Dynamic Risk Evolution and Adaptive Synchronization Control for Human–Machine–Environment Coupled Nuclear Emergency System: Based on Comprehensive On-Site Emergency Drills of Nuclear Power Plants

Abstract

As nuclear energy expands, nuclear emergency response systems increasingly exhibit strong human–machine–environment (H–M–E) coupling, long-duration operations, and multi-department coordination, in which minor disturbances can be amplified by feedback loops into cascading failures and loss of situational control. To address the inability of conventional static and linear methods to represent dynamic risk evolution and chaotic uncertainty, this study proposes an integrated “risk network–chaotic evolution–synchronization control” framework. Based on 12-year-old on-site comprehensive drill reports from a Chinese nuclear power base, we construct a directed H–M–E risk network in a semi-quantitative, qualitative–quantitative manner and identify critical nodes using a composite betweenness–PageRank risk metric. We further abstract the system into a three-dimensional nonlinear coupled dynamical model; phase portraits, Lyapunov exponents, and bifurcation analysis confirm threshold effects, period-doubling routes, and chaotic attractors, revealing nonlinear amplification under strong coupling. Finally, an adaptive chaotic synchronization controller driven by network coupling strength is designed. Simulations show all strategies suppress chaos and achieve synchronization, while the machine-dominated strategy offers the best speed–energy trade-off for emergency resource allocation.

1. Introduction

1.1. Background and Motivation

Nuclear emergency response serves as the final barrier in the defense-in-depth strategy for nuclear safety, encompassing emergency actions taken to control, mitigate, and alleviate the consequences of nuclear or radiological accidents. As the core component of the nuclear emergency system, the nuclear emergency response system acts as the institutional, material, and organizational vehicle supporting emergency operations. Comprising four key elements—operational mechanisms, technology, resources, and management—this system must efficiently execute the full spectrum of emergency tasks (including monitoring and early warning, situation assessment, decision-making and command, on-site response, personnel evacuation, protection organization, medical rescue, and environmental recovery) under extreme conditions characterized by high uncertainty, limited information, time constraints, and resource scarcity [1,2]. Its core objective is to ensure unobstructed emergency command chains, effective decision-making loops, and reliable organizational resilience [3,4], enabling rapid accident response and cross-organizational collaborative disposal through the synergistic empowerment of technology and management, thereby fulfilling the core requirements of nuclear emergency response [5,6]. The operational effectiveness of this system directly determines the outcome of consequence management for nuclear and radiological emergencies and serves as a critical benchmark for evaluating the overall performance of the nuclear safety system [7,8,9].

The operation of nuclear emergency response systems exhibits significant complexity, characterized by long time horizons, diverse participating entities, prominent resource constraints, multi-source heterogeneous information links with transmission delays, as well as tight coupling and multiple feedback loops among subsystems. Any delay, error, or supply-demand mismatch in a single link can be amplified through feedback mechanisms, triggering cascading risk propagation and potentially leading to systemic loss of control [10,11]. Unlike traditional risks based on discrete event probability superposition, such systemic risks stem from dynamically emerging uncertainties, vulnerability propagation, and the propensity for situational collapse arising from nonlinear interactions among internal system components [12]. Rooted in the system’s network structure and dynamic processes, the evolutionary path of such systemic risks often displays nonlinear and irreversible characteristics and may even exhibit an exponentially amplifying trend [13]. Typical cases such as the Fukushima nuclear accident have demonstrated that, under the combined effects of extreme natural disasters, multiple equipment failures, information constraints, and social pressure, traditional safety analysis frameworks centered on equipment and physical barriers can hardly fully reflect the strong coupling, strong feedback, and dynamic evolution among human, machine, and environmental factors in nuclear emergency response systems [7,10]. Therefore, as a critical vehicle for implementing the nuclear safety and emergency system [14], the focus of nuclear emergency capacity building should shift from traditional equipment and process validation to enhancing the system’s overall resilience and collaborative evolution capabilities.

At present, nuclear emergency preparedness efforts are centered on capability verification and quantitative assessment, driving the continuous improvement and iterative upgrading of nuclear emergency response systems through the analysis of drill data, process metrics, and response performance [15,16]. Nuclear emergency drills serve as a critical practical means to test, maintain, and enhance emergency capabilities and strengthen system resilience, with on-site comprehensive nuclear emergency drills in particular acting as an important basis for the proactive risk management of nuclear emergency systems [8]. The core objectives of these drills are to familiarize emergency personnel with procedures, validate equipment functionality, refine operational mechanisms, and identify potential risks, thereby ensuring a rapid and efficient response during actual accidents. A comprehensive drill typically involves hundreds of participants engaged in hours of collaborative decision-making and action, supported by large-scale monitoring data and sophisticated digital control systems under dynamically evolving plant and environmental conditions [17]. However, post-exercise reviews have repeatedly shown that a seemingly local deviation in equipment parameters, a delay in command transmission, or a lag in resource scheduling can trigger cross-domain cascading effects through intended or unintended human–machine–environment interaction interfaces, leading to global command coordination failure and response rhythm disruption in a short time [18,19,20]. This process, whereby local perturbations evolve into global disorder, epitomizes the nonlinear interactions and path-dependent characteristics of complex systems, which cannot be fully captured by static checklists or linear probability models.

Nevertheless, current mainstream drill assessment and risk analysis methods remain heavily reliant on qualitative benchmarking based on criteria checklists or subjective quantification derived from expert experience [21]. Such methods exhibit three fundamental limitations in addressing systemic risks: (i) at the structural level, they fail to characterize the intricate network correlations and causal feedback structures among risk factors [22]; (ii) at the dynamic level, they cannot reveal the evolutionary mechanisms and critical conditions of risks propagating through the network, amplifying, and even triggering systemic chaos [23]; and (iii) at the control level, their assessment results are often static snapshots, making it difficult to derive real-time intervention strategies that can achieve human–machine–environment collaborative stability for dynamic risk processes [24].

1.2. Related Work

Faced with these challenges, existing research on nuclear emergency system risks has primarily evolved along three mainstream paths. The first is the quantitative analysis of accident consequences based on Probabilistic Safety Assessment (PSA) and its derivative methods. Focusing on localized, discrete risk sources such as equipment failures and human errors, these methods conduct static, linear logical deductions using Event Tree (ET) and Fault Tree (FT) models [9,25,26]. While effective at identifying system vulnerabilities and quantifying accident frequencies and consequences at the engineering level, they often require extensive simplifications when dealing with scenarios involving compound external hazards and the variability of human responses during emergencies due to the high complexity and dimensionality of the scenario space. Hermanns et al. [27] further emphasized that contemporary PSA urgently needs to incorporate new dimensions of external hazards and dynamic accident progression. However, this significantly increases modeling and algorithmic complexity, leaving traditional static sequential frameworks still struggling to handle strongly time-varying, highly nonlinear human–machine–organizational interaction risks. The second research path introduces multi-dimensional indicator systems, incorporating soft factors such as technology, management, and communication into the evaluation. However, these approaches heavily rely on static scoring methods like the Analytic Hierarchy Process (AHP) or fuzzy evaluation, making it difficult to reveal how risks accumulate over time and undergo qualitative changes or to explain how minor initial perturbations are amplified within coupled networks and trigger cascading failures. From a behavioral operations research perspective, Argyris et al. [28] pointed out that emergency decision support faces behavioral challenges such as cognitive bias, groupthink, and decision paralysis under high pressure. These phenomena fall outside the measurement scope of static indicator systems, thereby limiting the applicability of such methods in real dynamic environments. The third path adopts System Dynamics (SD) and Multi-Agent Simulation (MAS) methods, which begin to incorporate the temporal dimension and H–M–E interaction complexity. However, these models are often overly macroscopic and lack fine-grained analysis of the underlying network topology, making it difficult to reveal emerging behaviors such as cascading failures and chaotic oscillations driven by network structure.

In recent years, complex network theory has emerged as a powerful tool for analyzing the topological structure, propagation paths, and structural vulnerabilities of multi-agent systems, finding widespread application in fields such as power systems [29], transportation systems [30], and cybersecurity and information systems [31]. Studies have shown that this theory can effectively identify critical nodes, key links, and systemic vulnerabilities through metrics like degree distribution, betweenness centrality, clustering coefficient, and PageRank, thereby revealing the “structural backbone” and key control points of risk propagation [32,33,34]. In the research of safety accidents and emergency management, complex networks have been employed to construct accident causation networks and fault propagation networks, explaining the cross-domain diffusion paths of risks [35,36,37]. Li et al. [38] and Feng et al. [39] utilized accident data and complex network theory to analyze risk causes and propagation mechanisms in complex industrial systems, demonstrating the cross-domain universality of constructing causal networks from accident data to analyze propagation mechanisms. In the field of nuclear safety and emergency response, existing studies have constructed failure event networks or safety barrier networks based on severe accident sequences to identify critical equipment and key operational steps under multi-failure scenarios [40,41]. Chen et al. [3] conducted invulnerability analysis from the perspective of the robustness of nuclear accident emergency response organizational networks, providing a network-based metric foundation for identifying critical positions and protection strategies. Kim et al. [4] modeled the information transmission and decision-making processes within nuclear emergency response organizations as a complex network; by analyzing the network’s topological structure and information flow evolution, they accurately identified potential vulnerabilities and key bottlenecks, offering theoretical support for enhancing emergency response efficiency and collaborative coordination. However, existing network research primarily focuses on engineering and mechanical factors, with insufficient attention paid to human and environmental elements. Methodologically, it remains limited to static topological characteristic analysis, node importance ranking, and invulnerability analysis, lacking in-depth integration with nonlinear dynamics and chaos theory.

Chaos theory in nonlinear dynamics provides an effective tool for revealing the evolutionary laws of systemic risks: it describes the butterfly effect, where minor perturbations are amplified, and elucidates the intrinsic randomness of systems that are highly sensitive to initial conditions and thus difficult to predict over the long term [42]. In multi-agent risk networks, where the dynamical behaviors of subsystems are mutually coupled, synchronization acts both as a mechanism for cross-domain risk propagation and a target for overall system stability [43]. It can suppress chaotic oscillations and guide the system from disorder back to order and also quantify emergency coordination into metrics such as error convergence speed, control precision, and implementation cost, thereby offering a quantitative basis for resource allocation.

Beyond conventional model-based synchronization schemes, more advanced chaotic synchronization methods have been proposed in recent years, including approaches tailored for short and noisy time series and time-reversible synchronization techniques that exploit both forward and backward evolution of observed trajectories [44,45]. Such methods can improve convergence speed and robustness when only limited time-series data are available, and thus offer a promising complement to model-driven synchronization in practical risk-monitoring and control scenarios.

In nuclear emergency scenarios, environmental perturbations, such as extreme weather or geological disasters, directly increase equipment failure probabilities and impair emergency response capabilities. The continuous deterioration of equipment status further elevates personnel’s cognitive load, inducing decision-making errors; these human operational errors, in turn, negatively impact equipment operational status, triggering secondary risks and ultimately forming a risk self-amplification mechanism within closed-loop structures [7,10]. It is important to emphasize that the introduction of chaos theory does not treat the nuclear emergency system as entirely random or uncontrollable but rather elucidates that under deterministic evolutionary rules and limited perturbation conditions, the system’s inherent structure may give rise to dynamic behaviors highly sensitive to initial conditions. Therefore, system capacity building should shift from traditional equipment and process validation to a progressive logic that first identifies critical thresholds, then stabilizes dynamic behaviors within safe operating regions, and finally implements collaborative control mechanisms to maintain system-wide coordination.

In summary, existing nuclear emergency research still exhibits three key limitations: First, complex network analysis often stops at static topological characteristics and stability assessment, failing to deeply integrate with nonlinear dynamic evolution and control strategies, which makes it difficult to explain risk threshold transitions and chaotic uncertainty. Second, chaos and nonlinear analysis mostly remain at the level of theoretical examples, lacking structured support and validation from real nuclear emergency drill data, resulting in a failure to establish a traceable mapping relationship between models and engineering practice. Third, research on collaborative control and resource allocation often relies on empirical rules or simplified linear assumptions; it fails to fully leverage network structure and propagation characteristics to achieve adaptive adjustment of control gains, making it difficult to obtain a comprehensive optimal solution with high timeliness, high precision, and low cost under complex coupling scenarios.

From a systems science perspective, accurately identifying the evolutionary characteristics and dynamic mechanisms of systemic risks during nuclear emergency drills, clarifying key coupling and threshold conditions, and designing effective control strategies to achieve overall situational stability and collaborative convergence have become urgent frontier topics in the field of nuclear safety and emergency management [1,42,46]. An analytical model that integrates network structure, dynamic behavior, and control strategy, formed by the fusion of complex network theory and nonlinear dynamics, is precisely the cutting-edge research direction in this field. Regarding nuclear transportation safety issues, existing research has shifted from single-risk assessment to a full-process analysis that covers risk identification, evolutionary modeling, and control intervention. For instance, Chen et al. [47] innovatively introduced a complex network evolution model into the risk management of spent-fuel road transportation and combined chaos synchronization control theory to realize the regulation of dynamic risks. This research approach provides a methodological reference for the nuclear emergency field to break through traditional paradigms and to conduct integrated research that links system structure, dynamic behavior, and control strategies. However, for high-risk specific scenarios such as nuclear emergencies, there remains a significant research gap in conducting systematic, full-chain research within a framework that encompasses risk network construction, chaos evolution analysis, and synchronization control implementation, all based on models constructed from real drill data.

Therefore, this paper proposes an integrated semi-quantitative analysis model for nuclear emergency drill data that combines the following: (i) a risk network representation; (ii) chaotic evolution analysis; and (iii) synchronization control design (hereafter referred to as the “risk network–chaotic evolution–synchronization control” framework), focusing on the dynamic evolutionary mechanisms and controllability of human, machine, and environmental (H–M–E) risks within nuclear emergency systems. First, based on multi-year, full-scope, comprehensive emergency drill data from a nuclear power base, event–event causal pairs are extracted from drill records, accident reports, command scripts, and on-site videos. A coding system encompassing 165 typical risk factors is established, and an H–M–E heterogeneous directed risk network with 389 triggering relationships is constructed. For node importance measurement, drawing on research on critical node identification in complex networks, this paper proposes a comprehensive risk centrality index that integrates betweenness centrality and the PageRank algorithm. This index simultaneously evaluates a node’s carrying capacity in risk propagation paths and its global influence in directed networks, thereby identifying key sources, hubs, and sinks in the network and providing a structural basis for subsequent dynamic modeling and control intervention. Second, addressing the characteristics of nuclear emergency systems such as strong coupling, strong feedback, and threshold effects, this paper abstracts the interactions among 165 micro-risk nodes into three subsystems: human (H), machine (M), and environmental (E). A three-dimensional continuous-time nonlinear dynamical model consistent with the network coupling logic is constructed. Using tools such as phase portraits, Lyapunov exponents, and bifurcation analysis, this study reveals the existence of period-doubling bifurcations and chaotic attractors under specific coupling parameters, elucidating the intrinsic mechanism by which minor perturbations are multiplicatively amplified within the H–M–E closed loop, ultimately leading to systemic instability. Finally, toward the engineering goal of guiding nuclear emergencies from disorder to ordered coordination, a synchronization control model based on a drive–response architecture is established. A piecewise error feedback synchronization controller is designed, incorporating the network’s comprehensive coupling strength as an adaptive parameter input to dynamically adjust control gains according to network connectivity and risk propagation intensity. Simultaneously, three control schemes—human-dominated, machine-dominated, and environment-dominated—are proposed. A quantitative comparison is conducted across multiple dimensions, including synchronization speed, control precision, energy cost, and efficiency, providing an operable semi-quantitative decision-making basis for nuclear emergency resource allocation.

The overall research pathway and the logical connections among these components are shown in Figure 1.

2. Materials and Methods

2.1. Risk Network Modeling

2.1.1. Risk Factor Exploration and Causal Network Construction

The risk network developed in this study is based on data provided by assessment teams from the East China Supervision Station of the Ministry of Ecology and Environment (National Nuclear Safety Administration) and by core experts from each emergency functional group at the participating nuclear power plant. The network draws primarily on summary and rectification reports from all on-site, comprehensive emergency drills conducted during the 12-year operating period of this nuclear power plant. These drills simulated a range of extreme conditions, including catastrophic external events, complete loss of off-site power, and critical equipment failures. These scenarios covered the entire emergency response process, from plant emergency preparedness and building-level response through site-wide response to off-site response. In addition, the drills included diverse sub-scenarios, such as the treatment of personnel injuries, shift handovers for emergency staff, orderly evacuation of non-essential personnel, and management of public communication, thereby reproducing the complexity of nuclear emergency management. The summary and rectification reports systematically cataloged and analyzed all issues revealed during the drills. For each issue, they specified the responsible departments, corrective actions, and implementation timelines, providing detailed and reliable input data for constructing the risk network.

To extract the causal factors embedded in problem items, to interpret the dynamics of these causes during emergency drills, and to ensure that the risk network is both rigorous and internally consistent, this study adopts a stepwise processing method and construction workflow. The specific steps are as follows. First, drawing on emergency drill command scripts, full-process exercise records, and on-site video footage, we systematically reviewed nearly 400 issues documented in 12 historical summary and rectification reports and identified 56 core problem items. This step provides a robust data foundation for subsequent causal factor extraction and risk analysis. Second, we conducted targeted identification of risk factors for each core issue. The identified risk factors were then grouped into three main categories: human factors (H), machine factors (M), and environmental factors (E). At the same time, we carried out precise identification, consistent classification, and standardized coding of the raw data. The coding table for risk factors is provided in Appendix A. Third, by leveraging natural language processing (NLP) techniques through a rule-based text-mining approach that identifies causal connectives (e.g., “due to,” “because,” “resulting in,” and “leading to”), we extracted cause–effect risk factor pairs within each core issue. This provided the basis for the subsequent construction of causal chains. Finally, the risk causal network system was constructed through a strictly progressive approach: beginning with simple causal chains that comprehensively cover one-to-one, one-to-many, and many-to-one causal relationships; then advancing to systematically organize complex causal chains, with a focus on integrating common nodes such as shared causes and shared effects to clarify the intrinsic connections among various risk nodes. Ultimately, the comprehensive construction of the risk causal network system is completed, achieving an effective transformation from fragmented risk nodes to a systematic risk network. This clearly presents the inherent risk correlation logic and transmission pathways for various emergency preparedness issues within the nuclear power base.

2.1.2. Constructing the Adjacency Matrix

The nuclear emergency risk network is modeled using a directed graph. Let the directed network of the nuclear emergency risk system be G = (V, E). For any two risk factor nodes $v_i,v_j\in V$, if risk trigger $v_i\to v_j$ exists, then the directed adjacency matrix is defined as shown in Equations (1) and (2):

$$a_{ij}=\left\{\begin{array}{c}1,(v_i\to v_j)\in E\\ 0,(v_i\to v_j)\notin E\end{array}\right.$$

(1)

$$A=\left[a_{ij}\right]\in {\left\{0,\left.1\right\}\right.}^{N\times N}$$

(2)

where $(v_i\to v_j)\in E$ denotes that there is a connected edge from node v_i to node v_j, $(v_i\to v_j)\notin E$ denotes that there is no connected edge from node v_i to node v_j. In a directed network, the out-degree $d_i^{out}$ and in-degree $d_i^{in}$ are defined as follows:

$$d_i^{out}=\sum _j{a}_{ij},\hspace{1em}{d}_i^{in}=\sum _j{a}_{ji}.$$

(3)

2.1.3. Network Risk Value Calculation

To address the subjectivity inherent in traditional expert scoring methods, this paper proposes a comprehensive quantitative risk metric for nodes based on heterogeneous network topology. The betweenness centrality index comprehensively evaluates a node’s transmission capacity along risk propagation pathways, while the PageRank method measures its global influence within directed networks.

(1)

Betweenness centrality

Betweenness centrality measures the degree of risk associated with a node. If a node lies on numerous shortest risk propagation paths, its failure will sever or accelerate the spread of risk within the network. The calculation formula is as follows:

$$BC(v_i)=\sum _{s\ne v\ne t\in V}{\displaystyle \frac{{\sigma }_{st}(v_i)}{{\sigma }_{st}}}$$

(4)

where ${\sigma }_{st}$ denotes the total number of shortest paths from node s to node t, ${\sigma }_{st}(v_i)$ represents the number of paths from node s to node t that pass through node $v_i$, and ${\widehat{C}}_B$ is the normalized value.

(2)

PageRank directed indicator

The PageRank algorithm quantifies the relative importance of nodes in a directed network. By leveraging a recursive principle whereby a node’s importance increases when it is targeted by already important nodes, the algorithm can be used to identify critical downstream recipients in risk chains as well as central upstream sources.

$$PR\left(v_i\right)={\displaystyle \frac{1-d}{N}}+d\sum _{v_j\in M\left(v_i\right)}{\displaystyle \frac{PR\left(v_j\right)}{d_j^{out}}}$$

(5)

In the formula, d denotes the damping factor (set to 0.85). N is the total number of nodes in the network. $M(v_i)$ is the set of nodes that point to node $v_i$, and $d_j^{out}$ is the out-degree of $v_j$. Within the nuclear emergency risk system, this index reflects the steady-state probability that a random walk will visit each risk factor under repeated risk propagation. Formula (5) indicates that when a device node is simultaneously operated by multiple high-risk personnel nodes or exposed to the combined influence of multiple environmental risk nodes, its PageRank value increases substantially. This indicates that such a node is highly susceptible to failure through cascading effects.

(3)

Normalization and aggregated risk score

To eliminate dimensional differences, min-max normalization is applied:

$$B{C}_i^{norm}={\displaystyle \frac{B{C}_i-\min (BC)}{\max (BC)-\min (BC)}},\hspace{1em}P{R}_i^{norm}={\displaystyle \frac{P{R}_i-\min (PR)}{\max (PR)-\min (PR)}}.$$

(6)

Define the risk centrality index $r_i$ as a weighted combination of normalized betweenness centrality and PageRank values:

$$r_i=\alpha B{C}_i^{norm}+(1-\alpha )P{R}_i^{norm},\hspace{1em}\alpha =0.5.$$

(7)

Among these parameters, $\alpha$ denotes the weighting coefficient, which is set to 0.5 in this study to balance the contributions of the two factors. Although the exercise data confirm the presence of causal links, the actual probabilities of fault propagation are difficult to estimate statistically for events with extremely low occurrence frequencies. Therefore, this study adopts the assumption that structure determines function and estimates edge propagation strengths from the inherent vulnerabilities (risk values) of the nodes. This constitutes a commonly used semi-quantitative approach in risk analysis when large-sample accident data are unavailable.

2.2. Development of a Kinetic Model for Nuclear Emergency Risk Evolution

To construct the macro-evolution equations, we abstracted the interactions among 165 nodes in the nuclear emergency system into three subsystems—human, machine, and environment—according to human–machine–environment systems engineering principles. The formulation of Equations (8)–(10) is inspired by the classical Lorenz chaotic system, but incorporates several physical modifications tailored to the characteristics of nuclear emergency scenarios, as summarized in

Table 1. Summary of system parameter corrections.

The Given Equation	Term	Physical/Management Interpretation	Mapping to Context	Theoretical Basis
The Evolution of Human Subsystems $\dot{x}$	$-ax$	Organizational Dissipation	Emergency organizations implement standardized procedures, regular training, and psychological interventions to ensure that initial human errors and psychological panic gradually subside over time.	Organizational Resilience
	$+y$	Stress Transmission	Critical equipment failures (for M01) directly increase operational workload and decision-making pressure, thereby leading to Emergency decision-making suffering from a lack of basis, errors, or delays (H36) and operational errors.	Cognitive Load Theory
	$+cyz$	Nonlinear Amplification	When equipment failure ($x_2$) and environmental deterioration ($x_2$) occur simultaneously, they cause a nonlinear, dramatic increase in human error rates (e.g., communication disruption compounded by extreme weather leading to command paralysis).	Synergetics
The evolution of machine systems ($\dot{y}$)	$-y$	System Self-recovery	The device’s inherent redundancy design, automatic switching logic, or fault isolation measures enable the risk to revert to a stable state.	Reliability Engineering
	$+bx$	Intervention Gain	The effectiveness of command decisions directly influences the accuracy of operations. Positive human interventions, such as correct emergency repairs, can stabilize equipment status; conversely, non-compliant operations or actions (H66) tend to exacerbate equipment failures.	HFE
	$-xz$	Environmental Inhibition	Harsh environments (such as high radiation, Noise E10) impair personnel’s ability to control equipment, resulting in diminished “positive human intervention on machinery”, and manifesting as negative feedback	Situation Awareness
Evolution of environmental subsystems ($\dot{z}$)	$-{\displaystyle \frac{1}{2}}z$	Environmental Decay	The physical attenuation of the disaster itself (such as the fire burning out or the flood receding) and the environment’s inherent capacity for recovery	Environmental Risk Assessment
	$+xy$	Secondary disasters caused by human–machine mismatch	Typical secondary disaster mechanism: Equipment failure compounded by improper personnel response ($+x_1{x}_2$), directly triggering severe environmental consequences such as radioactive release or explosion.	Cascading failure mode
	$-{\displaystyle \frac{1}{2}}y$	The isolation effect of engineering barriers	The intact equipment condition (e.g., Containment rupture M09) provides physical shielding against environmental consequences, directly suppressing the spread of environmental risks.	Principle of defense in depth

. In the following, we denote by x, y, and z the aggregated risk-posture intensities of the human (H), machine (M), and environment (E) subsystems, respectively.

Based on Table 1, we construct a three-dimensional continuous-time nonlinear dynamical system to analyze the coupled evolution of risk among the human (H), machine (M), and environment (E) elements in nuclear emergency scenarios.

$$f(t)={\left[x(t),y(t),z(t)\right]}^{\top }$$

(8)

Among these, the human factor ($x(t)$:H) risk-posture intensity provides an integrated measure of decision-making and command quality, coordination efficiency, psychological and physiological states, behavioral deviations, and the implementation of protective measures (for example, H36, H81, H68, and H86); the machine-related ($y(t)$:M) risk-posture intensity captures the reliability and failure levels of power supply, cooling, communications, platform systems, and other critical equipment (for example, M01, M03, M05, and M09); and the environment-related ($z(t)$:E) risk-posture intensity represents the combined impact of external natural and social disturbances, as well as on-site emergency conditions (for example, E01–E10). These three categories of variables are not single indicators, but rather aggregated assessments of risk factors within each category. They are used to characterize the dynamic evolution and coupled amplification of system-level risk. Explanations of these codes are provided in Appendix A.

(1)

The amplified impact of equipment and environmental factors on humans

$$\dot{x}=-ax+y+cyz$$

(9)

$-ax$: Under the combined influence of institutional constraints, training and education, oversight mechanisms, and self-regulation, human-factor risks often exhibit a decreasing trend. For example, when emergency training systems are refined (thereby mitigating the effects of hypotheses H02 and H03), and information transmission is standardized (reducing the influence of factors such as H30, H34, and H68), the trajectory of human-related risk is substantially suppressed. A higher value of parameter a indicates a stronger organizational capacity to mitigate human-related risk.

$+y$: Equipment failures and engineering system anomalies directly increase operational pressure and the probability of human error. For example, M0 and M0 substantially increase decision-making workload and on-site operational complexity, thereby increasing the risks of H3, H67, and reduced H81.

$+cyz$: Critical machine–environment interaction acts as an important amplification factor. In practice, when equipment malfunctions (M) and external disturbances (E) occur simultaneously, human-related risk does not simply add up but instead exhibits a multiplicative amplification effect. For example, the co-occurrence of E01 and M03 markedly increases the probabilities of H30, H35, and H68, thereby elevating the risks associated with H104 and H94.

(2)

Evolution of device posture under human–environment interaction mechanisms

$$\dot{y}=-y+bx-xz$$

(10)

$-y$: The equipment system incorporates recovery mechanisms—such as fault isolation, switching, and repair—through maintenance support, redundant design, and automatic protection, thereby reducing the risk of mechanical failure.

$+bx$: Human actions, management, and operations significantly impact equipment status. Sound organization and decision-making can suppress fault propagation; conversely, the presence of H36, H65, and H66 will exacerbate systemic risks arising from equipment failure propagation or improper handling.

$-xz$: This indicates that under adverse conditions (for example, E02, E04, and E10), the effectiveness of human response is reduced, thereby weakening the positive influence of personnel on equipment and, in extreme cases, turning it into a negative influence on equipment status. For example, E10 makes password confirmation difficult to perform (H33 and H68), thereby increasing the likelihood of misjudgment and operational error during critical switching and isolation procedures. These effects can either slow the reduction in M-class risks or accelerate their escalation.

(3)

Human–machine interaction leads to secondary environmental consequences.

$$\dot{z}=-{\displaystyle \frac{1}{2}}z+xy-{\displaystyle \frac{1}{2}}y$$

(11)

$-1/2z$: This indicates that environmental disturbances exhibit a certain degree of decay or controllability: for example, fires diminish after suppression, on-site order is gradually restored after control, and some external shocks recede from their peak over time. This property does not negate the persistence of disasters; rather, it abstractly captures the general feature that environmental conditions do not accumulate indefinitely.

$+xy$: Human–machine interactions may induce or aggravate environmental consequences and, thus, constitute a typical secondary-disaster mechanism: equipment failures are more likely to trigger events such as E07, E06, and E05 when they are mishandled or delayed, with the corresponding human-factor nodes including H36, H81, and H94.

$-1/2y$: This indicates that equipment status, through engineering barriers and control measures, can exert a suppressive or isolating effect on the environmental situation, for example, via containment structures, filtered discharges, and shielding or isolation systems. Based on the above description, the relationship between human, machine, and environment is illustrated in

Table 2. Human–machine–environment relationship chart.

	H	M	E
H	/	HM	/
M	MH	/	ME
E	EH	EM	/

and Figure 2.

Table 1 shows that HM denotes a relationship between H and M; similarly, ME denotes a relationship between M and E, EH denotes a relationship between E and H, and EM denotes a relationship between E and M. The symbol “/” in the table indicates that there is no cross-relationship within the same entity.

In summary, the nuclear emergency drill risk system is modeled as a three-dimensional nonlinear coupled system, with the following system dynamics equations:

$$\begin{array}{l}\dot{x}=-ax+y+cyz,\hfill \\ \dot{y}=-y+bx-xz,\hfill \\ \dot{z}=-{\displaystyle \frac{1}{2}}z+xy-{\displaystyle \frac{1}{2}}y,\hfill \end{array}a=16,b=40,c=2$$

(12)

here a > 0, b > 0, and c > 0 are system parameters that respectively denote the self-recovery or suppression strength of human-related risk, the coupling gain between humans and equipment, and the nonlinear amplification coefficient of machine–environment coupling. Parameters a, b, and c are not intended to represent precise physical quantities; instead, they characterize the relative strengths of self-regulation and cross-domain coupling under different organizational and technical conditions of emergency response. Therefore, this study focuses on identifying qualitative transition patterns and critical thresholds rather than on producing precise numerical predictions. Nuclear emergency response inherently exhibits feedback lags, incomplete information, cross-domain coupling, and threshold effects. For example, environmental degradation increases the probability of equipment failure, equipment failures aggravate personnel stress and the risk of operational error, and human degradation further reduces response efficiency, together forming a strongly nonlinear closed-loop system. At the macro level, such mechanisms may manifest as state variables that exhibit sensitive dependence and limited predictability. It is therefore justified to employ chaotic models as an abstraction.

2.3. Justification of Chaotic Abstraction for Nuclear Emergency Risk Systems

It should be emphasized that the introduction of chaos theory in this study does not imply that nuclear emergency processes are inherently random or uncontrollable. Instead, chaos is employed as an abstraction to represent deterministic but highly sensitive dynamics arising from strong nonlinear feedback, delayed information, and cross-domain coupling among human, machine, and environmental subsystems. Unlike stochastic models that attribute uncertainty primarily to external randomness, the proposed chaotic model highlights that substantial unpredictability can emerge endogenously from the system structure itself, even under fixed rules and bounded disturbances. This distinction is particularly relevant for nuclear emergency response, where deviations often originate from internally amplified feedback loops rather than exogenous noise alone. Within this framework, the chaotic system serves as a parsimonious macro-level representation of the H–M–E risk posture, built consistently with the underlying risk network, and provides a tractable basis for analyzing threshold phenomena, bifurcations, and the design of synchronization-based coordination strategies.

2.4. Analysis of Dynamic Evolution and Chaotic Behavior in Networks

To elucidate the mechanisms of risk propagation within the nuclear emergency response system, this section examines the dynamic evolution of the system under varying coupling parameters. Small perturbations in the radiation environment, in the psychological state of emergency personnel, or in the operational status of safety equipment may, through nonlinear feedback mechanisms, trigger disproportionately large and sudden escalations in overall risk, particularly when the facility operates near critical safety margins.

2.4.1. Definition of Lyapunov Exponents

System $\dot{F}=f(x,y,z)$ satisfies the variational equation for small perturbations:

$$\stackrel{·}{\delta (x,y,z)}=J(f(t))\delta (x,y,z),\hspace{1em}J={\displaystyle \frac{\mathcal{\partial }f}{\mathcal{\partial }(x,y,z)}}.$$

(13)

$\stackrel{·}{\delta (x,y,z)}$ denotes the infinitesimal perturbation vector of the system trajectory $f(t)$ at time t, which can be interpreted as the deviation between two nearby trajectories. $J(f(t))$ is the Jacobian matrix of the system’s dynamical equation f, evaluated along the trajectory $f(t)$. This equation describes how an initial infinitesimal perturbation ${\delta }_{x,y,z}(0)$ evolves over time along the reference trajectory $f(t)$. The evolution of the perturbation is determined by the linearized dynamics at the current system state, that is, by the Jacobian matrix. If the system is chaotic, perturbations in local directions may undergo exponential amplification or contraction. The Lyapunov exponent provides the mathematical tool for quantifying this local behavior.

The Lyapunov exponent is defined as:

$${\lambda }_k=\underset{t\to \infty }{\lim }{\displaystyle \frac{1}{t}}\ln {\displaystyle \frac{\delta f_k(t)}{\delta f_k(0)}}$$

(14)

${\lambda }_k$: The k-th Lyapunov exponent describes the long-term average exponential growth rate of perturbations along the k-th intrinsic direction of the system, as defined by the characteristic directions of the linearized system.

$\delta f_k(t)$: Represents the variation over time of the magnitude of the perturbation along the kth initial orthogonal direction. When the maximum Lyapunov exponent ${\lambda }_1>0$, the system exhibits sensitive dependence on initial conditions and is classified as chaotic.

2.4.2. Chaotic Behavior and Dynamical Evolution Characteristics

The identification of chaotic behavior in a system hinges on the presence of chaotic attractors and on Lyapunov exponents that are greater than zero. To validate the chaotic characteristics of the nuclear emergency risk system, phase space trajectories and Lyapunov-exponent spectra were computed using Equations (8)–(14), as shown in Figure 3 and Figure 4.

For the parameter set a = 16, b = 40, and c = 2, Figure 3 shows the phase space trajectories and their two-dimensional projections of the three-dimensional state variable (x,y,z), which describe the macro-level risk entropy of the human, machine, and environmental subsystems. The system trajectory does not converge to a single equilibrium point or a simple limit cycle but instead exhibits non-periodic oscillations around several regions, forming a chaotic attractor with bifurcated lobes and self-similar structure. This attractor indicates that, under strong nonlinear coupling, persistent and irregular risk oscillations emerge through cross-feedback among the three subsystems. The system hovers persistently near higher-risk states and struggles to return to a stable, low-risk regime. The patchy or banded structures observed in phase space reflect the presence of multiple quasi-steady regions, between which the system repeatedly transitions. This dynamic behavior embodies the characteristic of nuclear emergency systems being torn between extremes and resistant to complete stabilization. Figure 3 confirms, from a dynamical perspective, that under conditions of high uncertainty and strong coupling, the system is prone to entering chaotic states. Even if local indicators recover temporarily, small perturbations can still trigger the system to revert to high-risk oscillations. This mechanism explains why localized rectifications during actual emergency responses cannot guarantee long-term stability and highlights the necessity of implementing systematic, closed-loop, collaborative control.

Figure 4 presents the Lyapunov exponent spectrum of the nuclear emergency risk-dynamics system. The figure shows that, under the selected parameter combination, a positive maximum Lyapunov exponent (λ_max > 0) exists, and its value first increases and then exhibits small fluctuations as certain parameters are varied. This behavior reflects the system’s sensitivity to initial conditions and the exponential divergence of trajectories. This indicates that, during nuclear emergency drills, any minute disturbance within the system—such as a slight deviation in a verbal command, brief communication jitter, or minor variation in ambient illumination—can be amplified over time by nonlinear feedback mechanisms. The sensitivity of the emergency system to initial conditions is illustrated in Figure 5.

Figure 5 compares the divergence in the evolution of the system state over time under extremely similar initial conditions. The curves show that, in the early stages, the two trajectories almost coincide. However, after a certain period, they diverge rapidly and subsequently exhibit entirely distinct oscillatory patterns. Sensitivity to initial conditions is one of the core characteristics of chaotic systems. For nuclear emergency response systems, even small differences in state variables at the onset of drills or incidents, such as personnel workload, equipment integrity, or the level of environmental disturbance, can be rapidly amplified through the iterative action of the human–machine–environment feedback loop. This amplification can lead to entirely different response paths and consequence patterns. This underscores the critical importance of prioritizing seemingly minor issues at the early stages of actual emergency response. For example, a brief information delay or a slight procedural deviation may evolve into substantial deviations as the situation unfolds. Therefore, mechanisms for early anomaly detection and rapid course correction must be established in both drills and real-world scenarios to transform sensitivity to initial conditions into an opportunity for timely intervention.

2.4.3. Parameter Thresholds and Bifurcation Analysis

Under the parameter combination a = 16, b = 40, and c = 2, the system exhibits clear chaotic dynamics. To gain deeper insight into how different parameter configurations influence the system’s dynamical characteristics, particularly its transitions among stable, periodic, and chaotic states as key control parameters change, we conducted a bifurcation analysis. By systematically varying the coupling-strength parameters a, b, and c between the human–machine and environmental subsystems and plotting the corresponding bifurcation diagrams, we reveal the critical behavior and phase-transition pathways of the system state as the parameters change. This enables the identification of parameter regions that lead to system instability, as well as potential control windows. To visualize the specific system dynamics corresponding to different parameter values, the three-dimensional phase portraits for varying parameter values are shown in Figure 6.

Figure 6 illustrates the evolution of the system’s phase portrait as the parameters a, b, and c are varied. As the key coupling parameters increase, the phase portrait evolves from an initial stable equilibrium point or simple limit cycle structure to period-doubling oscillations and ultimately to structurally complex chaotic attractors. Within specific parameter ranges, quasi-periodic motion alternates with chaotic states. Here, parameter a represents the self-recovery or suppression strength of human-induced risk, b characterizes the human–machine coupling gain, and c reflects the degree of nonlinear amplification in the machine–environment coupling. The results indicate that when a is large while b and c are small, human-induced risk exhibits strong self-suppression, and the human–machine–environment coupling effect is weak, causing the system to converge towards stable states or simple periodic motion. As b and c increase, the coupling-induced amplification among the human–machine–environment subsystems intensifies, causing bifurcation of the original stable point. The system then undergoes successive period-doubling and enters a chaotic regime. When the parameters increase further, the system may exhibit window-like stable regions, demonstrating complex multistability. The dynamical results in Figure 5 indicate that the risk evolution of nuclear emergency systems exhibits pronounced threshold characteristics and chaotic behavior. Therefore, dynamic control mechanisms must be introduced to rapidly suppress mismatch errors and the divergence of risk once the system is activated.

The chaotic behavior revealed by the above analysis occurs only under specific parameter combinations. To systematically analyze how the system’s dynamical state evolves with respect to key control parameters and to fully identify the phase-transition pathway from stable periodic motion to chaos, it is necessary to compute and analyze the system’s bifurcation diagrams. The bifurcation behavior of the state variables x, y, and z under different parameter values is shown in Figure 7, Figure 8 and Figure 9, respectively.

Figure 7, Figure 8 and Figure 9 illustrate the bifurcation behavior of the state variables x, y, and z under steady-state conditions as the parameters are varied continuously. As the parameters increase, all three variables follow a typical bifurcation pathway: they evolve from a single equilibrium point to period-doubling oscillations, then transition to multi-periodic motion, and ultimately give rise to densely distributed chaotic attractors. Although the initial bifurcation points and detailed evolution differ among the variables, all three exhibit clear chaotic dynamical characteristics within specific parameter ranges.

2.5. Design of the Chaotic Synchronization Controller

2.5.1. Calculation of Risk System Network Coupling Strength

(1)

Topological structure coupling strength

The degree of a node is calculated by its out-degree $d_i^{out}$ and in-degree $d_i^{in}$:

$$d_i=d_i^{out}+d_i^{in}.$$

(15)

Further define the strength of structural coupling:

$${\gamma }_{\mathrm{deg}ree}={\displaystyle \frac{\overline{d}}{\max d_i}},\hspace{1em}\overline{d}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{d}_i.$$

(16)

$\overline{d}$ denotes the average level of connectivity, and $\underset{i}{\max }{d}_i$ denotes the connectivity scale of the strongest hub nodes. When the network is dominated by a few hubs (for example, H36 or H115 connected to numerous factors), a smaller value of $\overline{d}/\max d_i$ indicates that risk coupling is centralized and more sensitive to critical nodes. Conversely, a larger value of ${\gamma }_{\mathrm{deg}ree}$ implies denser overall network connectivity and stronger structural coupling.

(2)

Risk transmission coupling strength

To characterize the intensity of risk propagation along directed edges, we define the weight of each edge.

$$w_{ij}=\left\{\begin{array}{l}\alpha r_i+(1-\alpha )r_j,a_{ij}=1,\\ 0,a_{ij}=0,\end{array}\right.\hspace{1em}\alpha \in (0,1)$$

(17)

$a_{ij}=1$ indicates the existence of a directed connection from node i to node j; $r_i$ is the intrinsic risk level of the source node i; $r_j$ is the intrinsic risk level of the target node j; and $\alpha$ is a weighting coefficient that adjusts the relative importance of $r_i$ and $r_j$ in the edge weight. When $\alpha$ approaches 1, the edge weight is determined primarily by the source node’s risk level; when $\alpha$ approaches 0, it is determined primarily by the target node’s risk level.

Obtain the normalized upper bound:

$$S_{\max }=m\cdot r_{\max },\hspace{1em}{r}_{\max }=\underset{i}{\max }{r}_i.$$

(18)

$S_{\max }$: The maximum total edge weight sum that a network can achieve under the strongest propagation conditions.

$r_{\max }$: The highest risk value across all nodes.

m: The total number of directed edges in the network.

Define the strength of risk transmission coupling:

$${\gamma }_{risk}={\displaystyle \frac{S}{S_{\max }}},$$

(19)

where $S={\displaystyle {\sum }_{i.j}{w}_{il}}$ is the sum of the actual edge weights of all directed edges in the network.

(3)

Algebraic Connectivity Coupling Strength

Define the Laplacian $L_{out}$:

$$D_{out}=diag\left(d_1^{out},\dots ,d_N^{out}\right),\hspace{1em}{L}_{out}=D_{out}-A.$$

(20)

For $L_{out}$, compute eigenvalues $\left\{{\lambda }_k\right\}$. Sort the real parts and take the second smallest value ${\lambda }_2$ and the largest value ${\lambda }_{\max }$. Construct the normalized spectral connectivity metric:

$${\gamma }_{a\lg ebraic}={\displaystyle \frac{\left|{\lambda }_2\right|}{{\lambda }_{\max }}}.$$

(21)

This metric characterizes the network’s global coupling accessibility. Larger values of this metric indicate that the network’s overall structure is more resistant to fragmentation and that information and risk can propagate more readily across regions within the system.

(4)

Integrated coupling strength

The formula for calculating the comprehensive coupling strength is as follows:

$${\gamma }_{total}=0.3{\gamma }_{\mathrm{deg}ree}+0.3{\gamma }_{a\lg ebraic}+0.4{\gamma }_{risk}.$$

(22)

Use ${\gamma }_{total}$ as the input for quantifying network coupling complexity to control gain adaptive tuning.

2.5.2. Chaotic Synchronization Systems and Segmented Adaptive Controllers

In the context of nuclear emergency response, the drive–response synchronization framework is not intended to merely align two abstract chaotic trajectories. Instead, the drive (master) system represents a reference macro-level risk posture of the human (H), machine (M), and environmental (E) subsystems that is consistent with planned emergency procedures, resource deployment, and best-practice coordination under given constraints. The response (slave) system represents the actual evolution of the H–M–E subsystems under imperfect execution, delays, and mismatches. The synchronization error e(t), therefore, measures the deviation between the actual and desired system postures. The objective of the segmented adaptive synchronization controller is to suppress the growth of this error and to drive it towards zero, thereby improving situational alignment and cross-domain coordination. Even if the underlying nonlinear dynamics remain chaotic in an absolute sense, ensuring that the response trajectory closely tracks the reference trajectory and that the H–M–E subsystems evolve coherently rather than in a desynchronized manner is of direct practical value for nuclear emergency drills and real-time command decision-making. To describe the nonlinear dynamical evolution under human–machine–environment coupling in nuclear emergency drill systems, we construct a three-dimensional nonlinear driving system (master system). We then design a response system (slave system) with an identical structure but augmented with control inputs. The state variables of the driving system are denoted as

$$x(t)={\left[x_1(t),x_2(t),x_3(t)\right]}^{\top }.$$

(23)

The response system status is:

$$y(t)={\left[y_1(t),y_2(t),y_3(t)\right]}^{\top }.$$

(24)

The driving system is used to analyze the trajectory of the target or reference state, whereas the response system describes the dynamic evolution of the actual execution system under initially mismatched conditions.

(1)

Drive System

This paper employs the following three-dimensional nonlinear system as the driving system:

$$\left\{\begin{array}{l}{\dot{x}}_1=-a{x}_1+x_2+c{x}_2{x}_3,\hfill \\ {\dot{x}}_2=-x_2+b{x}_1-x_1{x}_3,\hfill \\ {\dot{x}}_3=-{\displaystyle \frac{1}{2}}{x}_3+x_1{x}_2-{\displaystyle \frac{1}{2}}{x}_2,\hfill \end{array}\right..$$

(25)

The initial value of the drive system is set to the maximum risk of each subsystem:

$$x_1(0)=10r_H^{\max },\hspace{1em}{x}_2(0)=10r_M^{\max },\hspace{1em}{x}_3(0)=10r_E^{\max }.$$

(26)

(2)

Response System

The response system configuration aligns with the drive system, but incorporates superimposed control inputs $u(t)={\left[u_1,u_2,u_3\right]}^{\top }$.

$$\left\{\begin{array}{l}{\dot{y}}_1=-a{y}_1+y_2+c{y}_2{y}_3+u_1,\hfill \\ {\dot{y}}_2=-y_2+b{y}_1-y_1{y}_3+u_2,\hfill \\ {\dot{y}}_3=-{\displaystyle \frac{1}{2}}{y}_3+y_1{y}_2-{\displaystyle \frac{1}{2}}{y}_2+u_3.\hfill \end{array}\right..$$

(27)

The response system’s initial value is set to the minimum non-zero risk among all subsystems:

$$y_1(0)=10r_H^{\min },\hspace{1em}{y}_2(0)=10r_M^{\min },\hspace{1em}{y}_3(0)=10r_E^{\min }.$$

(28)

2.5.3. Synchronization Error System Structure and Control Objectives

Define the synchronization error vector:

$$e(t)=y(t)-x(t)={\left[e_1(t),e_2(t),e_3(t)\right]}^{\top }, e_i(t)=y_i(t)-x_i(t).$$

(29)

To facilitate the evaluation of synchronization performance, the error norm is defined as follows:

$$\Vert e(t)\Vert =\sqrt{e_1^2(t)+e_2^2(t)+e_3^2(t)}.$$

(30)

The synchronous control objective can be formulated as designing a control law A such that for any given initial values A and B, the error system satisfies:

$$\underset{t\to \infty }{\lim }e(t)=0.$$

(31)

The above equation indicates that, after emergency control intervention, the three situational variables—human, machine, and environmental—tend to become consistent in their dynamical evolution. This enables the nuclear emergency drill system to achieve closed-loop coordination, gradually driving the system from an unstable synchronized state towards a coordinated synchronized state.

2.5.4. Design of a Segmented Linear Adaptive Feedback Synchronization Controller

Given that nuclear emergency response involves distinct initiation points for control measures, such as the establishment of an emergency command system, the completion of information aggregation, or the conclusion of control handover, this study adopts a segmented time-varying switch-control strategy. The control initiation time is denoted by $t_c$, and the control law is defined as follows.

$$u_i(t)=\left\{\begin{array}{l}-k_i{e}_i(t),t\ge t_c,\\ 0,t<t_c,\end{array}\right.\hspace{1em}i=1,2,3.$$

(32)

Phase $t<t_c$ represents the initial stage of an incident, or the natural evolution phase before command intervention, when information has not yet been consolidated, response decisions lack uniformity, and the action strategies of all parties remain uncoordinated. Phase $t<t_c$ marks the point at which the emergency command system, resource dispatch mechanisms, and process-control measures begin to take full effect. This is manifested in actions such as activating the emergency command platform, standardizing information channels, and implementing closed-loop management. Through active feedback mechanisms, deviations in the system state are corrected and brought back into alignment.

To incorporate the structural and propagation characteristics of the nuclear emergency drill risk network into the design of controller strength, this study maps the comprehensive coupling strength SS from Equation (22) to the control gain, as given by the following Equation (33):

$$k_1=a{\gamma }_{total},\hspace{1em}{k}_2=\beta {\gamma }_{total},\hspace{1em}{k}_3=\omega {\gamma }_{total}.$$

(33)

where $a$, $\beta$, and $\omega$ correspond to different levels of control. The higher the degree of network coupling k_i, the tighter the internal chain of risk influence within the system, leading to stronger propagation of errors across subsystems. Consequently, a larger feedback gain is required to achieve rapid error decay.

2.5.5. Lyapunov Stability Analysis

Select the following positive definite Lyapunov function and differentiate with respect to V(e):

$$\dot{V}(e)=e^T\dot{e}$$

(34)

Substituting Equation (34) into Equation (29) yields Equation (35).

$$\dot{V}(e)=e^T[F(e,t)-Ke]$$

(35)

Assuming that the nonlinear term in Equation (35) satisfies a Lipschitz condition, that is, there exists a constant L > 0, the resulting equation is given by:

$$\Vert F(e,t)\Vert \le L\Vert e\Vert .$$

(36)

Thus,

$$\dot{V}(e)\le L\Vert e{\Vert }^2-e^{T}Ke.$$

(37)

If the feedback gain matrix K satisfies:

$${\lambda }_{\min }(K)>L.$$

(38)

Then,

$$\dot{V}(e)\le -\left({\lambda }_{\min }(K)-L\right)\Vert e{\Vert }^2<0.$$

(39)

Therefore, the Lyapunov function is strictly decreasing, and the synchronization error converges asymptotically, meaning the response system achieves global asymptotic synchronization after control initiation.

$$\underset{t\to \infty }{\lim }e(t)=0$$

(40)

Based on Lyapunov stability analysis, the proposed piecewise error-adaptive feedback synchronization controller guarantees global asymptotic stability of the error system after control initiation. This effectively suppresses error propagation in chaotic regimes within the nuclear emergency drill risk system and achieves coordinated synchronization of risk states across the human, machine, and environmental subsystems.

2.6. Control Scheme Evaluation

Based on the control law $u_i=-k_i\times \gamma \times e_i$, constructed from the comprehensive risk-coupling intensity, this section establishes a performance evaluation model along four dimensions: synchronization speed, control accuracy, energy efficiency, and cost-effectiveness. This model is used to assess the overall effectiveness of different synchronization-control schemes across the three subsystems of human factors (H), machine factors (M), and environmental factors (E). Comprehensive scores for each scheme are obtained using a linear-weighting method. The comprehensive risk-coupling intensity is calculated as γ = 0.125079 using Equations (15)–(22). To this end, we design three control schemes with distinct resource-allocation strategies, whose specific structures and parameter configurations are detailed in

Table 3. Control scheme.

Control Plan	Preset k Value	Resource Allocation Ratio (H:M:E)	Actual Gain K × γ
Human-dominated	[60, 25, 15]	60%:25%:15%	[7.5, 33.13, 1.88]
Machine-dominated	[25, 60, 15]	25%:60%:15%	[3.13, 7.5, 1.88]
Environment-dominated	[25, 15, 60]	25%:15%:60%	[3.13, 1.88, 7.5]

.

2.6.1. Definition of Original Performance Metrics

(1)

Synchronization time and synchronization speed metrics

Let the synchronization time be defined as the first moment after the control initiation time $t_c$ when the total error norm $\Vert e_i(t)\Vert$ falls below the given threshold ε under the i-th control scheme:

$$t_{s,i}=\min \left\{t\ge t_c\Vert \Vert e_i(t)\Vert <\epsilon \right\}$$

(41)

In Equation (41):

$t_{s,i}$: Synchronous completion time of control scheme i;

$t_c$: At the controller startup time, $e_i(t)={\left[e_{1,i}(t),e_{2,i}(t),e_{3,i}(t)\right]}^{\top }$ represents the synchronization error vector for scheme i at time t.

$\epsilon >0$: Synchronization error threshold;

Define the synchronization speed metric as:

$$S_{sync,i}={\displaystyle \frac{1}{t_{s,i}-t_c+{\delta }_t}}\cdot C_{sync}.$$

(42)

In Equation (42):

$S_{sync,i}$: Synchronization speed metric value for scheme i;

${\delta }_t$ > 0: Minimal translation constant to prevent denominator from being zero;

$C_{sync}$ > 0: Scale factor, used to scale values to an appropriate range.

(2)

Final error and control accuracy metrics

The final error of the i-th scheme at the simulation’s end time T is defined as:

$$e_{final,i}=e_i(T)=\sqrt{e_{1,i}^2(T)+e_{2,i}^2(T)+e_{3,i}^2(T)}.$$

(43)

$e_{final,i}$: The total synchronism error norm of scheme i at t = T;

$T$: Simulation termination time

$e_{n,i}^2(T)$: Final-time error component of the i-th scheme on subsystem n (H/M/E)

To obtain a precision metric where smaller errors correspond to larger values, define:

$$S_{acc,i}={\displaystyle \frac{1}{e_{final,i}+{\delta }_e}}.$$

(44)

In Equation (44):

$S_{acc,i}$: Control accuracy index value for Scheme i;

${\delta }_e$ > 0: Preventing small constants that cause the denominator to become zero.

The smaller the final error $e_{final,i}$ and the larger the value of $S_{acc,i}$, the better the control scheme performs in terms of long-term synchronization accuracy.

(3)

Overall control of energy and energy efficiency indicators

Define the total control input vector for the i-th control scheme as:

$$u_i(t)={\left[u_{1,i}(t),u_{2,i}(t),u_{3,i}(t)\right]}^{\top }.$$

(45)

The corresponding control input norm is:

$$u_i(t)=\sqrt{u_{1,i}^2(t)+u_{2,i}^2(t)+u_{3,i}^2(t)}.$$

(46)

The total control energy is defined as:

$$E_{total,i}={\int }_0^T\left(u_{1,i}^2(t)+u_{2,i}^2(t)+u_{3,i}^2(t)\right)dt.$$

(47)

$E_{total,i}$: Solution i’s total controlled energy consumption throughout the entire simulation period;

$u_{k,i}(t)$: Option i Control input in dimension k (H/M/E);

T: Simulation termination time.

To reflect the principle that lower energy consumption is better, the energy efficiency metric is defined as:

$$S_{ene,i}={\displaystyle \frac{C_{ene}}{E_{total,i}+{\delta }_E}}.$$

(48)

In Equation (48):

$S_{ene,i}$: Energy efficiency indicator value for Plan i;

${\delta }_E$ > 0: A small constant (${\delta }_E$) to avoid numerical singularities at zero energy;

$C_{ene}$ > 0: Scale factor.

The smaller the total energy consumption, the larger the value of $S_{ene,i}$, indicating better synergistic effects achieved per unit of control input.

(4)

Cost–benefit indicators

To comprehensively evaluate the impact of synchronization speed and total energy consumption on system control performance, a cost–benefit metric is established to characterize the synchronization benefits achieved per unit of energy consumption per unit time. This metric is defined as follows:

$$S_{cp,i}={\displaystyle \frac{C_{cp}}{\left(t_{s,i}-t_c+{\delta }_t\right)\left(E_{\mathrm{total},i}+{\delta }_E\right)}}.$$

(49)

$S_{cp,i}$: Cost-effectiveness indicator value for Option i;

$C_{cp}$ > 0: Scale factor.

When two schemes achieve synchronization under similar synchronization error and reference trajectory conditions, the one that completes synchronization in less time and with lower energy consumption yields a higher value for $S_{cp,i}$, indicating a more favorable cost–benefit ratio.

2.6.2. Formula for Calculating the Comprehensive Score

To derive an overall optimal control scheme, it is necessary to comprehensively evaluate the importance of four metrics: synchronization speed, control accuracy, energy efficiency, and cost-effectiveness. Let the weight vector be defined as:

$$w={\left[w_{sync},w_{acc},w_{enc},w_{cp}\right]}^{\top }.$$

(50)

$w_{sync}$: Weight of synchronization speed;

$w_{acc}$: Weighting for control precision;

$w_{enc}$: Weighting for energy efficiency;

$w_{cp}$: Cost–benefit weighting.

Satisfies the normalization condition:

$$w_{sync}+w_{acc}+w_{enc}+w_{cp}=1$$

(51)

In accordance with the priority requirements for rapid restoration coordination and the maintenance of control accuracy during nuclear emergency response, this study assigns higher weights to synchronization speed and control accuracy in the performance evaluation, whereas energy efficiency and cost-effectiveness are given relatively lower weights. The specific weight distribution is as follows.

$$w_{sync}=0.35,\hspace{1em}{w}_{acc}=0.25,\hspace{1em}{w}_{enc}=0.20,\hspace{1em}{w}_{cp}=0.20.$$

(52)

The comprehensive score for the i-th control scheme is defined as:

$$S_i=w^{\top }{z}_i,\hspace{1em}S=Zw,$$

(53)

in Equation (53):

$$Z=\left[\begin{array}{cccc}{z}_{sync,1}& z_{acc,1}& z_{cne,1}& z_{cp,1}\\ z_{sync,2}& z_{acc,2}& z_{cne,2}& z_{cp,2}\\ ⋮& ⋮& ⋮& ⋮\\ z_{sync,N}& z_{acc,N}& z_{enc,N}& z_{cp,N}\end{array}\right],\hspace{1em}S={\left[S_1,\dots ,S_N\right]}^{\top },$$

(54)

here, S represents the composite score vector for all control schemes, where schemes with higher scores demonstrate superior overall performance across the four metrics.

3. Results

To validate the effectiveness of the proposed integrated model encompassing risk networks, chaotic evolution, and synchronous control. This section constructs a directed risk network for a nuclear emergency drill system.

3.1. Structural Characteristics and Key Node Identification of Nuclear Emergency Risk Networks

3.1.1. Risk Network Topology

Based on emergency drill data from nuclear power plants across China, a nuclear emergency risk network topology comprising 165 nodes and 389 directed edges was constructed, as shown in Figure 10.

As shown in Figure 10, the human factor (H), machine factor (M), and environmental factor (E) nodes exhibit a non-uniform, multi-clustered distribution. A small number of nodes with high in-degree and out-degree form network hubs and centers, leading to pronounced centrality in risk propagation. Once critical nodes malfunction, risk rapidly spreads along multiple paths, triggering system-level disorder. The network core simultaneously encompasses multiple highly connected human-factor nodes (H36 and H81) and machine-factor nodes (M01 and M05), which are densely interconnected with environmental nodes (E01 and E07). This indicates that nuclear emergency risk exhibits pronounced human–machine–environment closed-loop coupling characteristics. Environmental impacts are amplified through equipment linkages, ultimately manifesting as reduced human–machine coordination efficiency and accumulated misjudgments. This escalation leads to consequences such as personnel exposure to nuclear radiation (H86) and casualties (H115).

3.1.2. Node Risk Value Distribution and Critical Vulnerabilities

To measure the risk values of each node in the network, a comprehensive risk assessment method is employed. The risk values calculated for each node according to Formulas (3)–(7) are shown in Figure 11.

Figure 11 illustrates the distribution of node risk values derived from the risk-centrality index. The results reveal substantial differences in risk values across different node categories, with a characteristic long-tail distribution. A small number of actors exhibit markedly higher risk values than other nodes because they are connected to machine-related nodes. Several environmental nodes and the terminal consequence node H115 also stand out prominently. High-risk nodes typically exhibit both high betweenness centrality and large PageRank values. Among these, high-risk human-related nodes often correspond to recurring organizational and decision-making deficiencies in drills, such as H36, H30, H34, and H68. Once triggered, such deficiencies propagate and are amplified along multiple paths. High-risk machine-cause nodes are predominantly critical systems with strong support functions and limited redundancy (for example, M01, M03, M05, and M09). Their failure increases environmental risk E07 and personnel exposure risks H86 and H115. This indicates that relying solely on experience to select critical nodes is inherently limited. Introducing a risk-centrality metric based on topological coupling enables more systematic identification of vulnerabilities and more targeted prioritization of resources. Furthermore, when the weight coefficient α lies between 0.3 and 0.7, the relative ranking of high-risk nodes remains stable, demonstrating that this composite risk-centrality metric is robust to variations in the weighting parameter.

3.1.3. Outgoing and Incoming Distribution

The top ten nodes by in-degree and out-degree are detailed in

Table 4. Detailed statistics of the top ten in-degree and out-degree network nodes.

Top 10 Outdegree	List of Connected Nodes
H44	H100, H18, H87, H88, M28, M29, H106, H89, M35, H64, H49, H33
H03	H100, H87, H88, H32, H47, H51, M29, H106, H34, H37, H89
H04	H18, H88, M28, H39, M13, M26, M31, M32, M35, M37
H58	H55, H90, H32, M29, H37, H23, H48, H41, H62
H01	H61, H100, H18, H87, H88, H90, M21, M28
H02	H61, H88, H30, H32, H47, H51, H58, M29
H36	H81, H104, H80, H22, H52, H50, H83, H99
H21	M21, M28, M26, M31, M25, M04, M36
H35	H55, H67, H58, H105, H52, H36, H66
H66	M40, H87, H106, H89, H64, H49, H33
Top 10 indegree	List of connected nodes
H81	M03, M40, E08, E09, H55, H67, H80, H94, H52, M25, H13, H36, H48, H84, H43, H65
H115	H113, H114, H111, E07, H86, H81, H102, H107, H108, H94, H110, H84
M40	E02, M03, H67, H61, M28, M29, M26, H22, H110, H66, H85
H86	E07, H81, H104, H87, H88, H106, H107, H94, H56, H83, H98
H67	H78, H79, E09, H61, H35, H38, H40, H42, H41, M20
H35	M21, H30, H89, H49, M16, H29, H53, H45, H60, H70
H55	E09, H104, H58, H106, M31, H74, M36, H35, H76
H88	H01, H02, H03, H04, H05, H44, H70, H71, H72
H52	H34, M35, H64, H11, H36, H56, H35, H54
H106	H30, H03, H05, H11, H66, H44, H73

.

Table 4 shows that H44 and H03 serve as primary information hubs (out-degrees of 12 and 11, respectively), potentially acting as starting points within the network, whereas H81 and H115 function as key convergence centers (in-degrees of 16 and 12), occupying positions where resources and influence concentrate. Notably, H35 exhibits both a high out-degree (7) and a high in-degree (10), functioning as a crucial relay and regulatory node. At the same time, H88, as a common destination for multiple high-out-degree nodes, demonstrates terminal importance. The network exhibits marked heterogeneity, with node connections spanning H, M, and E categories, and M40 stands out as a cross-category interface. These highly connected nodes collectively form the core structure of the network and exert a controlling influence on information flow and resource transfer. These results provide a preliminary characterization of the structural model and key nodes involved in nuclear emergency risk propagation. However, whether this structural fragility will evolve into threshold transitions and chaotic uncertainty under dynamic feedback requires further validation through dynamical analysis.

3.2. Synchronized Control Effects

Based on the description in Section 2.4, the comparison of control errors with and without synchronization is shown in Figure 12.

Figure 12 compares the temporal evolution of the system error norm in the absence of control with that under the segmented feedback adaptive synchronization control strategy. Without control, the error may initially decrease briefly but then exhibit large oscillations or even amplification, resulting in unpredictable fluctuations. Once synchronization control is applied, the error decays rapidly and monotonically after control initiation (t = 10) and converges to a stable value close to zero within a finite time. The error vector e(t) represents the deviation between the response system (actual state) and the driving system (desired trajectory). The piecewise controller remains inactive for t < 10, allowing the system to evolve naturally under chaotic dynamics, during which the error may diverge in the high-dimensional phase space. When t ≥ 10, the control law is activated, suppressing e(t) through an adaptive gain k that is linked to the network’s overall coupling strength. This forms a negative feedback loop and achieves synchronization of the H–M–E triad of state variables. Figure 12 clearly shows that the controller effectively suppresses the continuous growth of error under chaotic conditions. For nuclear emergency response practice, building on established command structures and completed information aggregation, strengthening unified coordination, closed-loop execution, and dynamic collaboration can substantially reduce inconsistencies in responses across multiple departments and systems, thereby shortening the transition time from disordered emergency response to orderly coordination.

Figure 13 reveals the synchronous dynamical behavior under multi-subsystem coupling by comparing the state-evolution curves of the driving system and the response system in the three dimensions of human factors (H), machine factors (M), and environmental factors (E). During synchronization, the human-factor dimension exhibits large initial errors followed by monotonic asymptotic convergence. The machine-factor dimension achieves rapid and almost complete synchronization at the initial stage, whereas the environmental-factor dimension exhibits gradual convergence only after a period of small oscillatory decay. This divergence visually reflects the transient-response differentiation among the three subsystems, arising from their distinct intrinsic dynamical characteristics and from differences in the intensity of external disturbances. Overall, all three dimensions reach and maintain a highly coincident steady state within a finite time, thereby validating the global asymptotic stability and robustness of the proposed synchronization-control strategy. These results not only confirm, from a dynamical perspective, the potential for effective coordination in heterogeneous coupled systems but also provide theoretical foundations and methodological support for rapid situational alignment and coordinated risk control among human, machine, and environmental elements in nuclear emergency response systems.

3.3. Comparison and Comprehensive Evaluation of Control Schemes

Although Figure 12 and Figure 13 demonstrate that the proposed synchronization control strategy exhibits good overall convergence and robustness, in actual nuclear emergency collaborative response, different control resource allocation strategies (that is, differentiated investments across the human, machine, and environmental dimensions) will directly affect synchronization speed, control accuracy, and energy consumption costs. Therefore, this section carries out a quantitative comparison and comprehensive evaluation of three control schemes (human-dominated, machine-dominated, and environment-dominated, with configurations shown in Table 2) based on the performance evaluation model established in Section 2.5. The aim is to provide a basis for the selection and optimization of control strategies in engineering practice.

3.3.1. Comparison of Error Convergence Processes

Figure 14 shows the evolution curves of errors e1 (human-related), e2 (machine-related), and e3 (environment-related) over time under three control schemes, along with their convergence patterns in logarithmic coordinates.

As shown in Figure 14, the machine-dominant scheme (k = [25, 60, 15]) converges fastest across all dimensions, achieving near-instantaneous synchronization on the machine-factor dimension (M) immediately after control initiation. Its logarithmic error curve exhibits the steepest descent, reflecting a close match between the control–gain configuration and the deterministic characteristics of the machine-factor subsystem. The human-dominant scheme (k = [60, 25, 15]) exhibits smooth asymptotic convergence in the human-factor dimension (H), with initially large errors that later stabilize, but shows small residual fluctuations in the environmental dimension (E). This indicates a strong adjustment capability for human uncertainty but only limited direct suppression of environmental disturbances. The environment-dominant scheme (k = [25, 15, 60]) exhibits the slowest convergence in the environmental dimension (E), accompanied by pronounced oscillatory decay, reflecting the environmental subsystem’s sensitivity to external time-varying disturbances. This scheme, therefore, prioritizes disturbance compensation and robustness. All three schemes suppress the error below the prescribed threshold within a finite time, validating the effectiveness of the control structure. However, differences in the convergence dynamics reveal varying degrees of adaptation to the subsystem dynamics under different resource-allocation strategies.

3.3.2. Quantitative Analysis of Key Performance Indicators

Based on the four indicators defined in Section 2.4, the three schemes underwent normalization processing, yielding normalized composite scores for each dimension as shown in

Table 5. Control comprehensive score.

	Comparison Dimensions	Human-Dominated Solution	Machine-Dominated Solution	Environment-Dominated Solution
Parameter Design	Preset k value	[60, 25, 15]	[25, 60, 15]	[25, 15, 60]
	Actual gain k × γ	[7.50, 3.13, 1.88]	[3.13, 7.50, 1.88]	[3.13, 1.88, 7.50]
	Resource Allocation Ratio (Preset)	H:M:E = 60%:25%:15%	H:M:E = 25%:60%:15%	H:M:E = 25%:15%:60%
Synchronization Performance	Synchronize per unit time	12.98	11.95	11.32
	Post-control synchronization per unit time	2.98	1.95	1.32
	Final error	0.0	0.0	0.0
	Synchronization speed ranking	3	2	1
Energy consumption and efficiency	Total control energy	12,762.03	4203.06	7228.85
	Actual energy distribution	H:M:E = 88.5%:3.8%:7.7%	H:M:E = 34.7%:43.3%:22.0%	H:M:E = 18.3%:0.7%:81.0%
	Energy efficiency performance	Low	High	Moderate
	Comprehensive evaluation	0.6642	0.9817	0.8875
	Rank	3	1	2
	Features of the solution	Emphasizes human-centered adjustments, with high energy consumption but acceptable precision.	Balanced and fast, with optimal energy efficiency ratio	Fastest synchronization, but relatively high energy consumption

.

Table 5 shows that the control strategy based on differentiated allocation, using preset k values, yields pronounced contrasts among the three schemes in terms of synchronization performance and energy consumption costs. The environment-dominated scheme achieves the fastest synchronization, with a synchronization time of 11.32 time units, but at the cost of higher energy consumption. The human-dominated scheme exhibits the slowest synchronization, with a synchronization time of 12.98 time units and the highest energy consumption, and the actual energy input is excessively concentrated in the personnel dimension (88.5%), reflecting high human-factor regulation costs. The machine-dominated scheme achieves the best balance between synchronization speed (11.95 time units) and energy consumption (4203.06), requiring only 43.3% of the actual energy input in the machine dimension to achieve efficient synchronization. This demonstrates the advantages of the equipment subsystem in terms of rapid response and high determinism. A comprehensive evaluation indicates that the machine-dominated scheme (k = [25, 60, 15]) emerges as the preferred strategy for nuclear emergency collaborative control, with the highest overall score of 0.9817 and a good balance among rapidity, precision, and energy efficiency.

4. Conclusions

To address the limitations of conventional static and linear risk-assessment approaches in capturing the time-evolving, strongly coupled nature of nuclear emergency response, this study proposed and validated an integrated model of “risk network–chaotic evolution–synchronization control” using data from full-scope on-site emergency drills at a nuclear power base. The main conclusions are as follows.

(1)

A directed H–M–E nuclear emergency risk network was constructed, and key structural vulnerabilities were quantitatively identified. Based on drill records, incident reports, command scripts, and video materials, a directed risk network consisting of 165 risk-factor nodes and 389 causal trigger edges was established. Using the proposed risk centrality index, the network exhibits a clear heterogeneous and hub-dominated structure: a small number of nodes have high out-degree/in-degree and dominate the upstream triggering and downstream consequence aggregation. Quantitatively, the statistics of topological connectivity show that H44 and H03 act as major upstream dissemination hubs with out-degrees of 12 and 11, respectively, whereas H81 and H115 act as consequence convergence centers with in-degrees of 16 and 12, respectively. In addition, H35 simultaneously presents high out-degree and high in-degree, indicating a critical relay node that bridges multiple causal chains. These results demonstrate that nuclear emergency risk propagation is structurally centralized, implying that targeted interventions on hub and relay nodes can yield higher marginal returns than uniform resource allocation.

(2)

A topology-consistent three-dimensional nonlinear coupled dynamical model revealed threshold effects, bifurcations, and deterministic chaos in nuclear emergency risk evolution. By aggregating the 165 micro-level risk factors into three macro state variables representing human (H), machine (M), and environment (E), a continuous-time nonlinear system was developed to capture cross-domain coupling and multiplicative amplification effects. Numerical analysis indicates that the system can enter a chaotic regime under certain coupling intensities. In particular, for the representative parameter set a = 16, b = 40, and c = 2, the phase portrait forms a typical chaotic attractor, and the maximum Lyapunov exponent satisfies λ_max > 0, confirming sensitive dependence on initial conditions. Bifurcation analysis further shows that, as coupling parameters increase, state variables undergo the canonical route of equilibrium → period-doubling → multi-periodicity → chaos, with intermittent “window” regions of temporary stability, implying that nuclear emergency risk evolution is governed by critical thresholds and nonlinear state transitions rather than smooth linear escalation.

(3)

The proposed segmented adaptive synchronization controller effectively suppressed chaotic divergence and achieved fast H–M–E posture alignment. A drive–response synchronization architecture with a segmented feedback controller was designed, where control gains are adaptively tuned by the integrated network coupling strength. Simulation results demonstrate that, after control activation at t₀ = 10, the synchronization error norm changes from oscillatory divergence (uncontrolled case) to rapid monotonic decay and converges to a near-zero steady state within finite time. All three state dimensions (H, M, and E) achieve global asymptotic synchronization, verifying both the effectiveness of the controller structure and the validity of the Lyapunov stability condition used in the design.

(4)

Among three resource-allocation strategies, the machine-dominated control scheme achieved the best overall cost–performance, with explicit quantitative advantages.

Three differentiated control schemes were evaluated: human-dominated, machine-dominated, and environment-dominated. All schemes achieved zero final error in the simulations (final error ≈ 0.0) but differed significantly in synchronization time and energy cost: among the three schemes, the environment-dominated strategy synchronizes fastest (11.32 time units), followed by the machine-dominated (11.95) and human-dominated (12.98) strategies. However, the machine-dominated scheme yields the lowest total control energy (4203.06), versus 7228.85 (environment-dominated) and 12,762.03 (human-dominated), corresponding to energy reductions of 67.1% and 41.9%, respectively. In the weighted multi-metric assessment, the machine-dominated scheme ranks first (0.9817), ahead of the environment-dominated (0.8875) and human-dominated (0.6642) schemes.

These results indicate that while environment-focused interventions can yield the fastest synchronization, they incur higher energy costs, whereas human-focused interventions are both slower and significantly more expensive in terms of control energy. The machine-dominated strategy provides the most favorable trade-off between convergence speed and energy expenditure, consistent with the higher determinism and faster response characteristics of equipment-related channels in emergency control loops.

In the present study, synchronization control is implemented in a model-based framework, where the drive and response systems share the same set of nonlinear equations, and full-state information is available in simulation. In practical applications, however, only short and noisy time series of macro-level H–M–E indicators may be observable. Under such conditions, conventional model-based synchronization may converge slowly or become sensitive to data truncation. Thus, integrating time-reversible synchronization or other data-driven synchronization techniques with the proposed risk network–chaotic evolution framework, and validating them on real drill time series, will be an important direction for future research. Given that the entire analytical chain is built on a constructed causal network inferred from drill records, the results should be interpreted as semi-quantitative insights into structural vulnerabilities and dynamic behaviors, rather than as precise numerical risk predictions. Future work will incorporate probabilistic data and learning-based approaches to further quantify edge propagation probabilities.