多激励源混频构成的混沌

Chaos Structured by Mixing of Multi-Excited Source

摘要: 描写动态系统非线性微分方程的解，传统习惯于用时间函数u(t)表示，但这不是唯一的方式，特别是当u(t)的表达式解不出来的时候。现代的电路与系统理论，在动态系统中，选取合适的三个动态变量组成一个三维的相空间，三个变量相互间的非线性关系，可以用一条有界的空间曲线来描写，这就是三维的相图。对于N>3个变量可以组成 维的欧氏空间。这条有界的空间曲线，在数学上，不能用具体明显的参数式表达，我们不能求出其解析解。但可以用数值仿真画出它的图形解。如果这条有界的空间曲线在仿真的时间内是非周期的，这就是连续时间系统的轨道混沌。本文用频域的分析方法与功率平衡定理，研究混沌函数的诞生与属性。证明具有不同频率的多个激励源的混频，构成二阶微分电路也可以产生混沌。

Abstract: The solutions of dynamical system expressed with nonlinear differential equation usually is shown by using time function u(t). But this is not unique mode, when particularly u(t) cannot be solved. In the modern theory of circuit and system, we can select three dynamical variables in the nonlinear system to constitute 3-dimension phase space. The mutual nonlinear relation among three dynamical variables can be described by a bounded space curve. This is 3-dimension phase portrait. The nonlinear dynamical systems of regarding N>3 variation may constitute  -dimension Euclidean space. The bounded space curve cannot be represented by concretely explicit parametric form in math. It cannot be solved analytically by human. However, the graphic solution can be plotted by numerical simulation. If the bounded space curve is non-periodic in simulation interval, this is orbital chaos of continuous time system. This paper researches the produce and property of chaos by means of the analysis method of frequency domain and theorem of power balance. We prove that the second order differential circuit which is constituted by mixing of multi-excited source with different frequency also can produce chaos.