The Curvature Property of a Linear Dynamical System

Öz

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 $a=d=0$.

Anahtar Kelimeler

• Linear dynamical systems
• Riemann curvature tensor
• scalar curvature

The Curvature Property of a Linear Dynamical System

Öz

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 $a=d=0$.

Anahtar Kelimeler

• Linear dynamical systems
• Riemann curvature tensor
• scalar curvature

Kaynakça

1. 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.
2. Saunders, D. J. (1989). The Geometry of Jet Bundles. Cambridge University Press, Cambridge UK.
3. Morita, S. (2001). Geometry of Differential Forms. American Mathematical Society, Providence, RI, USA.
4. Ok Bayrakdar, Z., Bayrakdar, T. (2019). A geometric description for simple and damped harmonic oscillators, Turk J Math, 43: 2540 – 2548.