Simple Chaotic Jerk Flows With Families of Self-Excited and Hidden Attractors: Free Control of Amplitude, Frequency, and Polarity

Although chaotic systems with self-excited and hidden attractors have been discovered recently, there are few investigations about relationships among them. In this paper, using a systematic exhaustive computer search, three elementary three-dimensional (3D) dissipative chaotic jerk flows are proposed with the unique feature of exhibiting different families of self-excited and hidden attractors. These systems have a variable equilibrium for different values of the single control parameter. In the family of self-excited attractors, these systems can have a single all-zero-eigenvalue non-hyperbolic equilibrium or two symmetrical hyperbolic equilibria. Besides, for a particular value of the parameter, these systems have no equilibria, and therefore all the attractors are readily hidden. The proposed systems represent a rare class of chaotic systems in which a single system exhibits three different types of equilibria for different values of the single control parameter. In particular, for a single all-zero-eigenvalue non-hyperbolic equilibrium, the coefficient of the two linear terms provides a simple means to rescale the amplitude and frequency, while the introduction of a new constant in the variable x provides a polarity control. Therefore, a free-controlled chaotic signal can be obtained with the desired amplitude, frequency, and polarity. When implemented as an electronic circuit, the corresponding chaotic signal can be controlled by two independent potentiometers and an adjustable DC voltage source, which is convenient for constructing a chaos-based application system.