The Curvature Property of a Linear Dynamical System

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

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,

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