In order to explore the Lorenz-Haken model, we will concentrate on the flow curvature technique, a recently created method based on differential geometry. This approach treats a dynamical system's trajectory curve or flow as a curve in Euclidean space. Analytical calculations may be used to determine the flow curvature, which is the trajectory curve's curvature. The flow curvature manifold, which is related to the dynamical system of any dimension, is defined by the locations where the flow curvature is null. For the slow invariant manifold of the same dynamical system, the flow curvature manifold offers an analytical equation. The slow invariant manifold equation may be discovered using the flow curvature technique without the need of any asymptotic expansions. In this study, we compute the analytical equation of the slow invariant manifold for the three-dimensional Lorenz-Haken model using the flow curvature approach for the first time. This analytical equation, together with its visual representation in phase space, makes it possible to distinguish between the slow development of trajectory curves and the rapid one, which advances our knowledge of this slow-fast domain. This study also advances the field relative to earlier similar work. Aside from that, we utilize the Darboux theorem to demonstrate the slow manifold's invariance characteristic.

A slow-fast dynamical systems can be investigated qualitatively and quantitatively in the study of nonlinear chaotic dynamical systems. Slow-fast autonomous dynamical systems exhibit a dichotomy of motion, which is alternately slow and quick, according to experiments. Some investigations show that slow-fast dynamical systems have slow manifolds, which is supported by theory. The goal of the proposed study is to show how differential geometry may be used to examine the slow manifold of the dynamical system known as the mitotic model of frog eggs. The algebraic equation of the flow curvature manifold is obtained using the flow curvature technique applied to the dynamical mitosis model. Using the Darboux invariance theorem, we then argue that this slow manifold equation is invariant with regard to the flow.

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.

The slow invariant manifold is a unique trajectory of the dynamical system that describes the long-time dynamics of the system’s evolution efficiently. Determining such manifolds is of obvious importance. On one hand they provide a basic insight into the dynamics of the system, on the other hand they allow a reduction of dimension of the system occurs on the invariant manifold only. If the dimension of the invariant manifold is sufficiently low, this reduction may result in substantial savings in computational costs. In this paper, differential geometry based new developed approach called the flow curvature method is considered to analyse the Brusselator 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 two-dimensional Brusselator 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.

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.

In this paper, we analyze the pattern formation in a chemical reaction-diffusion Brusselator model. Twocomponent Brusselator model in two spatial dimensions is studied numerically through direct partial differential equation simulation and we find a periodic pattern. In order to understand the periodic pattern, it is important to investigate our model in one-dimensional space. However, direct partial differential equation simulation in one dimension of the model is performed and we get periodic traveling wave solutions of the model. Then, the local dynamics of the model is investigated to show the existence of the limit cycle solutions. After that, we establish the existence of periodic traveling wave solutions of the model through the continuation method and finally, we get a good consistency among the results.