South African-German WE-Heraeus Seminar: Nonlinear dynamics and anomalous transport in low dimension

The exponential rise in satellite activity over recent years has transformed Low Earth Orbit (LEO) into a congested and collision-prone region. This research aims to address this problem by identifying the natural deorbiting corridors of the resident space objects (RSOs) currently in LEO. Making use of a 3 degree-of-freedom, nonlinear, autonomous Hamiltonian that incorporates perturbations like the Earth’s oblateness, solar radiation pressure (SRP), lunisolar terms and more, this research numerically investigates the secular, averaged form of such a Hamiltonian, exploring how the resonant dynamics resulting from these perturbations influence the presence of chaos, specifically in the regions consisting of fast-deorbiting trajectories. To accomplish this numerical investigation, various tools of nonlinear dynamics are employed, such as the Smaller Alignment Index (SALI) for the identification of the chaotic behaviour of the system and the Lagrangian Descriptors method for the divulgence of the dynamical structures in the system’s parameter space. These tools will provide crucial numerical information that is necessary for the construction of analytical results.

We show how the usual derivation of the equations of motion for a classical field theory with non-holonomic constraints, constraints that depend on the derivatives of the field, fails. As a result, the usual method for gauge fixing in classical and quantum field theories fails for general gauges that depend on the derivatives of the fields, such as the Coulomb and Lorenz gauges. The point of failure occurs at the use of the transposition rule, $\delta(\partial_\nu A^\mu)=\partial_\nu(\delta A^\mu)$, in the derivation of the equations of motion from the extremization of the action; we show that the transposition rule does not hold for general non-holonomic constraints. We define the concept of integrability of non-holonomic constraints in field theory and prove that the usual transposition rule holds for theories with these constraints. We are thus able to recover the usual treatment of gauge fixing for gauges of this type. We apply our formalism to the specific examples of the Coulomb and Lorenz gauges.

Graphene is known for its exceptional material properties, both mechanical and electronic [1,2]. We study the mechanical and structural properties of graphene under the effects of temperature and uniaxial strain, using a Hamiltonian model describing the dynamics of a 2-dimensional monolayer graphene sheet [3,4,5]. We perform extensive numerical simulations of the model by using a symplectic integration scheme which allows fast numerical computations whose error is upper-bounded by a a small constant value for all times [6]. In particular, we quantify the stress-strain response of graphene at various temperatures and perform comparisons to other investigations which use different modelling techniques [7]. We also study the effects of strain and temperature on the distributions of bond lengths and bond angles in the lattice [8] and provide fitting expressions for these distributions using a combination of analytical and numerical observations of the material’s behavior. $$$$ References: $$$$ [1] Lee, Wei, Kysar, Hone. Science 321, 385-388 (2008).$$$$ [2] Du, Skachko, Barker, Andrei. Nature Nanotechnology 3, 491-495 (2008).$$$$ [3] Kalosakas, Lathiotakis, Galiotis, Papagelis. Journal of Applied Physics 113, 134307 (2013).$$$$ [4] Hillebrand, Many Manda, Kalosakas, Gerlach, Skokos. Chaos 30, 063150 (2020).$$$$ [5] Erasmus, Skokos, Kalosakas, preprint arXiv:2507.14709 (2025).$$$$ [6] Blanes, Casas, Farrés, Laskar, Makazaga, Murua. Applied Numerical Mathematics 68, 58-72 (2013).$$$$ [7] Shao, Wen, Melnik, Yao, Kawazoe, Tian. Journal of Chemical Physics 137, 194901 (2012)$$$$ [8] Kalosakas, Lathiotakis, Papagelis. Materials 15, 67 (2022).$$$$ Additional authors: Haris Skokos, Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town; Max Planck Institute for the Physics of Complex Systems. George Kalosakas, Department of Materials Science, University of Patras.

This study investigates the nonlinear dynamics and thermalization processes in a Newtonian fluid-filled porous medium subjected to bottom heating and periodic gravity modulation, aligning with the objectives of the South African-German WE-Heraeus Seminar on Nonlinear Dynamics and Anomalous Transport (Cape Town, 2–6 February 2026). Motivated by the need to understand complex thermal transport phenomena in geophysical and industrial systems, we analyze the interplay between thermal convection and periodic gravitational forcing. The system is modeled using the conservation of energy equation and hydrodynamic equations based on the Boussinesq–Darcy approximation. By applying the Galerkin-truncated approximation, we reduce the governing partial differential equations to a Lorenz-like system of first-order ordinary differential equations. This system reveals three equilibrium points, with stability contingent on key parameters such as the scaled Rayleigh number. In the absence of gravity modulation, the system exhibits a rich spectrum of behaviors, including steady convection, periodic convection, chaotic convection, reverse period doubling to chaos, and coexistence of steady and chaotic states. Introducing sinusoidal or non-sinusoidal periodic gravity modulation induces additional dynamics, including periodic convections, bistable periodic states, and period-doubling routes to chaos. These findings are validated through microcontroller-based experimental set-up, confirming the existence of diverse convective regimes. This work contributes to the seminar’s focus on thermalization by elucidating how periodic gravitational forcing influences nonlinear convective dynamics and anomalous transport in porous media, offering insights into thermal energy redistribution in complex systems.

When an epidemic or a new disease emerges, a key question is whether it will result in a single outbreak or multiple waves. In this work, we present a mathematical model that helps predict the occurrence of successive waves of a disease, based on the pathogenic load in the body and its virulence. The model accounts for two critical stages of disease progression, asymptomatic and pre-symptomatic and also incorporates vaccination as a preventive strategy. Moreover, several studies have shown that fear and nutritional habits among the population significantly influence the dynamics of epidemics. Based on these factors, we construct a time-varying epidemiological model and provide a mathematical analysis using the basic reproduction number. This approach can be applied to understand the evolution of diseases such as COVID-19, Cholera, Ebola, Monkeypox, Cervical Cancer, and others in sub-Saharan Africa. Our numerical simulations confirm that the model can reproduce periodic waves, demonstrating the potential for a disease to cause multiple outbreaks. Furthermore, we illustrate how control strategies targeting fear and nutritional habits can be implemented to ensure disease elimination after the first or second wave. Using available data, we apply the proposed model to analyse the wave patterns of Monkeypox in the Democratic Republic of Congo and South Africa and cholera in Cameroon since 2019.

Active turbulence, chaotic flow arising at low Reynolds number due to internal driving of fluids, provides not only the surprising insight that turbulent-like flow can emerge in biological systems, but also a rich variety of nonlinear dynamical behaviours. By uncovering the sequence of bifurcations and how the flow evolves as more energy is injected, we can understand the ultimate transition from regular to chaotic flow through subcritical bifurcations and transient chaos. Through this progression, the mixing properties of the flow are also fundamentally altered. Mixing efficiency decreases until the transition to turbulence, where the breakdown of stationary vortices results in dramatically better advective transport and consequently optimal mixing. Large-scale simulations and theoretical calculations combined provide new insight into how activity-induced chaos can enhance mixing in microscopic systems.

We attempt to describe the oscillon solution of the one-dimensional $\phi^4$ equation, \[ \phi_{tt}- \phi_{xx} - 2 \phi (1-\phi^2)=0, \] as a bound state of a kink and antikink. The variational approximation employing the Ansatz \[ \phi(x,t) = \tanh\left(\frac{x+R(t)}{b(t)}\right) - \tanh\left(\frac{x-R(t)}{b(t)}\right) - 1 \] reduces the partial differential equation to a Hamiltonian system for two collective coordinates: kink-antikink separation $R(t)$ and the width of each of the two constituents, $b(t)$. The conclusions from the dynamical system are verified in direct numerical simulations of the $\phi^4$ oscillon.

Originating as a concept in quantum mechanics, the notion of $\mathcal{PT}$ symmetry has proven to be highly capable, and has seen widespread adoption in various fundamental and applied fields. Posited as an alternative to the quantum mechanical condition of Hermiticity, which guarantees real spectra and the conservation of probability within a quantum mechanical system, $\mathcal{PT}$ symmetry defined on the basis of parity ($\mathcal{P}$) and time ($\mathcal{T}$) symmetries allows a broader class of physical phenomena to be mathematically characterized and described. Unlike the conventional Hermitian Hamiltonian description, physical systems are instead modeled as a composite of non-isolated (in contact with the environment) systems which allow for parity reflection and time-reversal symmetries. Provided such composite dynamical models are in equilibrium, with a balanced form of gain and loss between subsystems, they can be classified as halfway between those of isolated and non-isolated systems. In particular cases, under some special coordinate frame, a unique conservative Hamiltonian description is permitted. Since Hamiltonian systems are a primary domain for the study of chaos, $\mathcal{PT}$ symmetric systems too are natural candidates for exploratory analysis under the framework of chaos theory. An appropriate exposition of $\mathcal{PT}$ symmetry is therefore given, and followed by an investigative characterization of the dynamical behaviour of the $\mathcal{PT}$-symmetric system $$\ddot{x} + x + \alpha y + x y \dot{y} = 0$$ $$\ddot{y} + y + \alpha x - x y \dot{x}= 0.$$ By changing the characteristic gain and loss parameter $\alpha$, the exhibited chaoticity is quantitatively assessed through the application of chaos indicators such as the Maximum Lyapunov Exponent (mLCE) and the Small Alignment Index (SALI). $$$$ References: $$$$ Barashenkov, I. V., Pelinovsky, D. E., Dubard, P. Dimer with gain and loss: Integrability and PT-symmetry restoration. J. Phys. A: Math. Theor. 48 (2015) 325201 $$$$ Bender, C. M. 1992. PT Symmetry in Qauntum and Classical Physics. World Scientific Publishing Europe: London, UK $$$$ Bender, C. M. 2007. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70 (2007) 947-1018 $$$$ Konotop, V. V., Yang, J., Zezyulin, D. A. 2016. Nonlinear waves in PT-symmetric systems. Review of Modern Physics 88 (2016) $$$$ Lieberman, A. J., Lichtenberg, M. A. 1991. Regular and Chaotic Dynamics. Springer-Verlag: New York. $$$$ Skokos, Ch.: The Lyapunov Characteristic Exponents and Their Computation. Lect. Notes Phys. 790, 63-135 (2010)

This study presents a novel electromechanical model inspired by the coordinated motion of myriapods, designed to explore nonlinear dynamics and anomalous transport in coupled systems, aligning with the seminar’s focus on nonlinear lattices. We develop an array of mechanical arms driven by FitzHugh–Nagumo neuron circuits, where the governing differential equations are derived using Kirchhoff’s and Newton’s laws. Numerical simulations reveal that system stability is highly sensitive to parameter variations, particularly the stimulation current, which dictates distinct dynamic regimes: non-excitable, excitable, and oscillatory states. In the excitable state, the generated action potential (AP) induces significant actuation of a single mechanical arm, with magnetic signals amplifying the instantaneous displacement amplitude of the arm. When extended to an array of coupled electromechanical systems in the excitable state, the model demonstrates synchronized AP propagation across neurons, maintaining amplitude stability in the permanent regime, as confirmed by numerical analysis. This behavior mirrors the coordinated, non-rotational motion of myriapod legs, with the propagation velocity of nerve impulses quantitatively matching the mechanical displacement velocity. Our work contributes to the seminar’s objectives by offering a biologically inspired framework to study nonlinear lattice dynamics, highlighting robust signal propagation and collective behavior in coupled systems, with potential applications in understanding anomalous transport in complex networks.

We show that axoplasmic pressure waves governed by the improved Heimburg–Jackson hydrodynamic equation evolve as periodic soliton pulses, in which the membrane capacitance effectively ensures the coupling between transmembrane voltage (modeled by the modified Hodgkin–Huxley cable equation) and the pressure waves. Time independent wavefronts that serve as steady state solutions of the transmembrane voltage are analytically obtained, with it threshold values vital in determining wavefronts propagation in the entire nerve fiber. An increase in numerical value of the membrane coupling coefficient generally decrease the amplitude and speed of the propagating electromechanical wavefronts, thereby leading to the effective control of neuronal information and other mechanosensory processes. This work finds application in the field of anesthetics because the amplitude and speed of propagating wavefronts which transmit pain, can be controlled by varying the parameters of the system.

Ultracold quantum gases achieve exquisite platforms to explore few- and many-body phenomena in low dimensions and with extreme control. Being the most magnetic element of the periodic table, dysprosium presents strong interatomic dipole-dipole interactions. Contrasting with the standard contact interactions, the dipolar interactions are long-range and anisotropic. The relative strength between both types of interactions can be tuned via modification of the scattering length near Feshbach resonances. Within the last years, these properties led to exciting novel discoveries. Some of these arising exotic phenomena are supersolidity, topological ordering and the formation of droplets or droplet crystals. Our experiment aims in particular to a reduction of the dimensionality of the dysprosium quantum gas by restricting it to two dimensions. For this we use an accordion lattice for tailorable quasi 2D traps and an objective setup to probe and perturb the atomic cloud with sub-micron resolution. The overall goal is to explore and understand not only which exotic phases form in quantum gases under the influence of dipolar interactions in lower dimensional space, but also: How these orders arise? What are the underlying phase transitions and how do the states evolve dynamically when crossing them? And with particular interest: What is the role of anisotropic dipolar interactions on thermalization pathways in reduced dimensionality, also including the role of topological defects.

The method of Lagrangian Descriptors (LDs) is a computational tool used to visualize characteristic features of nonlinear dynamical systems and their phase space structures [1]. The technique is straightforward to implement, as it assigns a positive scalar value to the initial condition of each orbit. In this study, we apply LDs to two prototypical dynamical systems: the 2D Hénon–Heiles system [2] and the 2D standard map [3]. For the Hénon–Heiles system, computations are carried out using a symplectic integrator scheme [4]. To analyze the dynamics of these systems, we compute several quantities, including LD values, their gradients, and the maximum Lyapunov exponent. Although LDs are not themselves chaos indicators, they can be used to distinguish between chaotic and regular orbits by introducing two derived measures: the difference and the ratio of LDs of neighboring orbits [5]. Finally, we evaluate the effectiveness of these classifications by comparing them with results from an established chaos detection method, the Smaller Alignment Index (SALI) [6]. $$$$ References:$$$$ [1] Agaoglou M. et al., Lagrangian Descriptors: Discovery and Quantification of Phase Space Structure and Transport. Zenodo, 2020, pp. 1–3. $$$$ [2] Hénon M. and Heiles C., “The applicability of the third integral of motion: some numerical experiments”, Astronomical Journal 69 (1964), pp. 73–79. $$$$ [3] Zimper S., “Investigating the phase space dynamics of conservative dynamical systems by the Lagrangian descriptors method”. MA thesis. Cape Town, South Africa: University of Cape Town, May 2023, pp. 31. $$$$ [4] Blanes S. et al., “New families of symplectic splitting methods for numerical integration in dynamical astronomy”, Applied Numerical Mathematics 68 (2013), pp. 58–72. $$$$ [5] Hillebrand M. et al., “Quantifying chaos using Lagrangian descriptors”, Chaos: An Interdisciplinary Journal of Nonlinear Science 32 (2022), 123122. $$$$ [6] Skokos Ch. and Manos T., “The Smaller (SALI) and the Generalized (GALI) alignment indices: efficient methods of chaos detection”, Lecture notes in Physics, 915 (2016), pp. 129-181. $$$$ Acknowledgements: $$$$ I’d like to thank the Centre for High Performance Computing (CHPC) of South Africa for providing their computational resources, and the University of Cape Town. $$$$

The Poincare Surfaces of Section (PSS) have been deployed to study the dynamical behavior of low dimensional conservative systems. These have been largely selected based on the symmetry of the system. In this work, we consider carefully selected PSS with consideration of the system's symmetry and the equilibrium (fixed) points. We investigate dynamics in the Henon-Heiles and the Duffing systems.

One interesting nonlinear wave phenomenon is supratransmission [1], which allows plane waves to travel through gaps in frequency bands. Beyond its many uses in engineering and science, this phenomenon elucidates the wave types that can exist in discrete systems [2-3]. Using a chain of pendulum subjected to a harmonic-driving source with a constant amplitude and parametric excitation, we produce discrete rogue waves [4] in this study via the nonlinear bandgap method. This finding paves the way for the integration of isolated rogue waves into a straightforward device. $$$$ References:$$$$ [1] F. Geniet and J. Leon, Energy transmission in the forbidden band gap of a nonlinear chain, Phys. Rev. Lett. 89, (2002) 134102. $$$$ [2] A. B. Togueu Motcheyo, M. Kimura, Y. Doi and C. Tchawoua, Supratransmission in discrete one- dimensional lattices with the cubic-quintic nonlinearity, Nonlinear Dyn 95, (2019) 2461. $$$$ [3]A. B. Togueu Motcheyo, J. E. Macias-Diaz, Nonlinear bang gap transmission with zero frequency in cross-stitch lattice, Chaos, Solitons and Fractals 170 (2023) 113349. $$$$ [4] A. B. Togueu Motcheyo, M. Kimura, Y. Doi and, Juan F.R. Archilla, Nonlinear bandgap transmission by discrete rogue waves induced in a pendulum chain, Physics Letters A 497 (2024) 129334. $$$$

We evaluate the $\mathcal{PT}$ symmetry-breaking threshold in a circular array formed by $N$ optical waveguides with gain and $N$ waveguides with loss. An exact solution is obtained for three necklace geometries. The first one consists of a strand of waveguides with gain and a coaxial strand with loss. Each guide is coupled to its left and right neighbours in the same chain and in addition, to the parallel element of the other chain. The second necklace is formed by two coaxial strands where each chain alternates waveguides with gain and loss. The last array considered consists of a single strand that displays a clockwise gain-loss variation. As this necklace is traced quarter-way around, the waveguide's loss coefficient decreases from its maximum value to zero and switches to gain -- which then grows to its maximum, decreases and switches back to loss. All three $\mathcal{PT}$-symmetric lattices are shown to exhibit a nonvanishing gain-loss threshold as $N \to \infty$.