^2+1)\end{array}\right.$$

(43)

Or

$$\left\{\begin{array}{c}{b}_n{k}_1^1+a_n=d_1^1\\ b_n{k}_1^2+a_{n-1}-1=d_1^2-1\\ \dots \dots \dots \\ b_n{k}_{n-1}^1+a_2-1=d_{n-1}^1-1\\ b_n{k}_{n-1}^2+a_1-1=d_{n-1}^2-1\end{array}\right.$$

(44)

From (43) or (44), the coefficients of the regulator of a self-organizing control system with one input and one output in the disaster class elliptical are calculated:

$$\begin{gathered} k_1^1={\displaystyle \frac{d_1^1-a_n}{b_n}}, k_1^2={\displaystyle \frac{d_1^2-a_{n-1}}{b_n}},\dots ,\hspace{1em}\hspace{1em}{k}_{n-1}^1={\displaystyle \frac{d_{n-1}^1-a_2}{b_n}}, \\ k_{n-1}^2={\displaystyle \frac{d_{n-1}^2-a_1}{b_n}}, \end{gathered}$$

(45)

The synthesis task is solved unambiguously to the end. The proposed approach is described in Algorithm 1. A self-organizing system will provide the specified management qualities.

Algorithm 1. An algorithm for the synthesis of a self-organizing control system in the class of disasters “elliptical optics” with one input and one output.

Input data: - Training sample with parametric uncertainty - Set parameters of desired transients - Initial state of the system x(0) ∈ ℝⁿ Output data: - Coefficients of the regulator ki1, ki2, ki,i+1 - Stable control law u(t) Step 1. Initialization Initialize the weights of Wi randomly Initialize the structure of the Lyapunov function V(x) Set the learning parameters: speed h, attenuation of the neighborhood α Step 2. The main learning cycle Convergence conditions are not met yet: Select the input vector x ∈ ℝⁿ from the training sample Calculation of the control effect Calculate: u(t) = −xi+13 − 3xi+1·xi2 − ki,i+1(xi2 + xi+12) + ki1·xi + ki2·xi+1 Assessment of aperiodic robust stability: Calculate dV(x)/dt using the Lyapunov gradient–velocity method. Accept the current controller parameters if dV(x)/dt < 0, and proceed to the next training sample. Otherwise (dV(x)/dt ≥ 0), perform parameter adaptation using the SOM: - Find the best matching unit (BMU): ○ For each neuron i in the SOM, compute di = ||x − Wi||. ○ Select the neuron with minimum di as the BMU. - Update regulator parameters: For neurons in the neighborhood of the BMU, compute the neighborhood function h(i, BMU). ○ Update the weights: ○ ΔWi = α · η · h(i, BMU) · (x − Wi), ○ Wi ← Wi + ΔWi. - Update the Lyapunov function: Recalculate ∇V(x) using the updated parameters Wi. Verify the stability inequalities (23), (30), and (42). Step 3. Completion Criteria If for all training examples dV(x)/dt < 0: Complete the training Save the coefficients of the regulator: ki1, ki2, ki,i+1 End of the algorithm

4. Experimental Results

This section presents a numerical evaluation of the proposed self-organizing control synthesis method. The objective of the experiments is to assess the ability of the proposed approach to maintain aperiodic robust stability under parametric uncertainty and to compare its performance with benchmark control methods.

4.1. Simulation Setup

The controlled plant is a nonlinear single-input single-output (SISO) system with cubic nonlinearities consistent with the mathematical model introduced in Section 3. Parametric uncertainty is introduced by allowing independent variations in system parameters within ±20% of their nominal values. The initial conditions are randomly selected within a bounded domain of the state space to test robustness with respect to initial state variations.

Performance evaluation is based on the following metrics:

(i)

Convergence time to a stable equilibrium;

(ii)

Maximum overshoot during transients;

(iii)

Oscillation index characterizing the presence of transient oscillations;

(iv)

The sign of the Lyapunov function derivative dV/dt along system trajectories.

4.2. Benchmark Controllers

To evaluate the effectiveness of the proposed method, its performance is compared with two commonly used control strategies:

(i)

A gradient-based adaptive control scheme;

(ii)

A classical PID controller tuned using the Ziegler–Nichols method.

These benchmark controllers are selected as representative examples of parametric adaptation and fixed-structure control, respectively.

4.3. Results

The obtained results demonstrate that the proposed self-organizing control method consistently eliminates oscillatory behavior near bifurcation points and ensures monotonic convergence to stable equilibria. The Lyapunov function derivative remains strictly negative throughout the simulations, confirming aperiodic robust stability.

Across all tested uncertainty realizations and initial conditions, the proposed controller maintained monotonic convergence without overshoot or sustained oscillations.

In contrast, the adaptive control scheme exhibits noticeable transient oscillations when system parameters approach bifurcation boundaries, reflecting its sensitivity to structural changes in system dynamics in Figure 1. The PID controller fails to maintain stability under significant parameter variations, leading to divergence or sustained oscillations.

These results confirm that embedding the control law within a structurally stable elliptic catastrophe framework provides clear advantages over conventional control approaches when operating under parametric uncertainty.

These dependencies show the ability of the proposed method to adaptively and locally change the control structure in response to current environmental conditions, while maintaining the aperiodic robust stability of the system. The results obtained confirm the applicability of the proposed method to complex nonlinear control objects and correspond to theoretical expectations based on Lyapunov’s criteria and bifurcation stability.

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

Figure 3 illustrates the block diagram of the proposed self-organizing control system for a nonlinear SISO plant. The nonlinear plant is driven by a control input u(t) and produces the system state vector x(t), which is assumed to be fully measurable. The control law is generated by an elliptic catastrophe-based nonlinear controller, whose structure is defined according to the synthesis procedure described in Section 3. The controller computes the control input u(t) using the current state variables and a set of adjustable parameters. A Lyapunov stability monitoring block continuously evaluates the derivative of the Lyapunov function dV/dt based on the current system state and controller parameters. This block acts as a stability constraint, ensuring that only parameter updates preserving aperiodic robust stability are allowed. The controller parameters are adapted by a self-organizing map (SOM)-based tuning module, which processes state trajectories and uncertainty-related information. Parameter updates generated by the SOM are accepted only if the Lyapunov stability condition dV/dt < 0 is satisfied. This closed-loop architecture enables autonomous bifurcation-driven transitions between stable operating regimes without external supervisory logic, providing self-organizing behavior and robustness under parametric uncertainty.

As shown in Figure 4, the Lyapunov function decreases monotonically and its derivative remains strictly negative. The phase portrait in Figure 5 illustrates smooth convergence to a stable equilibrium without oscillatory behavior.

5. Discussion

The conducted research addresses the problem of synthesizing a self-organizing control system within the framework of elliptic catastrophe dynamics, a representative class of catastrophe theory. The discussion focuses on interpreting the obtained simulation results and highlighting the advantages of the proposed approach in comparison with conventional control methods. The choice of elliptic dynamics is motivated by their ability to capture a wide range of nonlinear and multimodal behaviors typical of single-input single-output systems operating under parametric uncertainty.

It is important to clarify the role of the self-organizing map (SOM) within the proposed control framework. The SOM is not used to optimize a cost function or to approximate a value function, nor does it aim to learn an optimal control policy. Its role is strictly restricted to constrained parameter adaptation within analytically guaranteed Lyapunov-stable regions.

This distinguishes the proposed approach from adaptive dynamic programming (ADP) and approximate optimal control methods, where learning is driven by performance optimization objectives and value-function approximation. In contrast, neuro-control and policy-learning frameworks focus on learning a control policy directly from data, often relying on persistent excitation and additional stability assumptions.

The proposed method prioritizes structural stability rather than optimality. Learning-based adaptation is permitted only insofar as it preserves the aperiodic robust stability guaranteed by the Lyapunov analysis, thereby preventing destabilizing parameter updates.

As summarized in

Table 1. Comparison of the proposed method with conventional control approaches.

Method	Handles Bifurcations	Structural Stability	Aperiodic Transients	External Switching
PID	✗	✗	✗	✗
Adaptive Control	✗	✗	±	✗
Sliding Mode	±	✗	✗	✗
Proposed Method	✓	✓	✓	✗

✓ indicates full capability, ✗ indicates no capability, and ± indicates partial or conditional capability.

, the proposed control framework uniquely combines bifurcation awareness, structural stability, and aperiodic transient behavior without relying on external switching mechanisms. This combination is not simultaneously achieved by classical PID, adaptive, or sliding-mode controllers, which typically address only subsets of these requirements.

At the synthesis stage, a qualitative apparatus of catastrophe theory was used, which makes it possible to identify stable and unstable modes of operation of a controlled object. One of the key features of the approach is the use of the elliptical dynamics model as a basis for describing the space of states and transitions between them. This made it possible not only to describe the behavior of the system in the vicinity of singular points, but also to determine the conditions under which a qualitative transition of the system from one regime to another (bifurcation) is possible.

The synthesis results showed that using the elliptical dynamics model makes it possible to build a control structure capable of self-organization and adaptation without the need for external correction of the control action. The system is endowed with the property of internal stability and the ability to dynamically respond to changing external and internal parameters. This is especially important for real control objects where a priori uncertainty and non-linearities are present.

It is worth noting that the proposed approach is applicable not only to engineering control problems but also to related domains, including biological and socio-technical systems exhibiting nonlinear and adaptive behavior. At the same time, further research is required to investigate scalability to higher-dimensional systems, robustness under measurement noise, and formal performance guarantees beyond aperiodic stability. Addressing these aspects will contribute to a more comprehensive theoretical and practical foundation for catastrophe-based self-organizing control.

6. Conclusions

Despite the remaining challenges, the synthesis of self-organizing control systems based on models of disaster theory, such as elliptical optics, opens up prospects for a deeper and more flexible understanding of complex nonlinear objects with one input and one output. The combination of nonlinear dynamics methods and catastrophic control theory ensures the creation of adaptive and stable systems capable of self-organization in conditions of uncertainty and structural changes. Further development of this field is impossible without close cooperation between specialists in the field of control theory, applied mathematics and engineering. This interdisciplinary collaboration will be the key to creating next-generation intelligent control systems.