International Journal of Dynamical Systems and Differential Equations (8 papers in press)

Regular Issues

• Numerical Solution of Singular Autonomous Systems using the Fourth-Stage Geometric Mean Runge-Kutta Method  
  by Vijeyata Chauhan, Pankaj Kumar Srivastava 
  Abstract: The numerical treatment of singular problems is always seen to be intriguing, and its significance grows when it is raised in an autonomous system. This study proposes the development and implementation of a potent Runge-Kutta based fourth-stage explicit algorithm to numerically treat differential equations arising in the singular autonomous system. The basic properties of geometric mean have been brought into play to develop the algorithm. The convergence of the method has been established to prove the efficacy of the proposed technique. The consistency and stability of the method are also discussed. Two numerical illustrations are covered in the study and the results are compared with some other existing conventional methods, which confirms the importance of the method. The proposed method is found more efficient not only in terms of accuracy but also for applicability in first-order differential equations.
  Keywords: Explicit Runge-Kutta method; singular autonomous system; geometric mean; differential equations; increment function.
  DOI: 10.1504/IJDSDE.2025.10070393
   
  
• Modelling the Best Path Selection for Distributing Emergency Supplies in Earthquake Situations Using Ant Colony Optimisation Techniques  
  by Wenling Yu 
  Abstract: An optimal route selection model is proposed based on improved ant colony optimization (ACO) algorithm. Moreover, the effectiveness of the model is verified through simulation experiments. Firstly, the emergency materials distribution model of earthquake disasters, including route, starting, and ending points, is established to effectively evaluate the advantages and disadvantages of different routes. Secondly, the traditional ACO algorithm is enhanced by introducing new heuristic information, optimizing parameters, and increasing iterations to advance the convergence speed and accuracy of the algorithm and improve the shortcomings of the traditional ACO in its application. Finally, the simulation results show that the proposed algorithm and traditional ACO can calculate the optimal solution starting from node 0 after 27 cycles and 50 cycles. The experimental results reveal that the improved algorithm can find the optimal solution more quickly and has higher accuracy and stability.
  Keywords: Ant colony optimisation; algorithm; Earthquake disaster; Emergency material; Modelling; Optimal route.
  DOI: 10.1504/IJDSDE.2025.10071358
   
  
• Modeling Economic Cycle Fluctuations with Delayed Feedback Mechanisms: a Nonlinear AM-CNN-BiLSTM Approach  
  by Hong Zeng 
  Abstract: The real economy is subject to nonlinear influences such as consumer behaviour, enterprise investment dynamics, and delays in policy adjustments, all of which contribute to the complexity of economic system dynamics. To address the limitations of existing models, this study proposes a nonlinear framework incorporating delayed feedback mechanisms. Specifically, an AM-CNN-BiLSTM model is introduced, which integrates attention mechanisms, convolutional neural networks (CNNs), and bidirectional long short-term memory (BiLSTM) networks to capture sequential dependencies and enhance predictive accuracy. By simulating time-delay effects, the model effectively characterises the dynamic behaviour of economic systems. Experimental results demonstrate the presence of chaotic motion in the economic cycle system under certain parameter settings, as indicated by a maximum Lyapunov exponent of 0.1938. The proposed model exhibits strong predictive performance, achieving R2 = 0.9721, RMSE = 0.0552, and MAPE = 0.0235. These findings contribute to a deeper understanding of how delayed feedback mechanisms influence economic fluctuations and offer valuable insights for economic forecasting and policy formulation.
  Keywords: Nonlinear modelling; economic cycle fluctuations; delayed feedback mechanism; feature screening.
  DOI: 10.1504/IJDSDE.2025.10072006
   
  
• Topological Characterisation of 3D Digital Image Based on Topological and Betti Numbers  
  by Yibo Zhao 
  Abstract: The Euler-Poincare characteristic (EPC) is recognised as a key topological parameter in digital topology, commonly used to describe object connectivity and derive various quantities and functions. Its calculation is closely linked to Betti numbers, which represent the number of tunnels, cavities, and components. A new algorithm has been proposed for calculating the number of tunnels in a 3
  Keywords: Euler-Poincaré characteristics (EPCs); Topological numbers; Tunnels; Component.
  DOI: 10.1504/IJDSDE.2025.10072514
   
  
• The Information Perception Analysis of Complex Network based on Local Similar Clustering and BP Neural Network  
  by Meijing Song, Yajing Lu 
  Abstract: The social network’s development makes network public opinion (NPO) the most active expression of the public opinion in society, which can reflect the public opinion in cyberspace. NPO exerts more and more influence on the real society. It appears in the different stages of a variety of phenomena as well as issues in society, and has a significant impact on politics as well as public management in reality. How NPO evolves as well as the law of its evolution are studied in detail. Besides, a systematical analysis of the influence mechanism in its evolution is also made, which has crucial practical meaning. This exploration is based on a research angle that quite differs from previous studies, that is, the dynamic interactive network relationship among network members, government, media as well as Internet users.
  Keywords: Clustering; Complicated networks; Information public opinion; Local similarity.
  DOI: 10.1504/IJDSDE.2025.10072674
   
  
• Stochastic Modelling of Consumer Market Volatility an Improved Differential Equation Approach to Predicting Risk  
  by Xiang Chen 
  Abstract: In macroeconomic operations, consumption behaviour not only reflects the trends of economic variables such as residents income and market expectations, but also significantly influences policy regulation and industrial adjustment. In the context of increasing global uncertainty, traditional consumption forecasting methods demonstrate limited efficacy in modelling dynamic trajectories and structural fluctuations. This paper proposes a novel consumption fluctuation modelling approach, VAE-SDE, which integrates variational autoencoders (VAE) and stochastic differential equations (SDE). By extracting potential structural information from historical data through the VAE and mapping it to the SDE parameters, the method enables generative forecasting of consumption paths. This approach not only enhances the interpretability of the model but also improves its capacity for uncertainty modelling and path simulation.
  Keywords: Consumption forecasting; VAE; uncertainty modeling; SDE.
  DOI: 10.1504/IJDSDE.2025.10072739
   
  
• An Integrated Marketing Model based on Online Consumer Behaviour: a System Dynamics Stability and Classification Study  
  by Kan Lu, Mingting Huang 
  Abstract: In the complex realm of modern e-commerce, accurately modelling user interests and delivering personalised recommendations are essential for enhancing platform efficiency, user satisfaction, and business value. Traditional recommendation algorithms often struggle with key challenges such as capturing dynamic behavioural changes, effectively integrating multimodal features, and maintaining system stability during inference. To address these limitations, this study proposes the adaptive transformer and stability-enhanced network (ATRMST-Net). ATRMST-Net integrates a transformer-based sequential modelling backbone with a system dynamics-inspired stability control mechanism. A multimodal attention fusion module is designed to effectively aggregate heterogeneous user interaction data, enabling a richer understanding of user preferences. Furthermore, the model incorporates a temporal smoothness regularisation term and a Jacobian response control component to enhance robustness and mitigate the impact of noisy or volatile behaviours. Extensive experiments on multiple real-world e-commerce datasets demonstrate that ATRMST-Net consistently outperforms a range of competitive baselines across standard recommendation metrics. Ablation studies further confirm the individual contributions of each model component. Overall, this work provides a theoretically grounded and practically effective solution for building more stable, interpretable, and accurate recommendation systems in dynamic commercial environments.
  Keywords: personalised recommendation; multi-modal behaviour modelling; transformer with Stability Regularisation.
  DOI: 10.1504/IJDSDE.2025.10072888
   
  
• Dynamic Feedback Control Strategy of Financial Market based on Fractional Order Differential Equation  
  by Qin Wang, Huwei Li, Bilal Alatas 
  Abstract: In the face of increasingly complex and diverse financial markets, accurately identifying market fluctuations and effectively monitoring risks have become major challenges in financial regulation. Traditional differential equation models exhibit limitations when handling high-dimensional, nonlinear, and complex data. This paper introduces the A-TransHS framework, which integrates the strengths of differential equation-based prediction and deep learning technologies. It leverages the Transformer architecture's self-attention mechanism to extract temporal features from historical time series data and subsequently optimizes the parameters of mixed sub-fractional order differential equations. Experimental results on the Yahoo Finance and CBOE datasets demonstrate that the A-TransHS framework significantly outperforms traditional methods and other deep learning models in terms of short- and long-term predictive accuracy, as measured by RMSE, MAE, and MAPE. These findings highlight its strong potential for modelling financial market dynamics and enhancing risk management.
  Keywords: Mixed sub-fractional order differential equations; financial regulation; option price prediction; transformer.
  DOI: 10.1504/IJDSDE.2025.10073022