I. J. Mathematical Sciences and Computing, 2022, 1, 18-27 Published Online February 2022 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijmsc.2022.01.02 Copyright © 2022 MECS I.J. Mathematical Sciences and Computing, 2022, 1, 18-27 Application of Differential Geometry on a Chemical Dynamical Model via Flow Curvature Method A. K. M. Nazimuddin Department of Mathematical and Physical Sciences, East West University, Dhaka-1212, Bangladesh. E-mail: nazimuddin@ewubd.edu Md. Showkat Ali Department of Applied Mathematics, University of Dhaka, Dhaka-1000, Bangladesh. Received: 24 July 2021; Accepted: 15 September 2021; Published: 08 February 2022 Abstract: Slow invariant manifolds can contribute major rules in many slow-fast dynamical systems. This slow manifold can be obtained by eliminating the fast mode from the slow-fast system and allows us to reduce the dimension of the system where the asymptotic dynamics of the system occurs on that slow manifold and a low dimensional slow invariant manifold can reduce the computational cost. This article considers a trimolecular chemical dynamical Brusselator model of the mixture of two components that represents a chemical reaction-diffusion system. We convert this system of two-dimensional partial differential equations into four-dimensional ordinary differential equations by considering the new wave variable and obtain a new system of chemical Brusselator flow model. We observe that the onset of the chemical instability does not depend on the flow rate. We particularly study the slow manifold of the four- dimensional Brusselator flow model at zero flow speed. We apply the flow curvature method to the dynamical Brusselator flow model and acquire the analytical equation of the flow curvature manifold. Then we prove the invariance of this slow manifold equation with respect to the flow by using the Darboux invariance theorem. Finally, we find the osculating plane equation by using the flow curvature manifold. Index Terms: Trimolecular Flow Model, Slow Manifold, Flow Curvature Method, Invariance Property. 1. Introduction Many dynamical systems contain slow-fast structure where the dynamics of the system goes slowly but towards the slow invariant manifold, the trajectory of the system from any initial point rapidly relaxes. Slow invariant manifolds can exist in singularly as well as non-singularly perturbed systems [1, 2, 3]. To obtain the equation of the slow manifold, the classical Geometric Singular Perturbation (GSP) technique [4, 5, 6, 7] is one of the techniques among various methods which use asymptotic expansion and this technique is applicable only for a singularly perturbed dynamical system. Another new method to determine the equation of slow manifold is the Flow Curvature Method (FCM) [8, 9, 10, 11, 12] and this method is applicable for a system whether it is a singularly perturbed dynamical system or not. Ginoux and Rossetto [13] adopted this FCM to the heartbeat model to generate the slow invariant manifold equation. Using the FCM, Ginoux [14] determined the equation of the slow manifold of the L-K model. Anguelov and Stoltz analyzed the asymptotic solution behavior of the Brusselator chemical dynamical model [15]. Different and interesting types of pattern formation induced from the Brusselator model through the numerical investigation are observed in [16]. Using the numerical bifurcation analysis, solutions of Periodic traveling waves and their stability of the Brusselator model are investigated in [17, 18]. Recently, the FCM was used to determine the equation of the slow manifold of the Brusselator chemical dynamical model [19]. The main research objective of this article is to find the analytical equation of the slow invariant manifold of the Brusselator flow model at zero speed. We have used a differential geometry-based method, namely FCM that is best because this method can be applied on both singularly perturbed systems and non-singularly perturbed systems and does not use the asymptotic expansion technique. In this article, we consider a two-dimensional dynamical Brusselator model. The reaction-diffusion Brusselator prepares a useful model for the study of cooperative processes in chemical kinetics, such as trimolecular reaction steps arising from the formation of ozone by atomic oxygen via a triple collision. This system also governs in enzymatic reactions and in plasma and laser physics in multiple couplings between certain modes. We convert this system of two-dimensional partial differential equations into four-dimensional ordinary