Stability analysis of Navier–Stokes system

Abstract

Stability analysis of dynamical system is very useful and is able to classify the role of stable and unstable equilibrium points. In this work, Naiver–Stokes system has been studied by using KCC theory. The Jacobi stability and dynamics of the deviation vector near equilibrium points have been also studied. Further, the effect of bifurcation parameter on stability of Navier–Stokes system has been observed and found the limiting conditions for bifurcation. Numerical examples of particular interest have been taken to compare the results of Jacobi stability and linear stability. It is observed that Jacobi stability on the basis of KCC theory is more efficient than the linear stability.

New insights into a chaotic system with only a Lyapunov stable equilibrium

“…It can be observed from Figure 11 that for different parameters, the instability exponent increases with time, indicating the Jacobi instability of the equilibrium point Q ′ . 50 Meanwhile, from Figure 12, we can know that for typical parameter a = 0.006 when 0 < t ≪ 1, one has || (t)|| > t 2 . This shows that the trajectories near the equilibrium point Q(0.25, 0.0625, 0.096) are dispersing, explaining the Jacobi instability of the equilibrium Q in the system (1.1).…”

“…44,45 In other words, the Jacobi unstable trajectories of a dynamical system behave chaotically, because it is impossible to distinguish the trajectories that are very close at the initial moment after a finite time interval. 43 The Jacobi stability of some typical dynamical systems (Lorenz system, 46 Rössler system, 47 Chen system, 48 Chua circuit system, 49 Navier-Stokes system, 50 Rikitake system, 51 and others (see, e.g.,other studies [52][53][54][55] as well as their references) has been studied. They qualitatively described the chaotic evolution of the dynamical systems by analyzing the dynamics of the deviation vector.…”

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

“…With the help of the nonlinear and Berwald connections the five geometrical invariants can be constructed of which the second invariant plays an important role as it gives the Jacobi stability of dynamical system. Jacobi stability analysis for different systems like Lorenz system [12], Chua circuit system [13] and other systems [14][15][16][17][18][19][20][21] have been studied. According to the articles [22,23], one of the geometrical invariants that identifies the beginning of chaos is the deviation vector from the so-called Jacobi equation.…”

Kosambi–Cartan–Chern Stability in the Intermediate Nonequilibrium Region of the Brusselator Model

“…As mentioned in the Introduction, KCC theory has been used to analyze the geometric structure of differential equations. Because a dynamic system is often described by differential equations, KCC theory has been applied to the geometric aspects of various dynamic structures, including those of physical (e.g., [Kumar et al, 2019;Krylova et al, 2019;Alawadi et al, 2020;Liu et al, 2020;Klën, & Molina, 2020]), biological (e.g., [Antonelli et al, 1993;Yamasaki & Yajima, 2013;Antonelli et al, 2014;Antonalli et al, 2019;Kolebaja, & Popoola, 2019]) and general (e.g., [Gupta, & Yadav, 2017;Chen, & Yin, 2019;) systems. Moreover, it has been applied in mathematics to resolve the inverse problem of updating the general parameters of dynamic systems [Sulimov et al, 2018], and the geometric parameters of complex systems, including chaotic ones (e.g., [Oiwa, & Yajima, 2017;Huang et al, 2019;Chen et al, 2020;Feng et al, 2020;Liu et al, 2021]).…”