Proceedings PaperChaotic systems, irregular functions, and nonlinear time series
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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.