Stabilizing Chaotic Food Supply Chains: A Four-Tier Nonlinear Control Framework for Sustainability Outcomes

Abstract

Food supply chains play a critical role in advancing sustainability within today’s food systems. In this work, we construct a differential equation-based model with a four-layer supply chain framework that captures the intricate relationships among producers, manufacturers, distributors, and retailers while considering resource optimization, waste minimization, and supply–demand equilibrium. To better understand and predict supply chain behavior, we perform a series of model analyses. By applying chaos theory, we analyze the system’s equilibrium states and evaluate their local stability. Our findings reveal that manufacturers and retailers encounter significant difficulties when the system shifts into chaotic behavior. This can be made worse by future uncertainties. This entails formulating tailored strategies to mitigate risks. Hence, we design a set of nonlinear feedback control strategies to synchronize two chaotic supply chain networks. Theoretical validity is established using Lyapunov theory. Our simulation results confirm that the proposed strategy can eliminate synchronization errors. Furthermore, it allows for swift alignment and coordination between the networks. Overall, this synchronization method is both effective and easy to implement for managing risks and enhancing sustainability in food supply chains affected by chaotic dynamics.

1. Introduction

Chaotic dynamics models have been extensively applied across diverse areas, including industrial processes and social sciences. A variety of chaotic systems have been identified in the literature. For example, the Lorenz attractor was the first chaotic system which was discovered in 1963 [1]. Since then, many chaotic models have been developed and continuously refined through experiments and simulations. The Chua circuit, the first chaotic system to be implemented electronically, has a double-scroll attractor [2]. Likewise, the Rössler system, derived from chemical experimentation, has been shown to produce chaotic attractors as well [3]. Chaos phenomena have found widespread applications across various fields due to their unique characteristics. In electronics, chaotic circuits are employed to generate pseudo-random signals, which are crucial for secure and efficient communication systems. In the realm of finance, chaos theory is used to describe the nonlinear behavior of market dynamics, where sudden and dramatic fluctuations in prices often exhibit chaotic patterns. Although chaos appears random and unpredictable, it arises from deterministic systems that are highly sensitive to initial conditions. This sensitivity means that even minor variations in starting points can lead to vastly different outcomes, a concept commonly known as the “butterfly effect.” The inherent unpredictability of chaotic behavior gives rise to considerable uncertainty and potential hazards, underscoring the importance of implementing robust control and mitigation approaches. There has been growing interest among researchers in developing methods to manage and synchronize chaotic systems to minimize their adverse impacts. Many such systems are derived from models based on differential equations. Chaotic dynamical systems are a type of nonlinear system distinguished by their extreme sensitivity to initial conditions, as well as their persistent irregular and seemingly random behavior.

In this paper, we develop a supply chain model based on ordinary differential equations to analyze key links in the food supply chain—such as production, processing, transportation, storage, and sales—and their interconnections. We conduct both theoretical analysis and numerical simulations to explore how these links influence each other. The results help us identify weak links and uncertainties within the supply chain, as well as principles and strategies for addressing these uncertainties and reducing risks. Furthermore, we propose methods to coordinate the different links in the supply chain, thereby enhancing the overall system’s stability. Our goal is to design control methods that are effective, easy to use, and capable of ensuring the safe, reliable, and stable operation of the entire supply chain. With the proposed control strategy, the slave system’s output gradually aligns with that of the master system, ensuring that the synchronization error approaches zero asymptotically.

In this study, we make three main contributions. First, we develop an explicit four-tier, food-specific nonlinear dynamical model (manufacturer–primary distributor–secondary distributor–retailer). Second, we design three implementable nonlinear synchronization controllers. We establish global exponential convergence of the synchronization errors using Lyapunov analysis. Third, we connect the resulting stabilization mechanism to sustainability-relevant operational outcomes, including mitigating amplification effects (bullwhip-like fluctuations) and improving coordination/service reliability. In short, relative to prior chaos-control studies that focus on simpler or more abstract supply chain structures, our framework provides a food-grounded four-tier model and explicitly links stabilization to sustainability related outcomes.

The remainder of this paper is structured as follows: Section 2 introduces a four-layer nonlinear food supply chain model, defining all state variables and parameters, and discussing the conditions under which chaotic behavior occurs. We also demonstrate the chaotic behavior of the model in this section. Section 3 introduces a driver–response synchronization framework, stating the modeling assumptions and constructing a synchronization error system. Theoretical proofs and numerical simulations are provided. Section 4 discusses practical implementation issues and managerial implications and outlines future research directions.

2. Literature Review

In this section, we review relevant studies related to (1) foundational chaos control and synchronization theory; (2) Nonlinear dynamics and coordination in supply chain systems; (3) Uncertainty, disruptions, and sustainability in (food) supply chains. We then position the present study relative to closely related works.

2.1. Foundational Chaos Control and Synchronization Theory

In response to the pseudo-random dynamics inherent in chaotic systems, various methods for chaos suppression have been explored by researchers. One notable advancement came in 1990 with the development of the OGY chaos control method [4], which effectively stabilized chaotic dynamics. For many years, the complexity of chaos led to the widespread belief that systems with differing initial conditions could not be synchronized. This assumption was challenged by the groundbreaking work of Ref. [5], who proved that synchronization of chaotic systems is indeed possible. They were the first to propose a method that enabled identical chaotic systems, even with different initial conditions, to achieve synchronization. These foundational advances in chaos control and synchronization theory provide the analytical basis for applying nonlinear dynamical tools to complex systems such as supply chains.

2.2. Nonlinear Dynamics and Coordination in Supply Chain Systems

An efficient and resilient supply chain is essential for promoting economic growth. In recent years, supply chain management has garnered significant attention. To improve the competitiveness and efficiency of supply chain operations, it is essential to integrate multiple functional areas and align the movement of goods, information, and capital to satisfy customer expectations. In addition, managing today’s expansive and intricately linked supply chains presents significant challenges due to the complexity of interdepartmental coordination and the broad reach of supply chain networks. The growing demand from consumers for personalized products and quicker delivery times is making it more challenging to streamline supply chain operations. The central goal of supply chain management remains to fulfill customer needs by delivering high-quality service at reduced costs.

In the past ten years, a substantial body of research has emerged focusing on the design, modeling, and analysis of supply chains to better understand their complex dynamics [6,7,8,9,10,11,12]. For example, Ma et al. [13] explored both local and global dynamics in apparel supply chains by formulating a mathematical model under a global framework. Their simulations revealed phenomena such as coexisting attractors and multi-stability. Similarly, Askar [14] introduced a nonlinear pricing mechanism into a specific supply chain model, emphasizing the study of synchronization and multi-stability behavior. Du et al. [15] explored the interplay between consumer dissatisfaction and inventory levels in the context of omni-channel retailing, emphasizing their combined effect on determining optimal pricing strategies. In a related study, Xu and Jackson [16] examined customer perceptions of the product return process, shedding light on how return experiences shape consumer behavior. Saha et al. [17] analyzed the influence of pricing and service effort on demand, aiming to enhance the understanding of effective coordination mechanisms within the supply chain. Building on this line of research, Liu et al. [18] employed the Cournot duopoly model to investigate supply chain dynamics, with particular attention to the structural characteristics of the model’s basin of attraction. To promote sustainable supply chain development, Zhou and Liu [19] introduced a two-stage modeling approach and provided strategic insights for selecting effective initial conditions. Addressing dynamic inventory levels and shifting consumer preferences, Gallino and Moreno [20] conducted data-driven analyses that highlighted the importance of sharing accurate inventory availability to enhance the performance of Buy Online, Pick Up in Store (BOPS) programs. Ei Ouardighi and Kim [21] examined the trade-offs between wholesale pricing and profit-sharing contracts, offering comparative insights into their respective advantages. Meanwhile, Chakraborty et al. [22] explored pricing strategies across different scenarios to alleviate inter-channel conflicts, proposing revenue-sharing schemes to strengthen coordination and foster collaboration among supply chain participants.

2.3. Uncertainty, Disruptions, and Sustainability in (Food) Supply Chains

The intrinsic uncertainty within supply chains is manifested through dynamic variations in customer demand, inventory levels, and production costs. These stochastic fluctuations continually challenge the stability and operational efficiency of supply chain systems. To address this, Chen et al. [23] developed a stochastic equilibrium model that captures these probabilistic elements and provides insights into system behavior under uncertainty. With the growing prominence of omni-channel retailing, researchers have increasingly focused on its integration within supply chains. Gao and Su [24], for example, investigated the operational implications of Buy Online, Pick Up In Store (BOPS) systems on physical retail performance. To counteract disruptions caused by major external events, Zheng et al. [25] proposed an optimization framework for managing ordering and inventory decisions in both decentralized and centralized supply chain configurations. In the food industry, maintaining a resilient and efficient supply chain is especially vital. Serbia’s dairy industry has faced ongoing challenges, prompting researchers like Milić et al. [26] to develop a panel data framework incorporating technical efficiency indicators to enhance the sector’s competitiveness. This model assesses the influence of multiple variables on the profitability of milk and dairy production. More recently, growing academic interest has centered on the problem of resource inefficiencies and waste within the dairy supply chain. To assess productivity and efficiency across the European dairy industry, Ref. [27] applied a two-stage stochastic frontier analysis, providing reliable estimates of the sector’s technical efficiency. In another study, Chiaraluce et al. [28] conducted a bibliometric analysis and literature review, exploring how circular economy concepts are integrated into the agri-food industry. Their findings highlighted significant linkages between circular economy strategies and food system sustainability. In a related study, Ding et al. [29] developed a simulation model that integrates a contract penalty mechanism to explore collaborative green innovation among stakeholders in the livestock supply chain. Their findings offer valuable insights into enhancing value chains and ensuring the resilience and sustainability of livestock product supply networks.

Academic and industry interest in sustainable food supply chains has risen significantly over the past few years [30]. Ensuring a resilient, dependable, and sustainable food supply system is vital for global well-being. Projections from the United Nations in 2022 estimate that the world population will rise to between 9.4 and 10 billion by 2050 [31], highlighting the urgent need for strategic planning and sustainable solutions to meet the demands of this growth. Population changes will create new demands on supply chains. To handle shifts in population size and structure, supply chains must become more agile. At the same time, climate-related disruptions are making supply chains more unstable and unpredictable. To ensure enough food for the future, we need stronger and more sustainable food distribution systems.

Modeling studies of supply chains help us analyze and predict each link in the food supply chain [32]. They can show when demand or supply might change, which supports better production planning and inventory management. By examining these models, we can draw both quantitative and qualitative conclusions that offer valuable insights [33]. Models also help identify uncertain and risky factors in the supply chain, providing a foundation for developing strategies to prevent these issues.

Table 1. Summary of Selected Studies on Supply Chain Dynamics, Chaos, and Sustainability.

Author(s) & Year	Focus Area/Context	Method/Model Used	Key Findings/Contributions	Relevance to Present Study
Ma et al. (2022) [13]	Apparel supply chain dynamics	Mathematical model under global framework	Revealed coexisting attractors & multi-stability	Shows how chaos affects practical supply chains
Askar (2022) [14]	Nonlinear pricing in supply chains	Duopoly game model	Studied synchronization & multi-stability	Relevant for pricing/coordination in chaotic supply chains
Liu et al. (2022) [18]	Supply chain oligopoly dynamics	Cournot duopoly model	Investigated basin of attraction structure	Connects market competition to chaotic outcomes
Zhou & Liu (2023) [19]	R&D competition & sustainability	Two-stage modeling	Suggested effective initial conditions for sustainability	Offers sustainability strategies under chaos
Zheng et al. (2023) [25]	Disruptions in logistics routes	Optimization framework	Coordinated inventory policies under disruptions	Enhances resilience against chaotic disruptions
Milić et al. (2023) [26]	Dairy supply chain in Serbia	Panel data with efficiency indicators	Improved competitiveness analysis	Extends chaos/sustainability discussion to food sector
Ding et al. (2024) [29]	Livestock supply chain innovation	Simulation with penalty contracts	Improved collaborative green innovation	Adds mechanisms for resilience & sustainability

presents a summary of selected studies related to supply chain dynamics, chaos, and sustainability. The table highlights the focus area, modeling approach, and main findings of each study, as well as their relevance to this study.

2.4. Positioning the Present Study Relative to Closely Related Works

To more clearly position the present study within this literature, the following discussion contrasts our approach with several closely related studies. Compared with [25], which developed an optimization-based two-level model for designing inventory strategies, contingency routes, and price discount contracts under predictable logistics route disruption risks, our research focuses on modeling the dynamic behavior and coordination of a four-tier food supply chain as a nonlinear chaotic system. Ref. [25] analyzes centralized versus decentralized structures, logistics-path damping and acceleration coefficients, and vehicle capacity ratios to improve profits and reduce emissions under planned disruptions, whereas our paper proposes three explicit synchronization strategies for interactions among manufacturers, distributors, and retailers, demonstrates global exponential synchronization, and links these controls to management leverage and sustainability-oriented performance indicators such as bullwhip-effect reduction, improved service levels, reduced perishable waste, and lower reliance on expedited shipping. Therefore, while Ref. [25] focuses on route planning and inventory recovery from a static optimization perspective, our work provides a dynamical-systems-based framework for stabilizing and coordinating multi-tier food supply chains under complex fluctuations.

Compared with [34], who analyzed the synchronization and control of two identical abstract systems using a three-dimensional chaotic supply chain model with active controllers and linear feedback, this paper develops an explicit, food-specific, four-level model (manufacturer–primary distributor–secondary distributor–retailer) in which all states are interpreted as operational decision biases. Our model is more complex and reveals the interrelationships and interactions between different levels. This paper advances a more structurally rich, sustainability-oriented framework that captures the interactions and coordination mechanisms across multiple tiers of the food supply chain. By explicitly linking control strategies to managerial levers and performance indicators, it provides a clearer path from theoretical design to practical implementation.

Compared to [26], who studied the synchronization of a general three-stage chaotic supply chain model and designed an active linear feedback controller to achieve local or global exponential synchronization between two identical abstract systems, this paper constructs an explicit four-layer food supply chain model (manufacturer–primary distributor–secondary distributor–retailer) and proposes a series of control laws to address the synchronization problem. Our work introduces a structurally richer and more application-oriented framework.

In this paper, we propose several synchronization strategies and prove the global exponential convergence of synchronization errors using Lyapunov functions. Rigorous mathematical proofs ensure the effectiveness of the control strategies. Numerical simulation results demonstrate the usability of the control strategies. Compared with the literature, these results are more practical and reliable. We link chaos mitigation and synchronization with measurable outcomes, including reducing the bullwhip effect, service levels, perishable waste, and reliance on expedited transportation, thus positioning the control framework as a sustainable food supply chain management tool.

3. Model Development for a Class of Food Supply Chains

This section introduces a dynamic food supply chain model characterized by uncertainty and formulated through ordinary differential equations. Inspired by the three-level bullwhip-effect model and synchronization scheme proposed by Zhang et al. [35], we expand the structure to incorporate an additional tier, resulting in a four-level system. Accordingly, our model is governed by four differential equations.

Subsequent analysis reveals that the system exhibits chaotic and intricate behavior. As depicted in Figure 1, the model comprises four distinct levels: retailers, primary distributors, secondary distributors, and manufacturers. The network is both structured and complex, with well-defined channels of information and material flow linking each level of the supply chain. Note that downward arrows represent physical/material flow and upward represent information and order flow.

The supply chain comprises manufacturers, primary and secondary distributors, and retailers, each playing a distinct but interconnected role. Manufacturers are responsible for the creation of goods, encompassing design, production, and packaging. Once completed, these products are sold to primary distributors—intermediaries who serve as the manufacturers’ first point of market contact. They purchase goods in bulk and distribute them to downstream partners. Secondary distributors operate further along the chain, receiving goods from primary distributors and often supplying them to smaller retailers or directly to consumers. At the end of the chain, there are retailers who provide the final sales interface with end customers.

The four-tier supply chain considered in this study is representative of many real-world food distribution systems. For example, in fresh dairy, meat, and produce supply chains, manufacturers typically supply products to regional (primary) distributors, who further allocate inventory to local (secondary) distributors or wholesalers before reaching retailers. Demand signals, promotional information, and inventory feedback are passed on to upstream across these tiers with time delays. Furthermore, ordering and replenishment decisions at each level are influenced by both local conditions and upstream–downstream interactions. Our proposed model abstracts these interactions into a nonlinear dynamical framework. This allows us to study their instability, synchronization, and coordination issues.

These businesses are interconnected and interdependent, working together as a team to enhance overall efficiency. Distributors help manufacturers reach more customers by purchasing and storing products. In turn, manufacturers rely on distributors to deliver products quickly and conveniently to various stores and markets, creating a smooth flow of goods through the supply chain.

As people’s dietary preferences become more diverse, consumer purchasing habits have also changed significantly. Instead of buying large quantities at once, most consumers now prefer smaller, more frequent purchases to ensure freshness and variety. This shift presents both opportunities and challenges for the food supply chain.

Retailers are no longer just sellers of goods; they serve as a vital link between manufacturers and consumers. With direct access to shoppers, retailers can gather up-to-date and accurate consumer insights. By relaying this information to manufacturers, they help guide product development and production adjustments, thereby improving resource efficiency. This feedback loop enables continuous optimization and enhancement of the entire supply chain. It strengthens connections across the supply chain but also contributes to its complexity. Ensuring prompt and efficient delivery of food products now requires careful coordination among all supply chain participants.

In response to these challenges, we model a four-tier food supply chain consisting of manufacturers (tier 1), primary distributors (tier 2), secondary distributors (tier 3), and retailers (tier 4). In terms of methodology, in this paper, we employs a nonlinear dynamical framework to supply chain modeling. Within this framework, we follow three key steps. First, we construct a food-specific multi-tier nonlinear adjustment system with explicit operational interpretations. Second, we analyze the system’s stability and chaotic behavior. Third, we design synchronization control laws and establish global exponential convergence using Lyapunov-based analysis. Our methodology differs from prior optimization-based or equilibrium-focused supply chain models because we emphasize transient dynamics, instability mitigation, and coordination under complex fluctuations rather than static optimal solutions.

Table 2. List of Parameters in the Chaotic Supply Chain Model.

Symbol	Description	Role in System (1)
a	Retailer self-adjustment rate (time−1)	Strength of local damping in ẋ.
b	Secondary distributor self-adjustment rate (time−1)	Strength of local damping in ẏ.
m	Coupling/adjustment coefficient for interactions involving x and z	Governs response to retailer in ẏ and damping of z.
n	Coupling/adjustment coefficient for interactions involving y and w	Governs response to y in ż and damping of w.
r	End-tier information feedback strength	Links retailer–secondary and primary manufacturer in ẋ, w.
k	Upstream coordination strength between adjacent midstream tiers	Couples y–z and z–w in ẏ, ż.
x	Retailer adjustment variable (deviation of order/inventory rate at tier 4)	State variable in Equation (1).
y	Secondary distributor adjustment variable (tier 3)	State variable in Equation (1).
z	Primary distributor adjustment variable (tier 2)	State variable in Equation (1).
w	Manufacturer adjustment variable (tier 1)	State variable in Equation (1).

summarizes the key symbols, parameters, and state variables used in the supply chain dynamics model. Each symbol is described with its corresponding role and interpretation within the system. Let $w\left(t\right),z\left(t\right),y\left(t\right),x\left(t\right)$ denote, respectively, the dynamic responses (order/inventory adjustment rates) at the manufacturer, primary distributor, secondary distributor, and retailer levels, expressed as deviations from their equilibrium values. The proposed model is presented below:

$$\left\{\begin{array}{l}{x}^{\prime }=my-\left(n+1\right)x-aw,\\ y^{\prime }=rx-y-yz,\\ z^{\prime }= xy+(k-1)z,\\ w^{\prime }=xz-\left(k-1\right)w+bz,\end{array} \right.$$

(1)

where all parameters are assumed positive unless stated otherwise. In system (1), the state variables are ordered from downstream to upstream. Specifically, $x\left(t\right)$ denotes the retailer adjustment deviation (level 4), $y\left(t\right)$ the secondary distributor adjustment deviation (level 3), $z\left(t\right)$ the primary distributor adjustment deviation (level 2), and $w\left(t\right)$ the manufacturer adjustment deviation (level 1).

The model consists of three components: (i) self-correction within each level that pulls the state back toward equilibrium, (ii) cross-level coupling that captures how each level responds to signals from neighboring levels (via shared information and/or material-flow changes), and (iii) nonlinear interactions that represent amplification, friction, and mutual constraints when multiple levels deviate simultaneously.

Next, we explain each differential equation term by term.

(i).

Retailer tier ($\dot{x}$).

The retailer dynamics are given by

$$\dot{\mathit{x}}=\mathit{m}\mathit{y}-(\mathit{n}+\mathbf{1})\mathit{x}-\mathit{a}\mathit{w}\mathbf{.}$$

Here, $-(n+1)x$ represents the retailer’s local stabilization measures (including order corrections and inventory rebalancing). These measures aim to mitigate deviations and bring them towards equilibrium. The term my reflects the retailer’s response to the status of secondary distributors. The term aw represents the upstream impact from manufacturer deviations, which may alter downstream behavior and status.

(ii).

Secondary distributor tier ($\dot{y}$).

The secondary distributor dynamics are

$$\dot{y}=rx-y-yz.$$

The linear term $y$ represents local damping (relaxation toward equilibrium). The coupling term $rx$ represents downstream feedback from retailers: deviations at the retailer level transmit upstream and influence the secondary distributor’s adjustment. The nonlinear term $yz$ captures midstream interaction effects; it is a bilinear cross-tier term whose impact depends on the signs and magnitudes of $y$ and $z$. Large simultaneous deviations can intensify coordination frictions (e.g., congestion, lead-time effects, allocation rules, or capacity conflicts), thereby reducing or, in some regimes, amplifying effective adjustments.

(iii).

Primary distributor tier ($\dot{z}$).

The primary distributor dynamics are

$$\dot{z}=xy+(k-1)z.$$

The linear term $(k-1)z$ captures how the coordination mechanism alters the persistence of the primary distributor’s deviation: if $k<1$, it strengthens damping so $z$ decays more quickly; if $k>1$, it weakens damping and can reinforce $z$, allowing the deviation to persist or grow. The nonlinear term $xy$ represents downstream interaction effects: when retailer and secondary-distributor deviations occur simultaneously (especially with the same sign), their combined signals reinforce midstream coordination pressure and amplify the primary distributor’s adjustment response.

(iv).

Manufacturer tier ($\dot{w}$).

The manufacturer dynamics are

$$\dot{\mathit{w}}=\mathit{x}\mathit{z}-(\mathit{k}-\mathbf{1})\mathit{w}+\mathit{b}\mathit{z}.$$

The term $(-(k-1\left)w\right)$ represents the manufacturer’s local stabilization (i.e., linear damping toward equilibrium, assuming ($k>1$)). The term (bz) captures feedforward influence from the primary distributor to the manufacturer (e.g., upstream transmission of backlog/inventory status and replenishment demand). The nonlinear term ($xz$) represents an interaction effect: when retailer-level deviation and primary distributor deviation occur simultaneously, the resulting compounded signal can amplify production pressure and trigger a stronger upstream response under stress conditions (e.g., sudden replenishment demand combined with midstream imbalance). For clarity,

Table 3. Term-to-action interpretation for Equation (1): operational meaning and managerial levers.

Equation Term	Operational Interpretation (Plain Language)	Managerial/Policy Lever (What It Means to “Increase/Decrease” the Term)
m y$\mathrm{in} \dot{x}$	Retailers make adjustments based on the status/information of secondary distributors (product availability, service levels, and replenishment coordination).	Growth (amplification) can be promoted by faster replenishment updates, stronger supplier-side inventory coordination, and quicker response actions. Conversely, growth can be reduced (damped) by strengthening smoothing mechanisms—such as tighter order caps and/or longer review cycles.
$-(n+1)x$$\mathrm{in} \dot{x}$	Retailers self-correct/regress to equilibrium.	Strengthening smoothing—via more conservative ordering and tighter ordering constraints—increases $n$ (i.e., stronger damping/less responsiveness). In contrast, more proactive and faster retailer responses decrease $n$ (i.e., weaker damping/greater responsiveness).
$-a w \mathrm{in} \dot{x}$	Upstream manufacturer deviations can influence retailers’ adjustments through allocation constraints, supply disruptions, and shortage signals.	Supply risk can be reduced through dual sourcing, safety stocks, and flexible contracts; however, available supply may be reduced under strict quota systems or tighter upstream restrictions
$rx$ $\mathrm{in} \dot{y}$	Secondary distributor responds to retailer signal (information feedback strength).	Sharing POS data, communicating orders frequently, and engaging in collaborative planning increase $r$, whereas delays, noisy signals, or limited information sharing decrease $r$.
$-y$ in $\dot{y}$	Secondary distributor local stabilization (internal smoothing).	Enhance internal control (to achieve more effective shock absorption); weakening it will make it exhibit a stronger responsiveness.
$xy$ in ż	Primary distributor reacts strongly when downstream deviations co-occur (joint downstream pressure).	The effect is alleviated by smoothing downstream orders, improving demand coordination, and increasing buffers; it is exacerbated by aggressive/over-reactive downstream policies and high volatility.
$(k-1$)z in ż	Persistence/mitigation of primary distributor bias based on coordination mechanisms.	Improve coordination through stable allocation, reliable delivery dates, and consistent planning rhythms (adjusted through strategies).
$xz$$\mathrm{in} \dot{w}$	Manufacturer pressure increases when retailer and primary-distributor deviations co-occur.	Reduce pressure by stabilizing downstream and midstream signals; increase pressure in cases of emergency replenishment and midstream imbalance.
$-(k-1)w$$\mathrm{in} \dot{w}$	Manufacturer stabilization/capacity correction toward equilibrium shaped by coordination parameter k.	The situation can be improved by increasing capacity flexibility, developing stable production plans, and ensuring reliable procurement, whereas poor coordination can increase instability.
$bz$ $\mathrm{in} \dot{w}$	Feedforward from primary distributor to manufacturer (backlog/inventory needs driving production).	Increase the b-value by implementing closer upstream planning (linked to distributors’ inventory/backlog situations, such as by adopting stricter replenishment trigger mechanisms); decrease the b-value by eliminating buffers or extending planning cycles.

summarizes the term-to-action interpretations for Equation (1).

In modern, industrialized food systems, most of the food we consume reaches consumers through supply chains, and many products are perishable (e.g., dairy or fresh produce). In perishable-food supply chains, even small demand shocks can propagate upstream due to frequent replenishment, limited visibility, and delayed responses. When each tier perceives a deviation from expected demand, it adjusts its orders and transmits information to other tiers. This feedback can amplify fluctuations, leading to losses such as overordering driven by forecasting errors and emergency shipments triggered by shortages. These effects increase costs, waste, and CO₂ emissions. In the four-tier model, the state variables represent normalized deviations from an operating equilibrium (e.g., $x\left(t\right),y\left(t\right),z\left(t\right)$, $w\left(t\right)$ as manufacturer-to-retailer adjustment deviations), and time t can be interpreted as a decision cycle clock (e.g., weeks). Parameters such as a and b act as relaxation rates (1/time), while coupling strengths (e.g., m, n, k, r) quantify how aggressively tiers respond to cross-tier mismatches. Consider a $10\mathrm{\%}$ retailer–distributor mismatch, so $w-z\approx 0.10$. With $m=11$, the implied corrective intensity is $m(w-z)\approx 1.1$ per week (prior to damping and other interactions), consistent with rapid replenishment adjustments. Depending on the overall feedback structure, such a strong response may stabilize the network or amplify oscillations, potentially shifting the system to a different dynamical regime.

Our numerical examples use a baseline parameter set (e.g., $=0.5,b=0.4,m=11,n=8,r=25,k=-2$). We selected these values because they represent moderate local damping together with sufficiently strong cross-tier feedback, so deviations neither decay immediately nor grow without bound. Under this baseline set, the model produces a chaotic attractor (as indicated by the numerical simulations), which is consistent with food supply chains where frequent updating and imperfect coordination can generate persistent, irregular oscillations.

Equation (1) models the supply chain using a system of differential equations, which exhibits a wide range of complex dynamic patterns. Under specific parameter settings, the system generates highly intricate chaotic behavior. To illustrate these dynamics, we provide a series of numerical simulations.

Figure 2 displays the system’s phase portraits projected onto various coordinate planes, offering visual insight into its evolution. The trajectories form chaotic attractors, highlighting the system’s underlying complexity and unpredictable nature.

To illustrate the chaotic trajectories of the system in three-dimensional space, we carried out numerical simulations. The resulting visualizations are shown in Figure 3. Specifically, the system’s dynamics are projected onto the (a) x-y-z and (b) x-y-w spatial configurations, highlighting the system’s intricate and complex behavior.

Chaotic systems are known for their extreme sensitivity to initial conditions—tiny differences in starting values can lead to dramatically different outcomes. To illustrate this characteristic, we conducted numerical simulations that track the system’s time evolution under slightly varied initial conditions.

Specifically, we selected two nearby initial states: (0.1, 0.2, 0.15, 0.3) and (0.15, 0.25, 0.1, 0.35). Although the differences between these two sets are minimal, the resulting system trajectories diverge significantly. As depicted in Figure 4, this divergence highlights the system’s pronounced sensitivity to initial inputs—a hallmark of chaotic behavior.

System (1) demonstrates complex dynamical patterns across a broad spectrum of parameter values. To explore these behaviors, we generate bifurcation diagrams by varying key parameters. Figure 5 illustrates how the system’s dynamics evolve as the parameter m changes, revealing a wide array of intricate behaviors. Similarly, Figure 6 captures the system’s response to changes in the parameter n, showcasing the richness and diversity of its dynamic regimes.

Due to space limitations, we include only two representative bifurcation diagrams (for parameters m and n). These parameters govern cross-layer responsiveness and coordination and as indicated by our sensitivity analysis, exert the strongest effect on variability near the baseline.

4. Synchronization of the Chaos Supply Chain Model

In this section, we examine the synchronization of two supply chain systems that begin with different initial conditions. Due to the chaotic and chaotic nature of these systems, their trajectories are highly sensitive to initial states and often exhibit pseudo-random dynamics. This unpredictability makes it difficult to forecast the behavior of such systems with precision.

To address this challenge and mitigate potential risks within chaotic supply chains, we apply a synchronization-based control strategy. When synchronization is achieved, the behavior of one system can be used to predict the behavior of another, even if they start from different initial states. This approach is particularly valuable in real-world applications where maintaining consistent behavior across multiple supply chains is essential.

To implement this, we construct a pair of chaotic supply chain models: the drive system (indexed by subscript 1) and the response system (indexed by subscript 2). The mathematical formulation of these systems is presented below.

$$\left\{\begin{array}{l}{x}_1^{\prime }=m{y}_1-\left(n+1\right)x_1-a{w}_1,\\ y_1^{\prime }=r{x}_1-y_1-y_1{z}_1, \\ z_1^{\prime }=x_1{y}_1+(k-1)z_1,\\ w_1^{\prime }=x_1{z}_1+\left(k-1\right)w_1,\end{array} \right.$$

(2)

and

$$\left\{\begin{array}{l}{x}_2^{\prime }=m{y}_2-\left(n+1\right)x_2-a{w}_2+s_1\left(t\right),\\ y_2^{\prime }=r{x}_2-y_2-y_2{z}_2+s_2\left(t\right),\\ z_2^{\prime }=x_2{y}_2+(k-1)z_2+s_3\left(t\right),\\ w_2^{\prime }=x_2{z}_2+\left(k-1\right)w_2-b{z}_2+s_4\left(t\right),\end{array} \right.$$

(3)

where functions $s_1\left(t\right), s_2\left(t\right), s_3\left(t\right),$ and $s_4\left(t\right)$ represent the control variables awaiting determination. This section aims to develop an appropriate control function to achieve synchronization between the drive and response supply chain systems. To assess the degree of synchronization, we analyze the difference between the two systems by subtracting the response system from the drive system, leading to the following formulation:

$$\begin{array}{c}{e}_1=x_2-x_1,\\ e_2=y_2-y_1,\\ e_3=z_2-z_1,\\ e_4=w_2-w_1.\end{array}$$

(4)

The variables $e_1,e_2,e_3$ and $e_4$ represent the deviations between the corresponding components of the drive and response systems. We now proceed to analyze the derivatives of system (4), from which we derive the following results:

$$\left\{\begin{array}{c}{e}_1^{\prime }=m{e}_2-\left(n+1\right)e_1-a{e}_4+s_1\left(t\right)\\ e_2^{\prime }=r{e}_1-e_2-y_2{z}_2+y_1{z}_1 +s_2\left(t\right)\\ e_3^{\prime }= x_2{y}_2-x_1{y}_1+(k-1)e_3 +s_3\left(t\right)\\ e_4^{\prime }=x_2{z}_2-x_1{z}_1 +\left(k-1\right)e_4 +s_4\left(t\right)\end{array} \right.$$

(5)

Equation (5) represents the error system, which captures the differences between the drive and response supply chain systems. The goal is to synchronize these systems such that their behaviors align over time. As synchronization is achieved, the error terms progressively decrease and eventually converge to zero—indicating that the error system’s trajectory settles at the origin.

In the following, we introduce a series of control strategies designed to guide the systems toward synchronization.

4.1. Synchronization Strategy 1

Synchronization Strategy 1 introduces a simple control law to align two chaotic supply chain systems (the drive and response systems) that start from different initial conditions. By designing a Lyapunov function, the authors prove mathematically that the error system—which captures the difference between the two models—is globally and exponentially stable under this control. In other words, despite starting with different trajectories, the two systems gradually synchronize, and their outputs converge to the same behavior over time. Numerical simulations confirm the theory, showing that the drive (master) system and the response (slave) system align smoothly, eliminating discrepancies and ensuring consistent behavior across chaotic supply chain networks.

In Equation (3), we adopt the following control law

$$\begin{gathered} s_1\left(t\right)=-m{e}_2+a{e}_4, \\ {s}_2\left(t\right)=-r{e}_1+y_2{z}_2-y_1{z}_1, \\ {s}_3\left(t\right)={-x}_2{y}_2+x_1{y}_1, \\ {s}_4\left(t\right)={-x}_2{z}_2+x_1{z}_1, \end{gathered}$$

(6)

The zero solution of Equation (5) is globally and exponentially stable. As a result, the drive and response systems described by Equations (2) and (3) are globally and exponentially synchronized.

Proof. 

We define the following Lyapunov function that is both positive definite and radially unbounded.

$$V=e_1^2+e_2^2+e_3^2+e_4^2.$$

By differentiating the Lyapunov function along the trajectory of system (5) under the control law defined in Equation (6), we derive the following expression:

$$\begin{array}{c}{\left.{\displaystyle \frac{dV}{ dt}}\right|}_{\left(5\right)}=2e_1{\dot{e}}_1+2e_2{\dot{e}}_2+2e_3{\dot{e}}_3+2e_4{\dot{e}}_4\\ = -2\left(n+1\right)e_1^2-2e_2^2+2(k-1)e_3^2+2\left(k-1\right)e_4^2\\ ={\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\\ e_4\end{array}\right)}^T\left[\begin{array}{cccc}-2\left(n+1\right)& 0& 0& 0\\ 0& -2& 0& 0\\ 0& 0& 2(k-1)& 0\\ 0& 0& 0& 2\left(k-1\right)\end{array}\right]\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\\ e_4\end{array}\right)\\ \\ \stackrel{ def }{=}\left(\begin{array}{ccc}{e}_1& e_2& e_3\end{array} e_4\right)Q_1{\left(\begin{array}{lll}{e}_1& e_2& e_3\end{array} e_4\right)}^T.\end{array}$$

In this context, $Q_1$ is a symmetric and negative definite matrix. Therefore, it follows that

$$e_1^2\left(t\right)+e_2^2\left(t\right)+e_3^2\left(t\right)+e_4^2\left(t\right)⩽\left[e_1^2\left(t_0\right)+e_2^2\left(t_0\right)+e_3^2\left(t_0\right)+e_4^2\left(t_0\right)\right]{\mathrm{e}}^{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(Q_1\right)\left(t-t_0\right)}.$$

□

The zero solution of system (5) is globally and exponentially stable. Consequently, the drive and response systems described in Equations (2) and (3) achieve global exponential synchronization.

To verify the effectiveness of the proposed control strategy, we conduct numerical simulations. We simulate the time evolution of two supply chain systems starting from different initial conditions and display their trajectories in a single figure. As illustrated in Figure 7, despite the initial discrepancies, the drive system (in red) and the response system (in blue) synchronize under the control law defined in Equation (6). The simulation results confirm the effectiveness of the designed control approach in achieving synchronization between the chaotic supply chain models.

4.2. Synchronization Strategy 2

Synchronization Strategy 2 applies an alternative control law to achieve alignment between the drive and response chaotic supply chain systems. Using this control, the error dynamics are again shown to be globally and exponentially stable through a Lyapunov function approach. This means that even when the systems begin with different initial states, their differences (errors) diminish over time and ultimately converge to zero. Numerical simulations illustrate that, under this law, the error trajectories steadily decline, confirming synchronization of the two chaotic models. The strategy demonstrates robustness and effectiveness, ensuring consistent system behavior despite initial disparities.

In Equation (3), we use the following control law:

$$\begin{gathered} s_1\left(t\right)=a{e}_4, \\ {s}_2\left(t\right)=-k{e}_1+y_2{z}_2-y_1{z}_1, \\ {s}_3\left(t\right)={-x}_2{y}_2+x_1{y}_1 \\ {s}_4\left(t\right)={-x}_2{z}_2+x_1{z}_1. \end{gathered}$$

(7)

Here, $q >r+m-\sqrt{\left(n+1\right)}$. Under this control, the zero solution of Equation (5) is globally and exponentially stable. As a result, both systems (2) and (3) reach global exponential synchronization.

Proof. 

Implementing the control law defined in Equation (7) within system (5) results in

$$\begin{array}{l}{e}_1^{\prime }=m{e}_2-\left(n+1\right)e_1\\ e_2^{\prime }=(r-q)e_1-e_2 \\ e_3^{\prime }=(k-1)e_3 \\ e_4^{\prime }=\left(k-1\right)e_4 \end{array}$$

We define a Lyapunov function that is both positive definite and radially unbounded to analyze the stability of the system.

$$V=e_1^2+e_2^2+e_3^2+e_4^2.$$

By computing the derivative of the Lyapunov function along the trajectories of system (5) under the control law given in Equation (7), we arrive at the following result:

$$\begin{array}{c}{\left.{\displaystyle \frac{dV}{ dt}}\right|}_{\left(5\right)}=2e_1{\dot{e}}_1+m{e}_1{e}_2+(r-k)e_1{e}_2+2e_2{\dot{e}}_2+2e_3{\dot{e}}_3+2e_4{\dot{e}}_4\\ = -2\left(n+1\right)e_1^2-2e_2^2+2(k-1)e_3^2+2\left(k-1\right)e_4^2\\ ={\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\\ e_4\end{array}\right)}^T\left[\begin{array}{cccc}-2\left(n+1\right)& {\displaystyle \frac{r+m-q}{2}}& 0& 0\\ {\displaystyle \frac{r+m-q}{2}}& -2& 0& 0\\ 0& 0& 2(k-1)& 0\\ 0& 0& 0& 2\left(k-1\right)\end{array}\right]\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\\ e_4\end{array}\right)\\ \\ \stackrel{ def }{=}\left(\begin{array}{lll}{e}_1& e_2& e_3\end{array} e_4\right)Q_2{\left(\begin{array}{lll}{e}_1& e_2& e_3\end{array} e_4\right)}^T.\end{array}$$

In this context, $Q_1$ is a symmetric, negative definite matrix. Therefore, it follows that

$$e_1^2\left(t\right)+e_2^2\left(t\right)+e_3^2\left(t\right)+e_4^2\left(t\right)⩽\left[e_1^2\left(t_0\right)+e_2^2\left(t_0\right)+e_3^2\left(t_0\right)+e_4^2\left(t_0\right)\right]{\mathrm{e}}^{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(Q_2\right)\left(t-t_0\right)}.$$

□

The zero solution of system (5) is globally and exponentially stable, which implies that systems (2) and (3) achieve global exponential synchronization. To validate the performance of the proposed control law, we carried out numerical simulations by examining the time evolution of the error system under differing initial conditions. Figure 8 presents the error trajectories between the drive and response systems. Despite the initial differences, the two chaotic supply chain models converge over time, confirming that synchronization is achieved using the control law defined in Equation (7). These simulation results clearly demonstrate the effectiveness of the proposed synchronization strategy.

4.3. Synchronization Strategy 3

Synchronization Strategy 3 introduces a third control law designed to guide the chaotic supply chain drive and response systems toward alignment. Similar to the earlier strategies, the stability of the error system is analyzed using a Lyapunov function, which demonstrates that the zero solution is globally and exponentially stable. This guarantees that any differences between the systems vanish over time, even if they start from distinct initial conditions. Numerical simulations validate the approach, showing that error trajectories converge to zero and synchronization is achieved. The results confirm that this strategy is reliable and effective in stabilizing and harmonizing the behavior of chaotic supply chain model.

Now, we apply the control law

$$\begin{gathered} s_1\left(t\right)=a{e}_4-p{e}_2, \\ {s}_2\left(t\right)=y_2{z}_2-y_1{z}_1, \\ {s}_3\left(t\right)={-x}_2{y}_2+x_1{y}_1 \\ {s}_4\left(t\right)={-x}_2{z}_2+x_1{z}_1, \end{gathered}$$

(8)

to Equation (3).

Here, $p >r+m-\sqrt{\left(n+1\right)}$. Then, the zero solution of Equation (5) is globally and exponentially stable. As a direct consequence, systems (2) and (3) become globally and exponentially synchronized.

Proof. 

Substituting the control law specified in Equation (8) into system (5) results in

$$\begin{array}{l}{e}_1^{\prime }=(m-p)e_2-\left(n+1\right)e_1\\ e_2^{\prime }=r{e}_1-e_2 \\ e_3^{\prime }=(k-1)e_3 \\ e_4^{\prime }=\left(k-1\right)e_4 \end{array}$$

We propose the following Lyapunov function, which is positive definite and radially unbounded, to analyze the system’s stability.

$$V=e_1^2+e_2^2+e_3^2+e_4^2.$$

Taking the derivative of the Lyapunov function along the trajectory of system (5) with control (8), we obtain

$$\begin{array}{c}{\left.{\displaystyle \frac{dV}{ dt}}\right|}_{\left(5\right)}=2e_1{\dot{e}}_1+(m-p)e_1{e}_2+r e_1{e}_2+2e_2{\dot{e}}_2+2e_3{\dot{e}}_3+2e_4{\dot{e}}_4\\ =-2\left(n+1\right)e_1^2-2e_2^2+2(k-1)e_3^2+2\left(k-1\right)e_4^2\\ ={\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\\ e_4\end{array}\right)}^T\left[\begin{array}{cccc}-2\left(n+1\right)& {\displaystyle \frac{r+m-p}{2}}& 0& 0\\ {\displaystyle \frac{r+m-p}{2}}& -2& 0& 0\\ 0& 0& 2(k-1)& 0\\ 0& 0& 0& 2\left(k-1\right)\end{array}\right]\left(\begin{array}{c}{e}_1\\ e_2\\ e_3\\ e_4\end{array}\right)\\ \\ \stackrel{ def }{=}\left(\begin{array}{lll}{e}_1& e_2& e_3\end{array} e_4\right)Q_3{\left(\begin{array}{lll}{e}_1& e_2& e_3\end{array} e_4\right)}^T.\end{array}$$

Since $Q_1$ is symmetric and negative definite, it follows that

$$e_1^2\left(t\right)+e_2^2\left(t\right)+e_3^2\left(t\right)+e_4^2\left(t\right)⩽\left[e_1^2\left(t_0\right)+e_2^2\left(t_0\right)+e_3^2\left(t_0\right)+e_4^2\left(t_0\right)\right]{\mathrm{e}}^{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(Q_3\right)\left(t-t_0\right)}.$$

□

The zero solution of system (5) is globally and exponentially stable. Consequently, systems (2) and (3) achieve global exponential synchronization.

To demonstrate the effectiveness of the proposed control strategy, we conducted numerical simulations by examining the time evolution of the error system under distinct initial conditions. Figure 9 displays the error trajectories of both the drive and response systems. Despite the initial differences between the chaotic supply chain models, synchronization is successfully attained using the control law defined in Equation (8). These simulation results provide strong evidence of the control law’s effectiveness in achieving synchronization.

Regarding synchronization speed, the results in

Table 4. Comparison of synchronization times for chaotic food supply chain control strategies.

Model	Time to Synchronization
Göksu et al. [34]	20–25
Zheng et al. [36]	2.6–8
Alsaadi et al. [37]	8
This paper—Strategy 1	2.5
This paper—Strategy 2	2.6
This paper—Strategy 3	2.5

show that the proposed strategies achieve faster convergence than previous chaotic supply chain synchronization models. Göksu et al. [34] reported synchronization times in the range of 20–25 time units, Zheng et al. [36] obtained 2.6-8 time units in a similar setting, and Alsaadi et al. [37] obtained about 8 time units. In contrast, our three strategies achieve synchronization in about 2.5–2.6 time units, which indicates that our control framework not only maintains strict stability guarantees but also significantly accelerates coordination between coupled food supply chains.

5. Sensitivity and KPI Analysis

In this section, we first perform a brief sensitivity analysis to assess the robustness of the proposed model. Starting from the baseline parameter set, we perturb each parameter one at a time by ±10% and ±20% while keeping all other parameters fixed, and we re-simulate the system under the same numerical settings. We then quantify how these perturbations change the long-run variability of the manufacturer-tier deviation (w(t)) using the standard deviation (std(w)) and the peak-to-peak amplitude. This analysis identifies which parameters have the greatest influence on volatility near the baseline and helps determine whether the qualitative conclusions are sensitive to small parameter changes.

Table 5. Local sensitivity ranking around the baseline (metric: average absolute % change in $\mathrm{s}\mathrm{t}\mathrm{d}\left(w\right)$ under ($\pm 10\mathrm{\%}$) and ($\pm 20\mathrm{\%}$) one-at-a-time perturbations).

Parameter	Avg. abs. % Change in std(w)	Influence (Qualitative)
n	43.694	High
m	24.723	High
r	17.268	Medium-High
k	13.592	Medium
a	2.3575	Low
b	0.02036	Very Low

summarizes the sensitivity results around the baseline parameter set, where each parameter is perturbed by ±10% and ±20% while all others are held fixed. The table reports the average absolute percentage change in (std(w)), indicating that (n), (m), (r), and (k) have the strongest influence on the variability (and hence stability) of (w(t)), whereas (a) and (b) have comparatively minor effects in this regime.

Now we conduct KPI analysis. Using simulated time series (after removing the warm-up period), we computed a set of stability and sustainability indicators for the selected KPIs under two scenarios: Scenario A (baseline) and Scenario B. Scenario A uses the baseline parameter set reported in the paper as the reference (uncontrolled) case. Scenario B uses the same dynamical system and initial conditions but reduces the coupling/response parameters m and n by 20%, representing weaker cross-layer response strength. After removing transient behavior, we compute and compare the KPIs for the two scenarios. The Improvement Pct value is defined relative to Scenario A: a positive value indicates that Scenario B reduces the KPI compared with Scenario A (i.e., Scenario B is better for KPIs where lower values are desirable), whereas a negative value indicates deterioration.

As shown in

Table 6. KPI comparison between Scenario A and Scenario B (positive improvement = lower KPI in Scenario B).

Metric	Scenario A	Scenario B	Improvement Pct (%)
Order variability: std(signal)	174.84	226.52	−29.556
Order variability: var(signal)	30,570	51,311	−67.847
Order variability: peak-to-peak	508.22	636.57	−25.256
Lead-time stability proxy: std(d/dt signal)	734.59	1039.20	−41.473
Spoilage proxy: time above + thrPos	150.45	150.30	0.0997
Spoilage proxy: area above +thrPos	24,403	31,389	−28.628
Service proxy: time below thrNeg	149.50	149.70	−0.1338
Service proxy: area below thrNeg	22,754	30,221	−32.812
Emergency shipment index: extreme-event count	6	4	33.333
Bullwhip proxy: var(upstream)/var(downstream)	107.62	128.42	−19.331

, Scenario B increases the overall volatility of the KPI time series, as reflected by higher standard deviation, variance, peak-to-peak amplitude, and greater rate-of-change volatility (lead-time stability indicator). Although the time spent above/below the threshold changes only slightly, the severity of oversupply/undersupply—measured by the area above/below the threshold—increases in Scenario B. Notably, Scenario B reduces the number of extreme events (emergency transport index) from 6 to 4, indicating fewer threshold-crossing peaks; however, the bullwhip-effect proxy (variance amplification) increases, suggesting stronger upstream amplification in Scenario B.

6. Conclusions

In this study, we investigate the dynamic behavior and coordination of a four-tier food supply chain consisting of manufacturers, primary distributors, secondary distributors, and retailers using a nonlinear chaotic framework. Our dynamical model captures the adjustment interactions across multiple tiers. We propose three synchronization control strategies and show that rigorous Lyapunov-based analysis can achieve global exponential convergence of synchronization errors. Numerical simulations further confirmed the effectiveness and robustness of our results.

Specifically, our results show that pseudo-random fluctuations generated by chaos can lead to instability, unpredictability, and loss of control. Minor disruptions in the supply chain—such as changes in raw material prices or deviations in demand forecasts caused by shipping delays—are amplified by chaotic behavior, undermining the system’s reliability and sustainability. A supply chain that remains in a chaotic state over time lacks stability, making it difficult to build long-term partnerships, which ultimately affects its competitiveness and sustainable development. Therefore, eliminating chaotic behavior is essential to enhancing the sustainability of supply chain systems. This study proposes a targeted control strategy based on a synchronization method to simultaneously improve the reliability and robustness of supply chains. Our synchronization method functions as a dynamic monitoring and feedback mechanism that enforces system stability, enhances efficiency and reliability, and mitigates risks, thereby fostering a more resilient supply chain.

Compared to existing chaotic and synchronous supply chain models, this paper offers main improvements. First, it constructs an explicit four-level structure specific to the food industry—manufacturer, primary distributor, secondary distributor, and retailer—with each state and parameter having a clear logistical interpretation and modeled as a state deviating from operational equilibrium. Second, it proposes three fully explicit synchronization strategies, derives their synchronization error dynamics, and establishes global exponential convergence using explicit Lyapunov analysis, rather than relying on informal or purely local arguments. Third, it directly links these control laws to implementable management tools and sustainability-related performance indicators, thus positioning synchronization as a practical tool for coordinating and managing sustainable food supply chains, rather than a purely theoretical exercise.

Despite our contributions, several limitations are present in this paper. The model focuses on a deterministic nonlinear framework with little stochastic disturbances such as random demand shocks, supply disruptions, and transportation delays. In addition, parameter values are assumed to be homogeneous and time-invariant, and behavioral factors such as bounded rationality or learning are not explicitly incorporated.

Future supply chain management will face even more challenges and opportunities. While technological development such as blockchain and artificial intelligence (AI) can significantly improve supply chain operational efficiency [38,39], it also introduces new risks and increased complexity. In future research, we can delve deeper into the impact of intelligent technologies on supply chain operations. Future research could also incorporate more stochastic elements and adaptive parameter adjustment across supply chain tiers. Empirical calibration using industry data in research would further enhance its practical relevance. Last but not least, future research could integrate environmental performance metrics within established sustainability frameworks, particularly the United Nations Sustainable Development Goals (e.g., SDG 12 on responsible consumption and production and SDG 13 on climate action), to more systematically evaluate sustainability-oriented outcomes in food supply chains [40].