Abstract
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.
Highlights
This has identified most of the elementary chaotic jerk systems with quadratic terms, which enable three different families of self-excited and hidden attractors in three different types of equilibria, i.e., a single all-zero-eigenvalue non-hyperbolic equilibrium, two symmetrical hyperbolic equilibria, and no equilibria
NUMERICAL RESULTS Table 1 lists equations of three simple chaotic jerk systems VE1-VE3, depending on the values of a single control parameter c, which corresponds to a7 in (1), for different families of self-excited and hidden attractors
REGIONS OF DYNAMICAL BEHAVIOR FOR SELF-EXCITED AND HIDDEN ATTRACTORS As shown in Table 1, the parameter c is the controller of dynamical behavior of the three chaotic jerk systems VE1, VE2, and VE3, as different values of c change types of equilibria and attractors
Summary
A three-dimensional (3D) chaotic system is expressed by a set of three coupled first-order ODEs, e.g., [1], whereas a chaotic jfeorrkmsy.x.s. t=emf i(sxe, xxp, rxe)ss[e2d]. The single control parameter performs as a constant controller to select the required dynamics These systems can have a self-excited chaotic attractor with an all-zero-eigenvalue non-hyperbolic equilibrium, a selfexcited chaotic attractor with two symmetrical hyperbolic equilibria, and a hidden chaotic attractor with no equilibria. This has identified most of the elementary chaotic jerk systems with quadratic terms, which enable three different families of self-excited and hidden attractors in three different types of equilibria, i.e., a single all-zero-eigenvalue non-hyperbolic equilibrium, two symmetrical hyperbolic equilibria, and no equilibria. NUMERICAL RESULTS Table 1 lists equations of three simple chaotic jerk systems VE1-VE3, depending on the values of a single control parameter c, which corresponds to a7 in (1), for different families of self-excited and hidden attractors.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
