分段线性化二阶DPCM传动系统的动力学研究
Dynamical Study of a Second Order DPCM Transmission System Modeled by a Piece-Wise Linear Function
摘要：
本文分析了二阶差分脉冲编码调制（DPCM）传输系统在考虑量化器非线性特性时的性能。以这种方式，定性的新特性的DPCM系统已被解开，这不能被观察到和解释，如果量化器的非线性被忽略。为了这项研究的目的，被认为是一个分段线性不可微量化特性。所得到的DPCM模型是迭代方程（即映射）的形式，其中逆映射不是唯一的（即不可逆映射）。因此，不可逆映射的数学理论特别适合于这种分析，以及非线性动力学的更经典的工具。这项研究使我们能够从理论上证明不可微映射与可微映射相比的一些新性质。在简要回顾了不可逆映射之后，将所提出的不可逆映射的方法和工具应用于DPCM系统。提出了一种计算DPCM映射分叉曲线的新算法。通过对参数和相平面的研究，指出了不同的非线性现象，如分岔曲线的重叠导致多稳定性、混沌行为或具有分形边界的多个盆地。所有观察到的现象表明，即使在恒定的输入信号的情况下，非常复杂的动力学行为，这里讨论。
英文摘要：
This paper analyses the behaviour of a second order DPCM (Differential Pulse Code Modulation) transmission system when the nonlinear characteristic of the quantizer is taken into consideration. In this way, qualitatively new properties of the DPCM system have been unravelled, which cannot be observed and explained if the nonlinearity of the quantizer is neglected. For the purposes of this study, a piece-wise linear nondifferentiable quantizer characteristic is considered. The resulting model of the DPCM is of the form of iteration equations (i.e. map), where the inverse iterate is not unique (i.e. noninvertible map). Therefore the mathematical theory of noninvertible maps is particularly suitable for this analysis, together with the more classic tools of Non Linear Dynamics. This study allowed us in addition to show from a theoretical point of view some new properties of nondifferentiable maps, in comparison with differentiable ones. After a short review of noninvertible maps, the presented methods and tools for noninvertible maps are applied to the DPCM system. An original algorithm for calculation of bifurcation curves for the DPCM map is proposed. Via the studies in the parameter and phase plane, different nonlinear phenomena such as the overlapping of bifurcation curves causing multistability, chaotic behaviour, or multiple basins with fractal boundary are pointed out. All observed phenomena show a very complex dynamical behaviour even in the constant input signal case, discussed here.
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