article · Measurement and Control
This research analyses the nonlinear dynamic behaviour of a discrete-step Degn-Harrison chemical system without diffusion. By discretising the continuous-time chemical reaction system using the explicit Euler scheme, a two-dimensional nonlinear map is produced. The behaviour of this map depends on both the reaction parameters and the step size of the discretisation. The study establishes the presence and stability conditions of the interior equilibrium point and explores codimension-one and codimension-two bifurcations, including period-doubling, Neimark-Sacker, and strong resonance cases. Normal forms are derived and non-degeneracy conditions are verified to ensure the validity of the theoretical results. Numerical simulations, including bifurcation diagrams, phase portraits, and maximum Lyapunov exponents, confirm the transition from regular to chaotic dynamics. The findings provide a framework for analysing similar nonlinear systems in applied sciences and improve the understanding of discretisation-induced complexities in chemical kinetics.
Understanding how continuous chemical systems behave when converted into discrete models is crucial for accurate computer simulations. This work helps researchers recognise how discretisation steps can introduce artificial chaotic behaviours or bifurcations. By providing a rigorous mathematical framework, it ensures that simulations of chemical kinetics and similar nonlinear processes in applied sciences remain reliable and predictable.
The abstract does not indicate a direct commercialisation pathway or specific real-world product applications for this theoretical mathematical analysis. It provides a foundational framework for researchers studying nonlinear dynamics and chemical kinetics, but remains at an early, purely theoretical stage of research without defined industry users or market-ready technologies.
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This research offers a well-developed analysis of the nonlinear dynamic behavior of a discrete-step Degn-Harrison system without diffusion. The continuous-time chemical reaction system is discretized by employing the explicit Euler scheme which results in a two-dimensional nonlinear map whose behavior depends on both the reaction parameters and the discretization step size. The presence and stability conditions of the interior equilibrium point are first established, and then the system’s bifurcation structure is explored in depth. Special attention is paid to codimension-one and codimension-two bifurcations, period-doubling, Neimark-Sacker and strong resonance cases (1:2, 1:3, and 1:4). To give a comprehensive characterization of these phenomena, we get the normal forms of these phenomena and check the non-degeneracy conditions of such phenomena, such that the validity of the theoretical results is ensured. Numerical simulations ensure the theoretical findings, such as bifurcation diagrams, phase portraits, and the maximum Lyapunov exponent, to give a picture of the switching from regular to chaotic dynamics. The outcomes of this study not only deepen the understanding of discretization-induced complexities in chemical kinetics as well as offering a structure for analyzing similar nonlinear systems within applied sciences.
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DOI: 10.1177/00202940261446465
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