ID 原文 译文
4163 基于传统的 Qi 混沌系统，通过增加控制参数和改变非线性项相结合的方法构造了一种具有复杂混沌特性的新型统一混沌系统。 Based on the traditional Qi chaotic system, a novel unified chaotic system with the complex chaotic characte-ristics was constructed by adding the control parameters and modifying the nonlinear terms.
4164 首先，分析了所提系统的动力学行为，数值仿真了相图、时域波形图、Poincare 截面图和功率谱图； Firstly, basic dynamical cha-racteristics of chaotic system were analyzed, and phase portrait, time domain waveform diagram, Poincare mapping andpower spectrum diagram were numerically simulated.
4165 其次，讨论了系统参数变化时 Lyapunov 指数谱、分岔图和混沌信号幅度对整个系统的影响，分析表明，所提统一混沌系统能产生 4 种新的具有多参数恒 Lyapunov 指数谱特性的双翼混沌吸引子，并存在多种全局和局部的非线性调幅参数； Secondly, system parameters influence on chaotic system was dis-cussed through Lyapunov exponent spectrum, bifurcation diagrams and chaotic signal amplitude. It was found that theunified chaotic system can generate the four new types of two-wing chaotic attractors with the multi-parameter invariableLyapunov exponent spectrum characteristics. Meanwhile, there exist the functions of the global and local nonlinear am-plitude modulation parameters.
4166 再次，以所提系统的第一种混沌吸引子为例，引入 2 种新的非线性函数，实现了网格多翼的扩展； Thirdly, taking the first chaotic attractor of system as an example by introducing the twonew types of nonlinear functions, the expansion of grid multi-wing attractor was realized.
4167 最后，搭建所提统一混沌系统的硬件电路，实验观察到 4 种新混沌吸引子，与数值仿真结果一致，验证了所提系统的可行性。 Finally, the hardware circuit ofnovel unified chaotic system was constructed. The four new types of chaotic attractors are observed ex
4168 深入研究了特征算子的谱表示与特征展开。 The spectral representation and expansion based on eigen-operators were deeply studied.
4169 给出了特征微分方程格林函数与厄尔密特微分算子及厄尔密特积分算子的关系式，以及厄尔密特微分算子与厄尔密特积分算子两者互逆的关系式； The relations be-tween Green's function of characteristic differential equation and Hermitian differential and integral operators were given.The inverse relations between Hermitian differential operator and Hermitian integral operator were also studied.
4170 给出了厄尔密特微分算子的谱表示，指出有限区间斯刘特征方程不能用于实现无穷维的谱表示式，厄尔密特微分算子的谱表示比诺伊曼研究简单清楚得多，具有优越性； The spectral representation of Hermitian differential operators was given. It was also shown that the S-L eigen-equation can-not be used to realize the spectral representations of infinite dimensions in a finite interval. The method is much simplerand clearer than that of Neumann, and has advantages.
4171 给出了厄尔密特积 分算子的特征展开（特征分解），具有理论一般性与全面性的优点，对文献[2]中将其用于研究特征谱表示的不正确论述进行了更正； The eigen-expansion (eigen-decomposition) of the Hermitian in-tegral operator was given, which has the advantages of theoretical generality and comprehensiveness. The incorrect dis-cussion in Wang et al [2] was correct that it is used to study the representation of characteristic spectrum.
4172 给出了最优特征展开中长球面波函数命名的物理与几何意义。 The physical andgeometric meanings of naming for the long spherical wave function in optimal eigen-expansion were given.