A New Family of Chaotic Systems with Different Closed Curve Equilibrium

Abstract

Chaotic systems with hidden attractors, infinite number of equilibrium points and different closed curve equilibrium have received much attention in the past six years. In this work, we introduce a new family of chaotic systems with different closed curve equilibrium. Using the methods of equilibrium points, phase portraits, maximal Lyapunov exponents, Kaplan–Yorke dimension, and eigenvalues, we analyze the dynamical properties of the proposed systems and extend the general knowledge of such systems.

New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization

“…Very recently, Zhu and Du [13] discovered and studied a new family of systems with different equilibrium (as shown in Figure 3) described bẏ…”

“…Many chaotic systems with different shapes of attractors have been reported, such as chaotic systems with butterfly attractors (see, e.g., [5]) and systems with multiscroll chaotic attractors (see, e.g., [6]). Recent developments include some different types of chaotic systems with no equilibrium points (see, e.g., [7]), with a single stable equilibrium (see, e.g., [8]), with a line of equilibrium points (see, e.g., [9]), with a circular equilibrium (see, e.g., [10]), with circle and square equilibrium (see, e.g., [11]), with rounded square loop equilibrium (see, e.g., [12]), and with different closed curve equilibrium (see, e.g., [13]). Furthermore, it has also been applied in many different areas including information processing (see, e.g., [14]) and chaotic masking communication (see, e.g., [15]).…”

A hybrid chaotic map for communication security applications

“…In this work, a new 4D chaotic system with only two quadratic nonlinearities is examined. In Zhu and Du, a new family of chaotic systems with different closed curve equilibrium is introduced. In this paper, the proposed system is tested by using equilibrium points, phase portraits, and maximal Lyapunov exponents.…”

Complex Dynamical Behaviors of Lorenz-Stenflo Equations

“…In 1996, the physicist Lennart Stenflo [18] obtained a new four-dimensional continuous-time dynamical chaotic system by adding a new variable w to the Lorenz system to describe the complex dynamical behaviors of the atmospheric acoustic-gravity waves. The Lorenz-Stenflo system can be described by the following ordinary differential equations [18]:…”

“…where a, b, c, r are parameters of system (1) and a > 0, b > 0, c > 0, r > 0. a is the Prandtl number of system (1), c is the Rayleigh number of system (1), b is the geometric parameter of system (1), and r is the rotation parameter of system (1). The Lorenz-Stenflo system is a four-dimensional continuous-time dynamical system that can describe the dynamical behaviors of the atmospheric acoustic-gravity waves in a rotating atmosphere [18]. It is important for us to study acoustic gravity waves because they are associated with minor weather changes and large-scale phenomena.…”