Workshop on Dynamical Systems: Theory, Modelling and AI 19–21 November 2025 

Tentative Programme

Day-1 programme

(The Day 1 of the workshop will be announced in due course.)

[Mathematical Methods for the treatment of Differential Equations (DEs)]

Lie Symmetries and Group Analysis of DEs, Integrable Systems and Painlevé Analysis, Geometric Theory of DEs and Modern Techniques for the treatment of DEs

Day-2 programme

(The Day 2 of the workshop will be announced in due course.)

[Differential Equations in Natural World and in Physics]

Ecological and Epidemic Modeling with Nonlinear Systems, Economic Dynamics and Market Models via DEs, DEs in Climate and Environmental Science

Day-3 programme

(The Day 3 of the workshop will be announced in due course.)

Modern Computational Approaches

Computer Algebra Systems in DEs Research, Neural DEs and Physics-Informed Neural Networks, From Data to DEs Models and Reduced Order Modeling & Surrogate Modeling with AI

Workshop Abstracts

(The program of the workshop will be announced in due course.)

Title: Ultraspherical Spectral Methods

Speaker: Nick Hale (Stellenbosch University)

Abstract: Spectral methods based on orthogonal polynomials provide exceptional accuracy for differential equations, especially when very high precision is required. This talk introduces Chebyshev spectral methods for ordinary differential equations and outlines their basic principles before presenting the ultraspherical spectral method, which generalises the approach through systematic polynomial basis conversions. The resulting sparse, well-conditioned formulation offers a unified framework for ordinary, partial, integro-, and fractional differential equations. The talk concludes with recent advances in the Ultraspherical Element Method, which combines spectral accuracy with the geometric flexibility of finite elements for solving PDEs on complex domains.

Title: The Invariance, Conservation Laws and Integration of Difference Equations

Speaker: Abdul Hamid Kara (University of the Witwatersrand, Johannesburg)

Abstract: A vast amount of work has been done to extend the ideas and applications of symmetries to difference equations (∆Es) in a number of ways. In some cases, the ∆Es are constructed from the DEs in such a way that the algebra of Lie symmetries remains the same. As far as conservation laws of ∆Es goes, the work is more recent. Here, we construct symmetries and conservation laws for some ordinary ∆Es, utilise the symmetries to obtain reductions of the equations and show, in fact, that the notion of ‘association’ between these concepts can be analogously extended to, for now, ordinary ∆Es. That is, an association between a symmetry and first integral exists if and only if the first integral is invariant under the symmetry. Thus, a ‘double reduction’ of the ∆ E is possible.

Title: Bayesian inference of interaction rates in a metabolite-bacteria network using time-series counts

Speaker: Jack Jansma (Department of Mathematical Sciences, Centre for Invasion Biology, Stellenbosch University, Matieland 7600, South Africa)

Abstract: The human gut hosts a vast and diverse set of microbes that indirectly interact with each other through consuming and producing compounds, called metabolites. Disruptions in this network between gut microbes and their human host can contribute to the onset and progression of various disorders, including obesity, inflammatory bowel syndrome and Parkinson’s disease. Understanding the intricate and dynamic interactions between microbes, metabolites and the host is essential for developing microbiota-targeted interventions to improve human health. To this end a precise mathematical framework is crucial to capturing the complex dynamics of the system.

Here, we develop a dynamic network model of coupled ordinary differential equations and present a computational workflow that integrates computer algebra with Bayesian inference for model identification. Our approach infers interaction rates—quantifying metabolite consumption and production—from experimental time-series count data within a Bayesian framework, incorporating prior knowledge and uncertainty quantification. This workflow enables in silico predictions of system behaviour under perturbations and offers a robust method to integrate high-dimensional biological data with mechanistic models. By refining our understanding of gut microbial dynamics, this framework facilitates the assessment of microbiota-targeted therapeutic interventions.

Keywords: Gut microbiota, Bayesian inference, Ordinary differential equations

Title: Applications of Equivalence Transformations to Group Classification 

Speaker: Sergey V. Meleshko (Suranaree University of Technology, Thailand)

Abstract: The presentation is devoted to the application of equivalence transformations to the group classification of differential equations. It includes our experience in finding generalized equivalence transformations and performing preliminary group classification. As the construction of an optimal system of subalgebras is an essential part of using equivalence transformations for group classification, this topic will also be discussed. Part of the talk will focus on overdetermined systems of partial differential equations. Both partial differential equations and integro-differential equations will be considered. The presentation includes applications to the Boltzmann equation, the construction of invariant solutions for Gromeka–Beltrami flows, and the analysis of flows of chemically reacting gases in Eulerian and Lagrangian variables (for both unsteady and steady flows).  

Title: A coagulation toy model for silicosis

Speaker: Fernando P. da Costa (Univ. Aberta, and CAMGSD, Técnico, Univ. Lisboa, Portugal)

Abstract: We present a system with a countable number of ordinary differential equations of coagulation type that can be considered a simple model for the silicosis mechanism. Silicosis is a respiratory disease due to the ingestion of quartz dust and their accumulation in the lungs. The mathematical model consists in an ODE for the concentration of free quartz particles, an ODE for the concentration of macrophages without quartz (cells of the immune system that identify, capture and try to expel entities strange to the body), and a countable number of ODEs each one describing the concentration of macrophages with a number $i\in\mathbb{N}$ of captured quartz particles. We briefly describe basic results such as existence, uniqueness, regularity, and semigroup results of the set of solutions. Then we study the dynamics of the infinite dimensional system in the case of a particular class of rate coefficients that allows for the decoupling of the full infinite dimensional system into a finite dimensional one and a lower triangular infinite system. By the analysis of the finite dimensional subsystem, we conclude that it has a saddle-node bifurcation, possessing two equilibria when a bifurcation coefficient which is the ratio of the input rate of quartz to the rate of creation of empty macrophages is below a critical value, and no equilibria above that value. The stability properties of the bifurcating branches of the full infinite dimensional system are studied.

Title: A Solow-Swan framework for economic growth with memory effect

Speaker: Mathew O. Aibinu (University of Regina, Canada)

Abstract: The Solow-Swan equation is a cornerstone in the development of modern economic growth theory and continues to attract significant scholarly attention. This study incorporates memory effects into the classical Solow-Swan model by introducing a formulation based on the Caputo fractional derivative. A comparative analysis is conducted between the integer-order and fractional-order versions of the model to examine the influence of fractional dynamics on capital accumulation. The findings reveal that the inclusion of a fractional-order derivative significantly affects the trajectory and long-term stability of capital, offering a more flexible and comprehensive framework for modeling economic growth processes.

Title: Laguerre spectral collocation

Speaker: Emma Nel (Stellenbosch University, Stellenbosch)

Abstract: In this work, we investigate the use of Laguerre spectral collocation methods for solving differential equations on the semi-infinite line. While spectral methods are known for their high accuracy, the unbounded domain and the exponential weighting associated with Laguerre polynomials present notable challenges. For example, the direct computation of the entries in the Laguerre spectral differentiation matrix leads to underflow and overflow issues in IEEE-754 arithmetic for large degrees N, even though the entries themselves are well-behaved. We discuss recent progress in addressing these numerical difficulties and explore approaches for improving convergence. By employing these techniques, one can achieve significantly higher values of N in Laguerre spectral methods, ultimately leading to better numerical results.

Title: The effect of long range interactions on energy spreading in nonlinear, disordered Klein-Gordon chains

Speaker: Jean-Jacq du Plessis (University of Cape Town)

Abstract: We investigate the dynamics of energy spreading and thermalisation in a one-dimensional nonlinear Klein-Gordon lattice with long-range interactions and disorder. Pairwise in- teraction terms in the potential are scaled by 1/r^α, where r is the distance between lattice sites and α ≥ 0 is a parameter which governs the range of interactions. Using measures such as the participation number, second moment and entropy, we study how energy spreading through the lattice depends on the range of interaction by varying the α parameter. In addition, we study the normal modes of the linearised system to understand their influence on the energy spreading dynamics.

Title: Painlevé Analysis, higher-order rogue waves and dispersive solitons for a new generalized nonlinear evolution equation using a Hirota's N-soliton method

Speaker: Sachin Kumar (University of Delhi)

Abstract: We investigate the dynamics of energy spreading and thermalisation in a one-dimensional nonlinear Klein-Gordon lattice with long-range interactions and disorder. Pairwise in- teraction terms in the potential are scaled by 1/r^α, where r is the distance between lattice sites and α ≥ 0 is a parameter which governs the range of interactions. Using measures such as the participation number, second moment and entropy, we study how energy spreading through the lattice depends on the range of interaction by varying the α parameter. In addition, we study the normal modes of the linearised system to understand their influence on the energy spreading dynamics.

Title: Shock wave interactions for rate-type materials

Speaker: Akshay Kumar (University of Hyderabad)

Abstract: In this talk, the rate-type material model, whose solution consists of shock and rarefaction waves as derived in [3], is considered to discuss the shock wave interactions that occurred in the double Riemann problem with non-constant initial data. It is observed that the solution to the double Riemann problem is ultimately (after interaction of shock waves) given through either a 1 -shock wave and a 2 -rarefaction wave or 1 -shock wave and 2 -shock wave or 1 -rarefaction wave and 2 -shock wave subject to the initial data.

Title: Approximate Solutions of Geomagnetic Dynamical System: Extended Center Manifold and  Criterion for Synchronization.

Speaker: Mohamed Tantawy (Faculty of Engineering, October 6 University, Giza, Egypt)

Abstract: In this work, we aim to derive approximate solutions for the nonlinear geomagnetic dynamical system by constructing the Picard iteration scheme, with numerous works exploring its various chaotic features. We have proven the convergence of the theorem for this scheme, which ensures that our iterative process will be close to the exact solution. Global bifurcation of the system components within the parameter space is considered. Numerical evaluations of the solutions are performed, accompanied by the visualization of both three-dimensional (3D) and two-dimensional (2D) phase portraits. The 3D portraits reveal hyperchaos, a phenomenon arising from the extreme sensitivity of a system's dynamical evolution to small changes in initial conditions. A criterion for the synchronization of the real and imaginary parts of the system is to establish a necessary condition. An approach is developed for constructing the extended center manifold, which is pivotal in analyzing the system's behavior.

Title: λ -symmetries for a simple second-order nonlinear equation.

Speaker: Slungile Tshibase (University of KwaZulu-Natal, Durban)

Abstract: Lie symmetries are very useful because they can be used to solve differential equations. However, many equations do not admit Lie symmetries. As a result, we need to consider extensions; the one we consider here is the idea of λ-symmetries. These symmetries generalise Lie symmetries by introducing a function λ to allow reduction of equations that do not admit Lie symmetries. We will study the λ-symmetries of second-order ordinary differential equations of the form y’’=f (x, y) through a classification perspective. We will provide functional forms of f that will admit λ-symmetries, and we will show how these symmetries can be used to solve those equations.

Title: Mathematical Analysis of blood flow in a narrow artery having multiple stenoses

Speaker: Sanjeev Kumar (Dr. Bhimrao Ambedkar University, Agra)

Abstract: A study of the effects of blood flow parameters in narrow arteries having multiple stenoses is discussed in this work, where the blood is considered as a non-Newtonian Kuang-Luo (K-L) fluid model, with no-slip conditions at the arterial wall. The main properties of the K-L fluid model are that the plasma viscosity and yield stress play a vital role. These parameters make this fluid remarkably similar to blood; however, the flow characteristics change significantly when we change these parameters. We have derived a numerical expression for the blood flow characteristics, such as resistance to blood flow, blood flow rate, axial velocity, and skin friction. These numerical expressions have been solved by MATLAB 2021 software and discussed graphically. Furthermore, these results have been compared with Newtonian fluid, and observation was observed that resistance to blood flow and skin friction is decreased when blood is changed from non- Newtonian to Newtonian fluid.