论文部分内容阅读
利用非线性电子元件和谐振回路构成的非线性系统,可以模拟非线性微分方程解的性质。非线性微分方程:x+γx+αx~2=A+Bcosω_i的解x(t)随参数B,ω_i的变化将经历各种系列的倍周期状态,如2×2~n系列,2×3~n系列等等。当调节参数B,ω_i使解x(t)处于倍周期分岔前瞬间,加入小信号λcosω_st,研究方程:x+γx+αx~2=A+Bcosω_it+λcosω_st的解η(t)。结果表明了非线性微分方程对小信号具有选频放大的普适性质。在模拟小信号对非线性微分方程的解的影响时,为使小信号的作用在非线性系统中充分表现,采用了双LRC振子系统,小信号直接作用于起主要作用的非线性元件上。双LRC振子系统的回路方程和上面第二式是同类型的非线性微分方程。所以该非线性系统可以模拟其解η(t)。实验观测发明:系统对小信号放大倍数>40;被放大的小信号频率与分岔状态频率之间的关系同理论一致,并且,倍周期分岔状态频率越小,放大的奇数倍频率分量也越多。作者还对实验结果、影响精度等问题作了初步讨论。
Using nonlinear systems consisting of nonlinear electronic components and resonant circuits, the properties of solutions to nonlinear differential equations can be modeled. The nonlinear differential equation: The solution x (t) of x + γx + αx ~ 2 = A + Bcosω_i will go through various series of periodic states with the change of parameters B and ω_i, such as 2 × 2 ~ n series, 2 × 3 ~ n series and so on. When the parameter B, ω_i is adjusted so that the solution x (t) is at the instant before the doubling period bifurcation, a small signal λcosω_st is added to study the solution η (t) of the equation: x + γx + αx ~ 2 = A + Bcosω_it + λcosω_st. The results show that the nonlinear differential equation has the universal property of frequency selective amplification for small signals. In simulating the effect of small signals on the solution of nonlinear differential equations, a dual LRC oscillator system is used to make the small signal behave well in nonlinear systems. Small signals act directly on the nonlinear elements that play a major role. The loop equation of the double LRC oscillator system is the same type of nonlinear differential equation as the second equation above. So the nonlinear system can simulate its solution η (t). The experimental observations show that: the system for small signal magnification> 40; amplified small signal frequency and the relationship between the bifurcation frequency is the same theory, and, the smaller the frequency of the doubling period bifurcation, the amplified odd multiple frequency components more. The author also made a preliminary discussion on the experimental results, the impact accuracy and other issues.