Narrow Results

The triangular form of a chaotic system and how to obtain it from the Lie derivative are introduced. Based on this, both the synchronization of Genesio–Tesi and Coullet systems and the generalized synchronization of Lü and Sprott-B systems are realized in two feedback ways through designing nonlinear controllers. Results of numerical simulations in MATLAB validate the effectiveness of the given synchronization methods further.

A new conjugate Lorenz-type system is introduced in this paper. The system contains as special cases the conjugate Lorenz system, conjugate Chen system and conjugate Lü system. Chaotic dynamics of the system in the parametric space is numerically and thoroughly investigated. Meanwhile, a set of conditions for possible existence of chaos are derived, which provide some useful guidelines for searching chaos in numerical simulations. Furthermore, some basic dynamical properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions and the compound structure of the system are demonstrated with various numerical examples.

Time-delayed feedback has been introduced as a powerful tool to control unstable periodic orbits or control unstable steady states. In the present paper, regarding the delay as a parameter, we investigate the effect of delay on the dynamics of Lü system with delayed feedback. After the effect of delay on the steady states is analyzed, Hopf bifurcation is studied, where the direction, stability and other properties of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Finally, we provide several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

This paper reports the finding of a chaotic system with one saddle and two stable node-foci in a simple three-dimensional (3D) autonomous system. The system connects the original Lorenz system and the original Chen system and represents a transition from one to the other. The algebraical form of the chaotic attractor is very similar to the Lorenz-type systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the chaotic system has a chaotic attractor, one saddle and two stable node-foci. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions, and the compound structure of the system are analyzed and demonstrated with careful numerical simulations.

This letter introduces a hyperchaotic system from the Lü system [Lü et al., 2004] with a linear state feedback controller. This hyperchaotic system has more complex dynamical behaviors, and can generate 4-scroll hyperchaotic attractor and 2-scroll chaotic attractor under different control parameters. In particular, the system can also exhibit novel coexisting intermittent chaotic orbits. Theoretical analyses and simulation experiments are conducted to investigate the dynamical behaviors of the proposed hyperchaotic system.

In this letter we synthesize numerically the Lü attractor starting from the generalized Lorenz and Chen systems, by switching the control parameter inside a chosen finite set of values on every successive adjacent finite time intervals. A numerical method with fixed step size for ODEs is used to integrate the underlying initial value problem. As numerically and computationally proved in this work, the utilized attractors synthesis algorithm introduced by the present author before, allows to synthesize the Lü attractor starting from any finite set of parameter values.

Firstly, the dynamics of the Lü system having an invariant algebraic surface are analyzed. Secondly, by using the Poincaré compactification in ℝ³, a global analysis of the system is presented, including the complete description of its dynamic behavior on the sphere at infinity. Lastly, combining analytical and numerical techniques, it is shown that for the parameter value b = 0, the system presents an infinite set of singularly degenerate heteroclinic cycles. The chaotic attractors for the Lü system in the case of small b > 0 are found numerically, hence the singularly degenerate heteroclinic cycles.

By constructing a suitable Lyapunov function, we show that for the Lü chaotic system parameters in some specified regions, the solutions of the system are globally bounded.

In this paper, Lü system with a set of chaotic parameters is proved to be smoothly nonequivalent to Chen and Lorenz systems with any parameter. The analytical formula for the upper bound of Lyapunov dimension of attractors in Lorenz-type systems are presented under some suitable parameter conditions. These properties studied in this paper may contribute to a better understanding of the Lorenz-type systems.

In this paper, the existence of homoclinic orbits of the equilibrium point $(0,0,0)$ is demonstrated in the case of the Lü system for parameter values not reported by G. A. Leonov. In addition, some simulations are shown that agree with our theoretical analysis.

In this paper, an efficient novel dual permutation–substitution structure-based color image encryption algorithm is proposed. Initially, the Secure Hash Algorithm-512 (SHA-512) is applied to the input image to generate the initial values for the Lü system dynamically. In the first stage of permutation, inter-color-component pixel shuffling is carried out with a circular pixel-swapping mechanism. In the second stage, intra-color-component pixel shuffling is executed, based on pseudorandom positions generated by the Lü chaotic system. Pixel values are changed in the substitution stage, based on float-valued chaotic sequences generated by the Lü system. The performance of the proposed algorithm is evaluated with metrics such as key space, key sensitivity, histograms, correlation coefficients (vertical, horizontal and diagonal), information entropy, number of pixel changes rate (NPCR), unified average changing intensity (UACI), peak signal-to-noise ratio (PSNR), mean absolute error (MAE), contrast analysis, encryption time and the National Institute of Standards and Technology Special Publication 800-22 (NIST SP 800-22) statistical test. The experimental results obtained and performance assessment show that the proposed scheme has produced good results.