DOI: 10.14457/tu.the.2020.75
title: Simple chaotic jerk, hyperjerk, and hyperchaotic hyperjerk circuits and systems: families of self-excited and hidden attractors

description: The thesis consists of two main parts. The first part proposes three elementary three-dimensional (3D) dissipative chaotic jerk flows with a unique feature of different families of self-excited and hidden attractors, as different values of a single control parameter result in three different types of equilibria. In the family of self-excited attractors, the proposed systems exhibit either a single all-zero-eigenvalue non-hyperbolic equilibrium or two symmetrical hyperbolic equilibria. In the family of hidden attractors, the proposed systems exhibit no equilibria at a particular value of the single control parameter, and therefore all attractors are readily hidden.In the first part of the thesis, the proposed systems represent a rare class of chaotic systems in which a single system exhibits three different types of equilibria through different values of a single control parameter. In particular, for a single all-zero-eigenvalue non-hyperbolic equilibrium, a coefficient of two linear terms provides a simple means to rescale amplitude and frequency, whereas a new introduced constant in a variable x provides a simple means of polarity control. As a result, a chaotic signal with the required amplitude, frequency, and polarity can be possible. For an electronic circuit implementation, two independent potentiometers and an adjustable DC voltage source can be used to control the corresponding chaotic signal, which provides a simple circuit realization for chaos applications.The second part of the thesis consists of two groups. The first group proposes a novel simple single-nonlinearity seven-term memristor-based chaotic hyperjerk (or snap) system. The system exhibits a line equilibrium and coexisting hidden attractors, and is simpler than three existing memristor-based chaotic hyperjerk systems in the sense that it comprises (i) only seven algebraic terms and (ii) a single term of (intrinsic memristive) nonlinearity. In some regions of parameter space, a chaotic attractor coexists with a line equilibrium or with a limit cycle of different periods. Basin boundaries of limit cycles and chaotic attractors have a complicated fractal structure. In some other parameter spaces, two limit cycles are coexisting. The proposed simple chaotic hyperjerk system can be applicable for many engineering applications e.g., chaos-based security and control applications. In the second part of the thesis, the second group proposes a new system in a rare type of four-dimensional (4D) seven-term no-equilibrium hyperchaotic systems and also compares with two existing systems of such a rare type. The proposed system offers ten concurrent advantages, five of which appear to be superior to both existing systems, i.e., (i) the first report of a fourth-order hyperchaotic hyperjerk system with no equilibria, (ii) a much simpler circuit based on only 21 electronic components, (iii) a higher value of the Lyapunov dimension at 3.2280, (iv) a higher value of the largest Lyapunov exponent at 0.2525, and (v) a large two-parameter space of hyperchaos. The other five advantages offer either equal or better features, i.e., (vi) two boostable variables for offset control, (vii) the number of nonlinear terms is two, (viii) no potential dangers of multistability, (ix) hyperchaos, chaos, and periodic behavior are possible, and (x) attractors are readily hidden attractors. The system is unique in the sense that there are no equilibria in both dynamical and hyperjerk forms.