分段连续系统的两种特征行为：V型阵发前奏锁相阶梯及半耗散性

【作者】 巢小刚；

【导师】 何大韧；

【作者基本信息】 扬州大学 ， 凝聚态物理， 2003， 硕士

【摘要】 本论文报道对一些分段光滑经典系统的研究，涉及两个分段光滑经典系统：一个带模拟开关的RLC电路模型和一个带耗散性元件的过电压保护电路模型。 在带模拟开关的RLC电路模型中发现V型阵发成为主要的从周期运动向混沌运动过渡的形式，这种阵发类型只能在分段光滑耗散系统中发生。除了V型阵发的已知特征，如由边界碰撞分岔导致周期轨道失稳，以及具有对数函数形式的平均层流相长度标度律等等之外，在这个系统中发现的一个V型阵发新特征是非传统魔梯形式的V型阵发前奏锁相阶梯。在这类由不连续映象描述的系统中，一般一个周期轨道经由V型阵发失稳之后必须经过一系列过渡高周期轨道才能过渡到混沌运动。描述这些过渡高周期轨道的特征量(如本文定义的穿越数)在这个特征量和控制参量构成的平面上形成一系列锁相台阶，所有的台阶构成所谓的V型阵发前奏阶梯。过去发现的V型阵发前奏阶梯都具有传统魔梯的形式。这是第一次在实际模型中发现非传统魔梯形式的V型阵发前奏锁相阶梯。为了验证这一数值结果的正确性，我们解析求解了穿越数为1/n的锁相台阶的参数位置，发现和数值结果符合得很好。我们的数值结果还说明在V型阵发前奏锁相阶梯之后出现的混沌吸引子就是映象函数的不连续边界的象集的归宿。这很可能是这个系统中V型阵发前奏锁相阶梯后的混沌吸引子的共同特征。 带耗散性元件的过电压保护电路模型由一个保守映象和一个耗散映象耦合而成的分段光滑映象描述，我们称该系统为“半耗散系统”。该分段连续系统既能展示不连续性导致的不可逆性以及由此而来的类耗散性，也能展示耗散映象所带来的传统耗散性，以及二者混合导致的“混合耗散性”。系统完全由传统耗散性或类耗散性主宰的参数、相空间范围和现象是局限的，然而这种局限行为往往对系统的特点有重要影响。系统大范围、长时间的行为经常由系统的混合耗散性支配。这是半耗散系统区别于传统耗散系统与类耗散系统的一个基本特征。该系统区别于类耗散系统的另一个特征在于：与类耗散系统向平面上的迭代禁区相比，该系扬州大学硕士学位论文统的禁区中的一部分是由耗散性导致的，我们称这部分禁区为“耗散性导致禁区”。这部分禁区对该系统中被称为“半耗散激变”特征现象中的逃逸映孔的定义起了关键作用。这种半耗散激变具有两个鲜明的特征:①.这种激变的机制是一个混沌类吸引子中突然出现一个具有混合耗散性的周期轨道，从而使原混沌类吸引子突然转变为一个混沌类瞬态;②.这种激变的逃逸孔洞是一个受到耗散性导致禁区的边界限制的“完全传统耗散性主宰区”。据我们所知，具有这两种特征的激变在国内外都还没有报导过。更多还原

【Abstract】 This thesis reports a study on the characteristics of some piecewise-smooth classical systems. The systems studied are a model of an RLC circuit with an analogy switch and a model of an electronic circuit with over-voltage protection and a dissipative element.It is discovered in the RLC circuit model that type V intermittency becomes the main route of the transition from periodic motion to chaos. This type of intermittency can happen only in piecewise-smooth dissipative systems. In addition to the known characteristics of type V intermittency, such as the mechanism of the border-collision bifurcation and the logarithmic scaling behavior of the averaged laminar lengths, a new characteristic of type V iritermittency discovered in this system is the so-called "prelude phase-locking staircase to type V intermittency", which does not show the traditional devil’s staircase form. In discontinuous maps, in general a periodic orbit, after losing its stability via a type V intermittency, can transmit to chaos only after a sequence of higher period orbits. The characteristic quantity (like the "transfer number" defined in this thesis) that describes the higher period orbits forms a sequence of phase-locked steps on the plane formed by it and the control parameter. All the phase-locked steps form a so-called prelude phase-locking staircase to type V intermittency. All such staircases discovered before have the form of the traditional devil’s staircase. This is the first time to observe a prelude phase-locking staircase to type V intermittency with other forms. In order to verify these numerical results, we analytically obtained the end positions of the phase-locked steps those have the transfer number 1/n and found a good agreement with the numerical results. Our numerical investigation also indicates that the chaotic attractor appeared after the prelude phase-locking staircase was end-result of the set of the images of the discontinuous border of the system function. This seems to be a common feature of the chaotic attractors those appear after the prelude phase-locking staircase in the system.The model of electronic circuit is described by a piecewise-continuous concatenation of a dissipative map and a conservative map, which is addressed as a "semi-dissipative system". The system can show the so-called quasi-dissipative properties caused by the noninvertibility that is induced by the discontinuity of the mapping, the traditional dissipative properties caused by the dissipative sub-map, as well as the so-called "mixed-dissipative property" caused by the mixture of them. Usually the area in parameter space, in phase-space and in the types of phenomena where traditional dissipative properties or quasi-dissipative properties dominate is confined, but the confined behavior often shows a great influence on the characteristics of the system. The system’s behavior usually is determined by the mixed dissipative properties in a long time or large ranges. This is one of the basic features of semi-dissipative systems. Another characteristic different from quasi-dissipative systems concerns the iteration gap. There is a piece of gap in the phase plane that is induced by dissipative property. It confines the "escaping hole" in a so-called "semi-dissipative crisis", which owns two particular properties: 1. The mechanism of the crisis is the sudden appearance of a periodic orbit with mixed-dissipative property inside a chaotic quasi-attractor; 2. The escaping hole can be defined as the area completely dominated by traditional dissipative property that is confined by the dissipative-induced gap. To our knowledge, such a crisis has never been observed yet.更多还原