Numerical Validation of a Fractional-Order Electronic Chaotic Oscillator Using Lyapunov Exponents

Date of Award

6-2024

Degree Name

Master of Science in Engineering

Department

Electrical and Computer Engineering

First Advisor

Damon A. Miller, Ph.D.

Second Advisor

Giuseppe Grassi, Ph.D.

Third Advisor

Sandun Kuruppu, Ph.D.

Fourth Advisor

Rick Meyer, Ph.D.

Keywords

Chaotic systems, differential equations, dynamical systems, fractional-order calculus, fractional-order integrator, Lyapunov characteristic exponents

Access Setting

Masters Thesis-Abstract Only

Restricted to Campus until

6-1-2034

Abstract

Chaotic systems exhibit exponential sensitivity to initial conditions and non-periodic bounded oscillations. Lyapunov characteristic exponents (LCEs) quantify how quickly nearby trajectories separate in a chaotic attractor. The derivative operators of fractional-order differential equations have a n on-integer order. A n electronic circuit approximation of a fractional-order integral operator was developed and used to implement a fractional-order chaotic system. The approximation optimizes the slope of the integrator frequency magnitude response in a narrow band. Three different methods were used to estimate the circuit LCEs from state voltage samples. The first two methods are direct methods, tracking the exponential divergence of nearby trajectories to estimate the LCE. The third reconstructs the system Jacobian using a neural network, and uses it to estimate the system LCE. The calculation of a positive maximal Lyapunov exponent confirmed that the electronic circuit implemented a chaotic attractor. An additional method based on a driven discrete map was used to further confirm chaotic behavior.

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