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On the Differential Geometry of Flows in Nonlinear Dynamical Systems

[+] Author Affiliations
Albert C. J. Luo

Southern Illinois University - Edwardsville, Edwardsville, IL

Paper No. DETC2007-34887, pp. 261-273; 13 pages
doi:10.1115/DETC2007-34887
From:
  • ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C
  • Las Vegas, Nevada, USA, September 4–7, 2007
  • Conference Sponsors: Design Engineering Division and Computers and Information in Engineering Division
  • ISBN: 0-7918-4806-X | eISBN: 0-7918-3806-4
  • Copyright © 2007 by ASME

abstract

In this paper, in order to investigate the relation between two flows given 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. For a small time interval, the change rate of the normal distance between the reference and compared flows on 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 for the non-contact and lth -order contact flows in two different dynamical systems are introduced. Through the new functions, the geometric relations between two flows in two dynamical systems imposed in the same phase space are investigated without contact between the reference and compared flows. The compared flow passing through, returning back from and paralleling to the surface of the reference flow is discussed first. The tangency and passability for a compared flow to the surface of a reference flow with the lth -order contact are presented. The dynamics for the compared flow with such a contact to the reference surface is briefly addressed. Finally, the brief discussion of applications is given. The contact and tangential singularity between two flows are different. The kth -order contact of the compared flow to the reference surface indicates that the compared flow to the reference surface is of the kth -order singularity. However, the compared flow with the kth -order singularity to the reference surface does not mean that the compared flow to the reference surface is of the kth -order contact.

Copyright © 2007 by ASME

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