Abstract
A hyperchaotic system is introduced, and the complex dynamical behaviors of such system are investigated by means of numerical simulations. The bifurcation diagrams, Lyapunov exponents, hyperchaotic attractors, the power spectrums, and time charts are mapped out through the theory analysis and dynamic simulations. The chaotic and hyper-chaotic attractors exist and alter over a wide range of parameters according to the variety of Lyapunov exponents and bifurcation diagrams. Furthermore, linear feedback controllers are designed for stabilizing the hyperchaos to the unstable equilibrium points; thus, we achieve the goal of a second control which is more useful in application.
Highlights
In recent years, hyperchaotic systems have been extensively studied because of exhibiting at least two positive Lyapunov exponents
This paper makes further analysis on chaotic and hyperchaotic system based on the new four-dimension system which has been joined in the state feedback control
One of which is that we will analyze and compare both chaotic and hyperchaotic Lyapunov dimensions and give the corresponding chaotic attractors and hyperchaotic attractors; the other one is that we introduce a second controller to the chaotic or hyperchaotic system which has already controlled by introducing a state feedback controller to delay or control hyperchaos, realizing the goal of a second control
Summary
Hyperchaotic systems have been extensively studied because of exhibiting at least two positive Lyapunov exponents. The dynamics of the system expand in more than one direction and generate a much more complex attractor compared with the chaotic system with only one positive Lyapunov exponent, which is studied by Rech et al [1, 2]. Correia and Rech [16] have introduced a state feedback control to the first equation based on Wang and proposed a method that considers the magnitude of the second largest Lyapunov exponent to numerically characterize points with hyperchaotic behavior, in which two parameters are simultaneously varied, and they obtained some practical and theoretical significance in the practice. This paper makes further analysis on chaotic and hyperchaotic system based on the new four-dimension system which has been joined in the state feedback control. The results are summarized, and the future directs are indicated
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