STUDY OF CHAOTIC CIRCUITS, THEIR DESIGN, AND APPLICATIONS

Abstract

Chaos is an exceptional phenomenon occurring in non-linear dynamic systems. Our lives are full of non-linear dynamic systems, such as the Solar System, the weather, the stock market, the human body, plant and animal populations, cancer growth, spread of pandemics, chemical reactions, the electrical power grid, the Internet, etc. the behaviour of which can be best modelled as chaotic system. More and more chaotic systems are being invented in an effort to simplify their algebraic representation but with high degree of complexity in output. With the development of non-liner dynamics and chaos theory, the hardware realization of chaotic systems has become a useful way to use them in real world applications for reliable and portable communication devices. In this direction, this thesis is primarily concerned with the hardware optimized design of chaotic systems. The idea is to design a circuit with minimum possible number of active and passive components which obeys the governing equations of a chaotic system. In chaotic systems, a single or more non-linear terms are always present in the governing equations which is responsible for output’s complexity. Quadratic, exponential, hyperbolic are some of the representative nonlinearities present in chaotic systems. Generally, implementation of the governing equations of a chaotic system requires the addition, subtraction, scaling operations besides the non-linearity. The Operational Amplifier (OpAmp) is commonly used active block in hardware implementation of chaotic systems. Current mode active block, such as Current Feedback Operational Amplifier (CFOA), has capabilities of handling both current and voltage signals which culminates in compact realizations in comparison to the existing OpAmp counterparts. In addition to CFOAs, suitable active and passive components are used to realize the non-linearity(ies) present in the governing equations and any scaling required therein. The design of Rӧssler chaotic system having single quadratic non-linear term, is addressed first by presenting CFOA based realization. It also uses an analog multiplier (AM) namely AD633. The presented design is a compact realization in comparison to existing OpAmp based counterparts. A complete circuitry for adaptive control synchronization between two Rӧssler chaotic systems is also put forward. A chaotic system based on two quadratic non-linearities namely Pehlivan–Uyarŏglu Chaotic System (PUCS), is focused next. Four variants of PUCS are introduced which appear v to be distinct in the sense that there is no obvious transformation of one into another. Non linear dynamic properties of these variants are investigated and expatiated through bifurcation diagrams, Lyapunov exponents, Kaplan Yorke Dimension, nature of fixed points through eigenvalues and chaotic phase space diagrams. A CFOA based realization is put forward that can realize the existing PUCS and its proposed variants, by simply adjusting component values. In AD633, there is a possibility of realizing algebraic functions including single multiplication, subtraction of two multiplication terms, and accumulation. The versatility of AD633 is gainfully exploited in presenting generalized circuit topology to implement chaotic systems with quadratic type non-linearities. The use of AM is inevitable in hardware realization of chaotic systems with quadratic nonlinearity, leading to increased active block count. A new chaotic system with exponential non-linearity has been put forward. It is realized using CFOAs and diodes, thus reducing the count of active building blocks. This new chaotic system has been verified for different properties, including Lyapunov Exponents, Kaplan Yorke Dimension and dissipativity. With the increase in demand of higher complexity for security, the dimension of the chaotic system can be extended beyond three. Such higher dimensional systems, also known as hyperchaotic systems, can be useful in those applications. One such four dimensional hyperchaotic system with two quadratic type non-linearities has been proposed and tested for different properties, including Lyapunov Exponents, Kaplan Yorke Dimension and dissipativity. The circuit implementation using CFOAs and AMs is also presented. All the above designs are verified either through LTspice simulations or combination of LTspice simulations and experimental evaluations. An attempt has also been made to realize chaotic systems on digital platform namely Field Programmable Gate Arrays (FPGAs). Ten different chaotic systems have been compared based on hardware utilization and delay on the target FPGA device Artix 7. Besides numerical simulations in python for confirmation of the correctness of the implemented systems via observations, the functional verification of the synthesized designs has been done using inbuilt simulator of Xilinx Vivado design suite.