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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

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.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

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