Controlling system dimension: A class of real systems that obey the Kaplan–Yorke conjecture

doi:10.1073/pnas.2535197100

Controlling system dimension: A class of real systems that obey the Kaplan–Yorke conjecture

^*U.S. Naval Research Laboratory, Code 5673, Washington, DC 20375; ^‡Department of Structural Engineering, University of California at San Diego, La Jolla, CA 92093-0085; and ^§U.S. Naval Research Laboratory, Code 6360, Washington, DC 20375

Edited by Harry L. Swinney, University of Texas, Austin, TX, and approved October 16, 2003 (received for review August 13, 2003)

Abstract

The Kaplan–Yorke conjecture suggests a simple relationship between the fractal dimension of a system and its Lyapunov spectrum. This relationship has important consequences in the broad field of nonlinear dynamics where dimension and Lyapunov exponents are frequently used descriptors of system dynamics. We develop an experimental system with controllable dimension by making use of the Kaplan–Yorke conjecture. A rectangular steel plate is driven with the output of a chaotic oscillator. We controlled the Lyapunov exponents of the driving and then computed the fractal dimension of the plate's response. The Kaplan–Yorke relationship predicted the system's dimension extremely well. This finding strongly suggests that other driven linear systems will behave similarly. The ability to control the dimension of a structure's vibrational response is important in the field of vibration-based structural health monitoring for the robust extraction of damage-sensitive features.

Footnotes

↵ † To whom correspondence should be addressed. E-mail: pele{at}ccs.nrl.navy.mi.
This paper was submitted directly (Track II) to the PNAS office.
Abbreviation: LE, Lyapunov exponent.