0
Research Papers: Mathematical Theory

# On the Differential Geometry of Flows in Nonlinear Dynamical Systems

[+] Author and Article Information
Albert C. Luo

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805aluo@siue.edu

J. Comput. Nonlinear Dynam 3(2), 021104 (Mar 11, 2008) (10 pages) doi:10.1115/1.2835060 History: Received May 28, 2007; Revised August 23, 2007; Published March 11, 2008

## Abstract

In order to investigate the geometrical relation between two flows in two dynamical systems, a flow for an investigated dynamical system is called the compared flow and a flow for a given dynamical system is called the reference flow. A surface on which the reference flow lies is termed the reference surface. The time-change rate of the normal distance between the reference and compared flows in the normal direction of the reference surface is measured by a new function (i.e., $G$ function). Based on the surface of the reference flow, the $kth$-order $G$ functions are introduced for the noncontact and $lth$-order contact flows in two different dynamical systems. Through the new functions, the geometric relations between two flows in two dynamical systems are investigated without contact between the reference and compared flows. The dynamics for the compared flow with a contact to the reference surface is briefly addressed. Finally, the brief discussion of applications is given.

<>

## Figures

Figure 1

Differential flows of the two dynamical system changes for (a) infinitesimal time interval [t−ε,t) and (b) infinitesimal time interval (t,t+ε]. The reference and compared flows are x¯(t) and x(t). The normal vectors of the referenced surface are nSt−ε, nSt, and nSt+ε.

Figure 2

A three-dimesional view of (a) the flow xt passing through the reference surface for the reference flow x¯t and (b) the flow xt returning back from the reference surface for the reference flow x¯t. The shortest distance is ∣nStm⋅(xtm−x¯tm)∣ at time tm.

Figure 3

A three-dimesional view of (a) the flow xt passing over the reference surface St at (tm,x¯m) and (b) the flow xt tangential to the reference surface St at (tm,x¯m). The zero-order contact point is xtm=x¯tm at time tm.

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections