Yet another 3D quadratic autonomous system generating three-wing and four-wing chaotic attractors

A Novel 2D – Grid of Scroll Chaotic Attractor Generated by CNN

The complex grid of scroll chaotic attractors that are generated through nonlinear electronic circuits have been raised considerably over the last decades. In this paper, it is shown that a subclass of Cellular Nonlinear Networks (CNNs) allows us to generate complex dynamics and chaos in symmetry pattern. A novel grid of scroll chaotic attractor, based on a new system, shows symmetry scrolls about the origin. Also, the equilibrium points are located in a manner such that the symmetry about the line x=y has been achieved. The complex dynamics of system can be generated using CNNs, which in turn are derived from a CNN array (1×3) cells. The paper concerns on the design and implementation of 2×2 and 3×3 2D-grid of scroll via the CNN model. Theoretical analysis and numerical simulations of the derived model are included. The simulation results reveal that the grid of scroll attractors can be successfully reproduced using PSpice.

A new method for generating chaotic system with arbitrary shaped distributed attractors

In this paper, a new method for generating a chaotic system with arbitrary shaped (including heart-shaped, oval, circle, piecewise-linear, and cuboid) distributed attractors is proposed. In this article, a simple four-wing chaotic attractor is first presented by using a periodic piecewise function instead of a constant parameter in the Lorenz system, on the basis of which the chaotic system with arbitrary shaped distributed attractors in the plane can be constructed. This means that the distributed chaotic attractors can be arranged in an arbitrary shape in the plane. The chaotic system can generate any quantity of distributed chaotic attractors, and simulation results show that any desired number of positive Lyapunov exponents can be obtained. Therefore, the chaotic system will have more complicated dynamic characteristics. The dynamical mechanisms of this chaotic system are further investigated, and theoretical analysis and numerical simulation are in accordance with each other, which verifies the effectiveness of the approach. Lastly, the proposed chaotic system is used for image encryption. Numerical results show that the proposed scheme has an excellent performance.

Chen's attractor exists if Lorenz repulsor exists: the Chen system is a special case of the Lorenz system

In this paper, we show, by means of a linear scaling in time and coordinates, that the Chen system, given by x=a(y-x), y=(c-a)x+cy-xz, ż=-bz+xy, is, generically (c≠0), a special case of the Lorenz system. First, we infer that it is enough to consider two parameters to study its dynamics. Furthermore, we prove that there exists a homothetic transformation between the Chen and the Lorenz systems and, accordingly, all the dynamical behavior exhibited by the Chen system is present in the Lorenz system (since the former is a special case of the second). We illustrate our results relating Hopf bifurcations, periodic orbits, invariant surfaces, and chaotic attractors of both systems. Since there has been a large literature that has ignored this equivalence, the aim of this paper is to review and clarify this field. Unfortunately, a lot of the previous papers on the Chen system are unnecessary or incorrect.

Multiscroll attractors by switching systems

In this paper, we present a class of three-dimensional dynamical systems having multiscrolls which we call unstable dissipative systems (UDSs). The UDSs are dissipative in one of its components but unstable in the other two. This class of systems is constructed with a switching law to display various multiscroll strange attractors. The multiscroll strange attractors result from the combination of several unstable "one-spiral" trajectories by means of switching. Each of these trajectories lies around a saddle hyperbolic stationary point. Thus, we describe how a piecewise-linear switching system yields multiscroll attractors, symmetric or asymmetric, with chaotic behavior.