Chaotic systems, irregular functions, and nonlinear time series

Robert C. McCarty

doi:10.1117/12.162681

18 November 1993 Chaotic systems, irregular functions, and nonlinear time series

Robert C. McCarty

Proceedings Volume 2038, Chaos in Communications; (1993) https://doi.org/10.1117/12.162681
Event: SPIE's 1993 International Symposium on Optics, Imaging, and Instrumentation, 1993, San Diego, CA, United States

Abstract

With the advent of the studies of fractals, chaos, and non-linear dynamical processes by Mandelbrot, Falconer, Ruelle, and others, many of these results suggested to the researchers concerned with the mathematical modeling of non-linear physical behavior that there thus existed analytical bridges between the macroscopic and the microscopic world in the many scientific disciplines of particular concern. Of particular interest to the author is the use of irregular functions as forcing functions in the solution of such well known non-linear differential equations as Duffing's and Van der Pol's equations in the modeling of electrical or mechanical oscillations with temporarily changing frequencies. Irregular functions, as constructed by Weierstrass and Singh, are defined as those functions that are everywhere continuous, but nowhere differentiable. The use of Weierstrass's function as a forcing function for Duffing's non-linear equation is used to illustrate the behavior of a chaotic process generated by a frequency evolutionary process in time. A simple mechanical model of such a process is represented by the motion of a pendulum when the staff supporting the ball is shortened or lengthened during its execution of otherwise simple harmonic motion.

Citation Download Citation

Robert C. McCarty "Chaotic systems, irregular functions, and nonlinear time series", Proc. SPIE 2038, Chaos in Communications, (18 November 1993); https://doi.org/10.1117/12.162681

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available