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Non-autonomous Chaotic Circuit that Integrates a Variable Forcing Function on to a Single PCB
Keywords: Chaos Electronics, Nonlinear Systems and Dynamics, High Frequency
Initially, the area of chaos was primarily studied by mathematicians and physicist in order to help describe or model physical, chemical or naturally occurring phenomena. This motivation has now shifted towards taking advantage of the inherent properties found in chaos for various applications, such as communication systems, radar, random number generation (RNG), and noise signal generation. Some of the advantageous properties include continuous power spectral density for communication, radar and noise signal generation. Additionally, the topological mixing and long-term aperiodic nature of chaos is beneficial in RNG. Many of these chaotic systems are defined by ideal autonomous third order (or higher) ordinary differential equations. In order to gain these advantages in these applications, these systems often need to be implemented in electronics. The structures of autonomous systems often require a feedback path, which is negligible at lower frequencies. However, when the frequency is significantly increased, the finite propagation delays associated with real world devices can cause problems in recreating the ideal system’s dynamics. One method of trying to minimize the overall propagation delay through the feedback path is to avoid the use of operational amplifiers and realize a similar function using only transistors. This can reduce the design complexity if the final product is intended for an ASIC. However, some non-autonomous systems might more favorably scale with frequency better than an autonomous system. The structure of non-autonomous systems typically consists of a linear forcing function and some sort of passive nonlinearity. Many of these systems are described in literature using ideal mathematical definitions, but some of these were observed when analyzing a nonlinear system in laboratory testing. An example of these could be seen in the nonlinear circuits realized using transistors or diodes and tabletop function generators. These experiments are often used as a way to demonstrate the inherent nonlinearities in devices such as transistors or diodes. While these are effective at demonstrating the underlying nonlinear dynamics, they are often at very low frequencies (in the range of tens of Hertz) and require external equipment, which limits these systems for being used for potential applications where high frequency operation can be beneficial. In order to take advantage of these particular systems, the linear forcing function can be integrated onto a single PCB with these passive nonlinear circuits. There is a wide range of options to generate the linear sinusoidal oscillators. Some of the simplest methods to analyze include using operational amplifiers (op amp), which is sufficient for low up to moderate frequencies. For very high frequencies, a transistor-based approach would most likely be better; however, this is not as easy to analyze the circuit topology. Presented here is a demonstration of a non-autonomous chaotic nonlinear circuit where the forcing function is integrated onto the same PCB. While the chaotic subsystem is non- autonomus, the overall functionality of the full circuit behaves more similarly to an autonomous system including the forcing function. This implementation uses four op amps, a single transistor, resistors, and capacitors. No inductor is required and the sinusoidal oscillator is included on the board help to minimize the designs physical footprint, which is approximately 1in by 1.2in. The forcing function is generated using a twin-T op amp based oscillator and a nonlinear transistor circuit with two RC integrating constants. Using the appropriate resistor and capacitor values, the system can be operated from 2kHz up to 9.6MHz.
Benjamin K. Rhea,
Auburn University
Auburn University, AL

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