Application of Differential Geometry on a Chemical Dynamical Model via Flow Curvature Method

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Author(s)

A. K. M. Nazimuddin 1,* Md. Showkat Ali 2

1. Department of Mathematical and Physical Sciences, East West University, Dhaka-1212, Bangladesh

2. Department of Applied Mathematics, University of Dhaka, Dhaka-1000, Bangladesh

* Corresponding author.

DOI: https://doi.org/10.5815/ijmsc.2022.01.02

Received: 24 Jul. 2021 / Revised: 20 Aug. 2021 / Accepted: 15 Sep. 2021 / Published: 8 Feb. 2022

Index Terms

Trimolecular Flow Model, Slow Manifold, Flow Curvature Method, Invariance Property.

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.

Cite This Paper

A. K. M. Nazimuddin, Md. Showkat Ali," Application of Differential Geometry on a Chemical Dynamical Model via Flow Curvature Method ", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.8, No.1, pp. 18-27, 2022. DOI: 10.5815/ijmsc.2022.01.02

Reference

[1]Levinson, N., (1949). A second-order differential equation with singular solutions, Ann. Math,      50:127–153.

[2]Andronov, A.A., Chaikin, S.E (1937). Plane Theory of Oscillators, I, Moscow. 

[3]Tikhonov, A.N. (1948). On the dependence of solutions of differential equations on a small parameter, Mat. Sbornik N. S., 31:575–586.

[4]Fenichel, N. (1974). Asymptotic stability with rate conditions,  Indiana Univ. Math. J, 23:1109–1137. 

[5]Fenichel, N.  (1971). Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J, 21:193–225.

[6]Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31: 53–98. 

[7]Fenichel, N. (1977). Asymptotic stability with rate conditions II, Indiana Univ. Math. J, 26:81–93.

[8]Ginoux, J.M. and Rossetto, B. (2006). Differential geometry and mechanics applications to chaotic dynamical systems, Int. J. Bifurc. Chaos, 4(16): 887–910. 

[9]Ginoux, J.M., Rossetto, B. and Chua, L.O. (2008). Slow invariant manifolds as curvature of the flow of dynamical systems, Int. J. Bifurc. Chaos, 11(18): 3409–3430. 

[10]Ginoux, J.M. (2009). Differential geometry applied to dynamical systems, In: World Scientific Series on Nonlinear Science, Series A, 66, World Scientific, Singapore. 

[11]Ginoux, J.M.and Llibre, J. (2011). The flow curvature method applied to canard explosion, J. Phys. A Math. Theor., 44: 465203. 

[12]Ginoux, J.M., Llibre, J. and Chua, L.O. (2013). Canards from Chua’s circuit, Int. J. Bifurc. Chaos, 23(4): 1330010. 

[13]Ginoux, J. M., & Rossetto, B. (2014). Slow invariant manifold of heartbeat model, arXiv preprint arXiv:1408.4988.

[14]Ginoux, J. M. (2014). The slow invariant manifold of the Lorenz–Krishnamurthy model, Qualitative theory of dynamical systems, 13(1): 19–37.

[15]Anguelov, R. & Stoltz, S. M. (2017). Stationary and oscillatory patterns in a coupled Brusselator model, Mathematics and Computers in Simulation, 133(C): 39-46. 

[16]Alqahtani, AM. (2018). Numerical simulation to study the pattern formation of reaction–diffusion Brusselator model arising in triple collision and enzymatic, Journal of Mathematical Chemistry, 56(6):1543-1566.

[17]Nazimuddin, A.  K.  M. and Ali, M. S. (2019). Periodic  Pattern  Formation  Analysis Numerically  in  a  Chemical  Reaction-Diffusion  System, International  Journal  of  Mathematical  Sciences  and Computing (IJMSC), 5(3):17-26. 

[18]Nazimuddin, A.  K.  M. and Ali, M. S. (2019). Pattern Formation in the Brusselator Model Using Numerical Bifurcation Analysis, Punjab University Journal of Mathematics, 51(11):31-39. 

[19]Nazimuddin, A.  K.  M. and Ali, M. S.  (2020). Slow Invariant Manifold of Brusselator Model, International Journal of Mathematical Sciences and Computing (IJMSC), 6 (2): 79-87.

[20]Prigogine, I. and Lefever, R. (1968). Symmetry breaking instabilities in dissipative systems, The Journal of Chemical Physics, 48(4), 1695–1700.