The Curvature Property of a Linear Dynamical System

Year 2021, Issue: 28, 1288 - 1290, 30.11.2021

Tuna Bayrakdar , Zahide Ok Bayrakdar

https://doi.org/10.31590/ejosat.1014593

Abstract

Bu çalışmada iki-boyutlu, düzgün, otonom bir dinamik sistem üç-boyutlu bir Riemann manifoldu olarak değerlendirilmiş ve bir $\text{d}x/\text{d}t=ax+by$, $\text{d}y/\text{d}t=cx+dy$ lineer dinamik sisteminin skaler eğriliğinin pozitif olmadığı gösterilmiştir. Manifold skaler-
düzdür ancak ve ancak $b=-c$ ve $a=d=0$.

Keywords

Linear dynamical systems, Riemann curvature tensor, scalar curvature

The Curvature Property of a Linear Dynamical System

Abstract

In this work a two-dimensional smooth autonomous dynamical system is regarded as a three-dimensional Riemannian manifold and it is shown that the scalar curvature of a linear dynamical system $\text{d}x/\text{d}t=ax+by$, $\text{d}y/\text{d}t=cx+dy$ is non-positive. The manifold is scalar-flat iff $b=-c$ and $a=d=0$.

Keywords

Linear dynamical systems, Riemann curvature tensor, scalar curvature

References

• Vassiliou, P. J. (2000). Introduction: Geometric approaches to differential equations. In Vassiliou P. J. and Lisle I. G. (Eds.), Geometric approaches to differential equations (pp. 1-15). Australian Mathematical Society Lecture Series. 15, Cambridge University Press, Cambridge UK.
• Saunders, D. J. (1989). The Geometry of Jet Bundles. Cambridge University Press, Cambridge UK.
• Morita, S. (2001). Geometry of Differential Forms. American Mathematical Society, Providence, RI, USA.
• Ok Bayrakdar, Z., Bayrakdar, T. (2019). A geometric description for simple and damped harmonic oscillators, Turk J Math, 43: 2540 – 2548.