Application of the Flow Curvature Method in Lorenz-Haken Model

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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.


Received: 31 May 2019 / Revised: 9 Jun. 2019 / Accepted: 14 Jun. 2019 / Published: 8 Feb. 2020

Index Terms

Flow Curvature Method, Slow Manifold, Dynamical System, Differential Geometry.


We consider a recently developed new approach so-called the flow curvature method based on the differential geometry to analyze the Lorenz-Haken model. According to this method, the trajectory curve or flow of any dynamical system of dimension  considers as a curve in Euclidean space of dimension . Then the flow curvature or the curvature of the trajectory curve may be computed analytically. The set of points where the flow curvature is null or empty defines the flow curvature manifold. This manifold connected with the dynamical system of any dimension   directly describes the analytical equation of the slow invariant manifold incorporated with the same dynamical system. In this article, we apply the flow curvature method for the first time on the three-dimensional Lorenz-Haken model to compute the analytical equation of the slow invariant manifold where we use the Darboux theorem to prove the invariance property of the slow manifold. After that, we determine the osculating plane of the dynamical system and find the relation between flow curvature manifold and osculating plane. Finally, we find the nature of the fixed point stability using flow curvature manifold.

Cite This Paper

A. K. M. Nazimuddin, Md. Showkat Ali," Application of the Flow Curvature Method in Lorenz-Haken Model ", International Journal of Mathematical Sciences and Computing (IJMSC), Vol.6, No.1, pp.33-48, 2020. DOI: 10.5815/ijmsc.2020.01.04.


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