A new simple four-dimensional equilibrium-free autonomous ODE system is described. The system has seven terms, two quadratic nonlinearities, and only two parameters. Its Jacobian matrix everywhere has rank less than 4. It is hyperchaotic in some regions of parameter space, while in other regions it has an attracting torus that coexists with either a symmetric pair of strange attractors or with a symmetric pair of limit cycles whose basin boundaries have an intricate fractal structure. In other regions of parameter space, it has three coexisting limit cycles and Arnold tongues. Since there are no equilibria, all the attractors are hidden. This combination of features has not been previously reported in any other system, especially one as simple as this.

Coexisting Hidden Attractors in a 4-D Simplified Lorenz System

A new finding of the existence of hidden hyperchaotic attractors with no equilibria

This paper studies a four-dimensional (4D) memristive system modified from the 3D chaotic system proposed by Lu and Chen. The new system keeps the symmetry and dissipativity of the original system and has an uncountable infinite number of stable and unstable equilibria. By varying the strength of the memristor, we find rich complex dynamics, such as limit cycles, torus, chaos, and hyperchaos, which can peacefully coexist with the stable equilibria. To explain such coexistence, we compute the unstable manifolds of the equilibria, find that the manifolds create a safe zone for the hyperchaotic attractor, and also find many heteroclinic orbits. To verify the existence of hyperchaos in the 4D memristive circuit, we carry out a computer-assisted proof via a topological horseshoe with two-directional expansions, as well as a circuit experiment on oscilloscope views.

Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria

Sliding mode control is an important method used in nonlinear control systems. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as its insensitivity to parameter uncertainties and disturbances. In this paper, we derive new results based on the sliding mode control for the anti-synchronization of identical Qi three-dimensional (3D) four-wing chaotic systems (2008) and identical Liu 3D four-wing chaotic systems (2009). The stability results for the anti-synchronization schemes derived in this paper using sliding mode control (SMC) are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the SMC method is very effective and convenient to achieve global chaos anti-synchronization of the identical Qi four-wing chaotic systems and identical Liu four-wing chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper.

Anti-synchronization of four-wing chaotic systems via sliding mode control

Recently, a novel image encryption scheme based on improved hyperchaotic sequences was proposed. A pseudo-random number sequence, generated by a hyper-chaos system, is used to determine two involved encryption functions, bitwise exclusive or (XOR) operation and modulo addition. It was reported that the scheme can be broken with some pairs of chosen plain-images and the corresponding cipherimages. This paper re-evaluates the security of the encryption scheme and finds that the encryption scheme can be broken with only one known plain-image. The performance of the known-plaintext attack, in terms of success probability and computation load, become even much better when two known plain-images are available. In addition, security defects on insensitivity of the encryption result with respect to changes of secret key and plain-image are also reported.

Breaking a novel image encryption scheme based on improved hyperchaotic sequences

This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic attractors and a pair of diagonal double-wing chaotic attractors. More importantly, the system can generate a four-wing chaotic attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic attractors, major and minor diagonal double-wing chaotic attractors, and a four-wing chaotic attractor. Poincare-map analysis shows that the system has extremely rich dynamics. The physical existence of the four-wing chaotic attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.

A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system

On a new hyperchaotic system

A new hyperchaotic system which has two large positive Lyapunov exponents is presented and physically implemented. Spectral analysis shows that the system in the hyperchaotic mode has an extremely broad frequency bandwidth of high magnitudes, verifying its unusual random nature and indicating its great potential for some relevant engineering applications such as secure communications.

A new hyperchaotic system and its circuit implementation

In this paper, new properties of a four-dimensional chaotic system are investigated. These properties explain the behavior of the system and clarify why it can only generate two coexisting double-wing chaotic attractors but cannot produce a single four-wing chaotic attractor. It is shown that a new system with an extremely complex four-wing chaotic attractor and a larger positive Lyapunov exponent than the original system is formed by using these findings and introducing state feedback control to the system. Some basic dynamical behaviors and the complex structure of the new four-wing autonomous chaotic system are theoretically investigated. A detailed bifurcation analysis demonstrates the evolution process from local attractors to global attractors. The local attractors include two coexisting sinks, two coexisting single-wing periodic orbits and two coexisting double-wing chaotic attractors. The global attractors contain a diagonal double-wing periodic orbit, a diagonal double-wing chaotic attractor and a four-wing chaotic attractor. Spectral analysis indicates that the system in the four-wing chaotic mode has a very wide frequency bandwidth, confirming its random nature and its suitability to engineering applications such as secure communications.

A four-wing attractor and its analysis

In this paper, several smooth canonical 3-D continuous autonomous systems are proposed in terms of the coefficients of nonlinear terms. These systems are derived from the existing 3-D four-wing smooth continuous autonomous chaotic systems. These new systems are the simplest chaotic attractor systems which can exhibit four wings. They have the basic structure of the existing 3-D four-wing systems, which means they can be extended to the existing 3-D four-wing chaotic systems by adding some linear and/or quadratic terms. Two of these systems are analyzed. Although the two systems are similar to each other in structure, they are different in dynamics. One is sensitive to the initializations and sampling time, but another is not, which is shown by comparing Lyapunov exponents, bifurcation diagrams, and Poincare maps.

https://ujcontent.uj.ac.za/vital/access/services/Download/uj:15914/SOURCE1

Michaël Antonie van Wyk

Papers

A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system

On a new hyperchaotic system

A new hyperchaotic system and its circuit implementation

A four-wing attractor and its analysis

A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems