Abstract: On the basis of the classical three-dimensional Lorenz system, this paper increases a nonlinear controller to construct a new four-dimensional hyperchaotic Lorenz system. The dynamics of the new system conduct are studied by bifurcation analysis, Lyapunov exponent spectrum and phase chart fractal dimension. Numerical simulations show that the hyperchaotic Lorenz system can render different dynamics behavior of hyperchaos, chaos, periodicity and quasi-periodicity. Furthermore, a corresponding experiment circuit is designed, the mechanics conduct of the electrical system through a oscillogram is observed, the result basically unanimous with the numerical simulation results. Finally, the new hyperchaotic system is applied to image encryption, and computer simulation experiments confirm that the new hyperchaotic system can produce good cryptographic properties of pseudo-random sequence.

Key words: Lorenz system, hyperchaos, Lyapunov exponents, chaotic electric circuit, image encryption

摘要： 在经典三维Lorenz系统的基础上，增加一个非线性控制器，构造了一个新的四维超混沌Lorenz系统。通过数值计算，模拟分析了新系统的分岔图，Lyapunov指数随控制参数的变化，超混沌吸引子的相图，求出系统的Lyapunov指数及其吸引子分形维数。结果显示，通过改变新引入的非线性控制器的控制参数，可以使超混沌Lorenz系统分别呈现超混沌、混沌以及周期、拟周期等动力学行为。根据新Lorenz系统的状态方程，设计了与之相对应的实验电路，并在示波器中观察到电路系统的动力学行为，该结果与数值仿真结果基本吻合。将系统应用于图像加密，模拟实验结果表明，该系统能产生具有良好密码学特性的伪随机序列。

关键词: Lorenz系统, 超混沌, Lyapunov指数, 混沌电路, 图像加密