Tools to explore chaos are as far away as a personal computer or a pocket calculator. A few lines of simple equations in BASIC produce fantastic graphic displays. In the following computer experiment, the dimension of a strange attractor is found by three algorithms; Shaw's, Grassberger-Procaccia's and Guckenheimer's. The programs were tested on the Henon attractor which has a known fractal dimension. Shaw's and Guckenheimer's algorithms were tested with 1000 data points, and Grassberger's with 100 points, a data set easily handled by a PC in one hour or less using BASIC or any other language restricted to 640K RAM. Since dimension estimates are notorious for requiring many data points, the author wanted to find an algorithm to quickly estimate a low-dimensional system (around 2). Although all three programs gave results in the neighborhood of the fractal dimension for the Henon attractor, Dfracta1=1.26, none appeared to converge to the fractal dimension.

In previous papers by Awrejcewicz in 1986 and Narayanan and Jayaraman in 1991, it was claimed that the nonlinear oscillator with dry friction exhibited chaos for several forcing frequencies. The chaos determination was achieved using the characteristic exponent of Lyapunov which requires the right-hand side of the differential equation to be differentiable. With the addition of the dry friction term, the right-hand side of the equation of motion is not continuous and therefore not differentiable. Thus this approach cannot be used. The Filippov definition must be employed to handle the discontinuity in the spatial variable. The behavior of the nonlinear oscillator with dry friction is studied using a numerical solver which produces the Filippov solution. The results show that the system is not chaotic; rather it has a stable periodic limit cycle for at least one forcing frequency. Other forcing frequencies produce results that do not clearly indicate the presence of chaotic motion. / Graduation date: 2002

Our investigation was to study the feasibility of piece-wise synchronization of chaotic circuits. In conventional experiments of electronic-circuit chaotic synchronization, two circuits were real-time and continuous connected together. In our research, a computer was used as the master subsystem that output chaotic signals to a slave circuit to study the performance of synchronization. The circuit was based on Cuomo¡¦s design. Several methods of piece-wise control were tested to find out the key point of chaotic synchronization. The experimental results revealed that the most important synchronized waveforms were the chaotic orbits near the region that the attractor change quadrants. A conditional piece-wise synchronization method was developed based on our discoveries. Comparing to the periodic piece-wise synchronization method, our method is more robust to sustain the circuit noise. Another advantage is that, in our method, the experimental results fit the computer simulation quite well.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

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