Design And Dynamics Analysis Of Simple Chaotic Circuits And Systems

Chaotic phenomenon is a kind of random-like behavior of deterministic systems widely existing in nature.In the past half century,the research on chaos theory and its applications has made rapid development.The construction,dynamics analysis,and experimental verification of chaotic systems have become a hot topic in the past decade.Generally,simple chaotic circuits and systems with complex dynamical behaviors make more benefits to the promotion of chaos-based pseudorandom number generators and encryption applications in terms of feasibility and flexibility.The simplicity of chaotic circuits and systems can be considered in terms of the following three qualifications: 1)mathematical simplicity of system equation;2)physical simplicity of circuit topology;3)the system should be autonomous and has the most common implementation.It is difficult to achieve all the three simplest cases in practical design.Therefore,the study of simple chaotic circuits and systems with complex dynamics is a challenging and valuable subject in theory and applications.This thesis made an attempt to achieve a balance among the three qualifications and proposed some chaotic circuits and systems.The main contributions of this thesis are as follows.Based on the saturation output characteristic of op-amp,a piecewise-linear Chua’s diode is designed and further applied to a modified Chua’s circuit.Different from most of existing modified Chua’s circuits,the proposed circuit has both stable and unstable equilibria and is able to generate hidden attractors with extremely small attraction basins.Besides,a simple autonomous chaotic circuit is designed only using five discrete components.It is easy to be physically implemented and has robust chaos.The dimensional equations have six linear terms and only one invariant dead-zone nonlinearity.By connecting the memristive diode bridge and negative impedance converter in parallel,a novel second-order locally active extended memristor emulator is constructed.With the nonlinear hysteresis loop characteristic of memristor,a two-element simplest third-order autonomous memristive circuit is designed by simply connecting the memristor and one capacitor in parallel.In comparison with existing memristive chaotic circuits,the proposed circuit has the simplest topology,and exhibits complex behaviors of unipolar periodic and chaotic bursting oscillations along with coexisting attractors.Inspired by the coupled inductor–capacitor(LC)circuit and the saturation output characteristic of op-amp,a four-dimensional conservative chaotic system is proposed,which has only one piecewise nonlinearity and several linear terms.The conservative nature and coexisting motions are analysed by using Liouville’s theorem,Hamiltonian energy function,and Lyapunov exponents.Finally,pseudorandom number generator based on the conservative chaotic system is designed to generate high-quality pseudorandom bits.A simple three-dimensional autonomous chaotic system with only one quadratic nonlinearity and six linear terms is proposed.By introducing control functions to coefficients of specific linear term or quadratic term,the system can be extended as an amplitude-controllable,equilibrium-boostable chaotic system.Consequently,when taking the periodic pulse excitations as control functions,multi-scroll attractors are further generated via polarity symmetry.This approach can be further extended to variableboostable systems to generate different multi-scroll attractors and simplify the circuit implementations.A simple parametric control scheme for multi-scroll attractor generation is proposed and applied to five-term autonomous chaotic systems.Firstly,the three-segment piecewise-linear(PWL)saturation function is nested inside the sine nonlinearity to limit the amplitude of the inputs,i.e.,govern the number of periods existing in sine nonlinearity.When the nested sine-PWL function is introduced to jerk system,the number of system equilibria is governed by the single parameter,i.e.,the saturation value,which further selects the multi-scroll attractors.If introducing the nested sine-PWL function to Sprott-A system without equilibrium point,the saturation value can be used to extend the number of chaotic scrolls.Because the modified Sprott-A systems have an integer dimension of three,the evolutions of chaotic orbits are finally filled of all volume of the phase space and coin chaotic sea rather than an attractor.