张雪娟 

 

 

论文题目：双稳态系统、单稳态系统、耦合振子系和混沌系统的随机共振现象  

 

作者简介：张雪娟 ，女，1972年07月出生，1999年09月师从于北京大学钱敏教授，于2002年07月获博士学位。

                                       

                

摘  要

　

在非线性科学领域，一个重要的课题是研究确定性系统在随机力（即噪声）扰动下的动力学行为。在传统的观念上，噪声总是被认为起破坏作用的。但是近二十年来非线性理论和实验揭示，对许多非线性系统来说，适当的噪声干扰反而可以帮助系统提高信号输出的能力。这类现象称为随机共振。基于它在物理、化学、通讯以及工程等领域广阔的应用前景，近年来，随机共振引起了学术界的极大兴趣。有关随机共振的发生条件及发生机制正在进行广泛而深入的讨论。

本文从数值模拟和理论分析两方面研究双稳态系统、单稳态系统、耦合振子系以及混沌系统的随机共振现象及相应的动力学机制。

 

第一部分是对随机共振现象的数值模拟：

 

第一章研究的是双稳态周期驱动系统

,             　　  　      (1)

其中是Gauss白噪声。这是研究随机共振的典型模型之一。大量文献探讨了这一模型的随机共振机制，但文献中通常认为随机共振只发生在周期驱动强度为次阈值的范围（）内，而对过阈值周期驱动的情形（）则大都认为系统不再具有双稳态，从而随机共振不可能发生。本文通过考虑与原周期驱动系统等价的柱面上仅由噪声驱动的高维系统的动力学行为，发现即使对过阈值的周期驱动，系统仍有随机共振发生，并且其机制本质上和次阈值驱动时的随机共振机制是一样的。这一结果大大地拓宽了双稳态系统的随机共振发生区域，并且我们从一新的观点解释了随机共振的发生机制。事实上，不仅对，而且对，柱面上的确定性系统都有双稳态。但这时的双稳态已不是过去文献中所说的势函数的两极小值点，而是柱面上的两个稳定极限环。稳定极限环与不定极限环的相对位置对随机共振的发生及其效果起着关键作用。本部分的相关结果发表在Phys. Rev. E 65，011101(2001)上。

 

第二章研究的是圆周上无周期驱动的一阶单稳态系统

,　　　　　　　　　         (2)

和无周期驱动的局部耦合振子系

,   (3)

的随机共振现象。对系统（2），当控制参数时，系统有自随机共振发生，而当时系统无随机共振发生的结论，并从其确定性系统的相曲线出发对自随机共振的发生机制给予了一个清晰的阐述。进一步地，我们还讨论了功率谱尖峰频率和系统旋转数的关系，分析了两者并不吻合的原因。有关本部分的结果已发表在Phys. Rev. E 62, 6469(2000)上。对系统（3），i). 我们发现其随机共振的效果比单振子系（2）要好得多，体现在功率谱上，每个振子的功率谱谱峰高度比单振子系的要高得多，而且功率谱宽度变窄；ii). 尽管耦合系统没有受到外界的周期驱动，但由于每个振子的coherent输出通过耦合项的作用充当了它邻近振子的周期输入，我们认为系统出现的随机共振现象也可看成是由噪声和周期力协同作用而导致的，从而比单振子系时的效果要好得多；iii). 每个振子的功率谱尖峰频率相同，旋转数也相同，并且旋转数和尖峰频率在很大的噪声范围内吻合得很好。这些结果已在Phys. Rev. E 65,031110(2002)上发表。

 

在第二章的基础上，我们在第三章进一步讨论了系统（2）和（3）引入周期驱动后的随机共振现象。对有周期驱动的单振子系统，我们发现在一定的参数范围内，系统除了发生传统意义上的由噪声和周期力共同协作而导致的随机共振现象，还保留着无周期驱动时由噪声诱导的自随机共振现象。这两种随机共振在有耦合项存在时效果得到大大的加强。另外，i). 文献中通常认为，确定性系统有一定的能量阈值是随机共振发生的必要条件之一。但对有周期驱动的耦合系统来说，我们发现即使确定性系统没有稳态（能量阈值），在一定的周期驱动和噪声扰动下，系统也会有随机共振发生（称为无稳态随机共振）；ii). 当外部驱动频率和系统无周期驱动时的功率谱尖峰频率吻合时，系统发生真正意义上的共振现象（称为频率随机共振），并且有耦合时的共振效果比没有耦合时的要强得多。此结果已在Chin. Phys. Lett. 20, 202(2003)上发表；iii). 对有周期驱动的耦合系统来说，随机共振的发生与否可用旋转数与驱动频率的吻合与否来判断。比起文献中不论是计算平均功率谱和信噪比，还是用计算驻留时间的方法，这种用旋转数的方法可大大地节省计算量。

 

第四章首先讨论的是无周期驱动的二阶阻尼单摆方程(Josephson Junction方程)

　                     (4)

的情况。对这一系统，我们首先证明了确定性系统一维全局吸引子的存在性，并根据全局吸引子的类型将系统参数分为节点区、焦点区和极限环区。然后分别考察了这三个参数区域中的自随机共振的发生情况。我们发现在结点区系统发生势井间的随机共振，在焦点区发生势井内的随机共振，而在极限环区则没有随机共振现象发生。此结果已发表在J. Phys. A：math & Gen. 34,10859(2001)上。进一步地，我们还初步研究了系统（4）引入周期驱动后的动力学行为，发现在有些混沌区，系统也有随机共振现象发生。

 

第二部分是与随机共振有关的一些数学结果的证明：

 

在第五章，我们对第一章的双稳态周期驱动系统定性地分析了与其等价的柱面上自治的确定性系统的动力学行为。对次阈值周期驱动情形，我们利用Poincare-Bendixon定理，证明了对所有的驱动频率，系统在柱面上有且只有两个稳定极限环。对过阈值周期驱动的情形，我们利用张驰振荡的研究方法，证明了当时，极限环的存在唯一性；又利用分支问题的Liapunov第二方法证明了当时系统在柱面上有且仅有两个稳定环。这些结果已被Physica A所录用。

 

第六章是对第二章和第三章数值模拟发现的耦合振子系锁频现象(即各振子旋转数存在且相等)的理论证明。对无周期驱动的情况，前人通过将其对应的随机微分方程等价地看成一柱面上的随机过程，利用遍历定理，证明了在噪声扰动下，各振子的旋转数存在且相等。但对有周期驱动的情形，柱面上的随机过程已不再象无周期驱动时那样具有平稳性和遍历性。我们注意到如果考虑时间间隔为驱动周期的子随机过程，则它为具有唯一不变分布的渐近平稳的马氏序列，并且对应于一具有遍历性的平稳序列。这样，我们所关心的随机过程就可看成一系列具有不同初始时间和不同不变分布的渐近平稳序序按时间重新排序后所得到的过程。利用这一点，我们给出了耦合振子系在噪声和周期力共同作用下旋转数的理论与a.e.收敛公式，为第三章的数值模拟提供了理论依据。

 

第七章对有限状态的可逆Q过程，我们给出了Green-Kubo公式的表达式，得到输运系数等于条件扩散系数的期望的结论，并且输运系数可以用Q矩阵的个特征值来表达，并证明了可逆马氏过程的功率谱为Lorentz型的，从而不可能发生随机共振现象。此结果已在Phys. Lett. A 309，371－376（2003）上发表。

 

近来，在博士论文的基础上，我们进一步展开了噪声在诱导棘轮系统作定向运动(即分子马达)方面的工作。i). 在我们发表在Chin. Phys. Lett. 20，810（2003） 的一篇论文中，讨论了分子马达与随机共振之间的关系，发现对一些模型，二者之间存在很好的对应关系。我们指出，这二种现象一致性的根本原因是系统存在非平衡态环流。ii). 在分子马达的研究中，平均流是刻画噪声驱动系统定向运动的一个重要的量。但由于随机微分方程的轨道关于时间的导数往往是不存在的，文献中关于实际上是一个物理含义明确但数学上并没有严格定义的量。为进一步理论研究和数值模拟的需要，我们从向前条件速度出发，给出了平均流的严格数学定义。进而证明了平均流等于时间平均速度又等于概率流的这一有关分子马达的基本表达式。

 

　

 

Abstract

 

Making the dynamics of a system under the fluctuation of noise clear is one of the important subjects in nonlinear science. The influence of noise on a system used to be thought destructive. However, in 1980s, scientists discovered that, in many nonlinear systems, a certain amount of noise can result in the output of the system undergoing a resonance-like behavior as a function of the noise strength. Such a counterintuitive phenomenon was termed as stochastic resonance (SR). Since then, SR has aroused continuous interest due to its promising applications in many fields of natural sciences. For nearly two decades, the necessary conditions for SR to occur and the corresponding mechanism of SR have been extensively studied.

 

In this thesis, the phenomena of SR in different systems is investigated, which includes bistable systems, monostable systems, coupled systems with and without periodic driving as well as chaotic systems. The thesis is composed of two parts:

I.                   Physical description of SR observed in numerical simulations.

II.                Mathematical proof of some results displayed in the numerical simulations.

 

Part I (Chapter 1-4): numerical results about SR

 

Chapter 1 is used to discuss SR for the typical bistable system

         ,      　　    　          (1)

where  is the Gaussian white noise. In subthreshold regime, SR has been well explored by others with different techniques. For suprathreshold case, SR had been deemed as impossible. In this thesis, following the idea of embedding the non-autonomous equation in an autonomous system, we show the existence of SR in suprathreshold regime (as well as in subthreshold regime) via numerical simulation. The main point is that there exist two stable limit cycles of the deterministic system. It is found that the mechanisms of SR for both subthreshold and suprathreshold cases are essentially the same. The reason is that the system in these two parameter regimes has similar dynamical properties. It is shown that the onset of SR crucially depends on the relative positions of the stable and unstable limit cycles. The result was published in Phys. Rev. E 65, 011101(2001).

 

Chapter 2 offers results of SR for a monostable system and a locally coupled system.

       ,　　　　　　  　              (2)

,  .  (3)

Both are without periodic driving. For system (2), it is shown that in the monostable and semistable (and ) cases, coherent oscillations occur and autonomous SR exists. In the case , noise plays only a destructive role and therefore no SR occurs. We clarify the mechanism of autonomous SR via qualitatively analyzing the phase curves of the deterministic system. We also study the relationship between the peak frequency of the power spectrum and the rotation number of the system. The surprise is that these two quantities do not agree with each other in noisy case. The SR result of system (1) was published in Phys. Rev. E 62, 6469(2000). As for the coupled system (3), numerical results show the array-enhanced SR. Although there is no external periodic input, the coherent output of every oscillator plays a role of periodic input to its neighbors, so there is still a bona fide SR. The rotation number and the peak frequency are shown to agree with each other very well in a wide range of noise intensity. This fact is important for the discussion of SR in the periodic driving case. The results of system (3) were published in Phys. Rev. E 65, 031110(2002).

 

Based on Chapter 2, in Chapter 3, we further introduce periodic driving in systems (2) and (3) and explore the SR phenomena. For the single oscillator case, an intriguing phenomenon was found. In some parameter regimes, besides the conventional SR resulting from the interplay of noise and the periodic signal, there exists autonomous SR induced mainly by noise itself. The effects of these two SR phenomena are shown to be array-enhanced in the coupled case. Moreover, we find three special phenomena in the coupled system:

 i). Even though the deterministic system has no energy threshold, SR in the coupled system can also happen. Such a phenomenon used to be deemed as impossible if the deterministic system has no energy threshold.

ii). When the frequency of an external driving equals to the noise- induced peak frequency of the power spectrum without periodic forcing, real physical resonance happens, which is called frequency SR. This work was published in Chin. Phys. Lett. 20, 202 (2003).

iii). The happening of SR in the periodically driven case can be simply and effectively judged by the coincidence of the rotation number with the frequency of external periodic driving.

 

In Chapter 4, we give a systematic investigation of SR in the following under-damped single pendulum system without periodic driving

.              　　(4)

The one-dimensional global attractor of the deterministic system is demonstrated. Then the  parameter regime is compartmentalized into three regions according to the different types of stable equilibrium point on the global attractor. When the global attractor is formed by a stable node, a saddle point and the heteroclinic orbits between them, interwell SR is induced by noise. When the stable node on the attractor curve is replaced by a stable focus, intrawell SR occurs. As the global attractor becomes into a stable limit cycle, no SR happens. This work was published in J. Phys. A: math. & Gen. 34, 10859 (2001). Furthermore, the dynamics of system (4) with periodic driving is considered. The phenomena of SR in some chaotic regimes are also numerically manifested.

 

Part II (Chapter 5-7): mathematical theorems related to the occurrence of SR

 

In Chapter 5, we qualitatively analyze the dynamics of the bistable system (1) in deterministic case. Two stable limit cycles in subthreshold case is proved existent. In suprathreshold case, for sufficiently small signal frequency (), the dynamics of the system is just a relaxation oscillation on a cylinder, which means that the system has only one stable limit cycle. As  increases to a certain value, the system bifurcates two stable limit cycles again. This work has been accepted for publication in Physica A.

 

In Chapter 6, the frequency locking phenomenon of the locally coupled systems is proved. For the case of noise fluctuation alone, that every oscillator has the same rotation number has been proved in a.e. sense. The key point is to wind the solution of the system on a cylinder along a special direction. The process is thus ergodic and has a unique invariant measure on the cylinder. However, when periodic driving is included, the system winded on the cylinder is time-inhomogeneous, and the process has no longer invariant measure and the ergodicity. Consider the sub-process which takes place on the time sequence (, is the period of driving), which is time- homogenous and is asymptotically stationary with a unique invariant distribution. Thus the diffusion process on the cylinder can be regarded as a composition of these asymptotically stationary sequences sorted by time. The existence of the rotation number of the system is proved in  sense, and its a.e. formula according to a certain limit order is given.

 

In Chapter 7, the eigenvalues of the so-called Hamilton quantum formalism of a reversible master equation is endowed with an explicit probability implication by deducing the Green-Kubo formula of a reversible - process with finite states. Furthermore, the Lorentzian power spectrum of the output demonstrates that SR is impossible to occur in reversible Markvian systems. This work was published in Phys. Lett. A 309, 371-376 (2003).

 

Recently, based on the work of the doctorial thesis, we further investigate the constructive role of noise on inducing undirectional transport in Brownian ratchets:

 i). In one of our published papers (Chin.Phys.Lett.20,810(2003)), we discussed the relationship between molecular motor and stochastic resonance. It is found that for some ratchet systems, these two phenomena are closely related with each other. Such a consistency is just due to the existence of circular flux in non-equilibrium.

ii). In the research of molecular motor, an important quantity is the mean mobility . Though its physical concept is clear and its relation with the time-averaged velocity  as well as the probability current has been pointed out in the literature, the definition of  remains to be clarified. This is because the derivation of  which usually corresponds to a stochastic differential equation actually does not exist. For the further theoretical studies in ratchet effect, it is of essential importance to give a specific definition of the mean mobility . According to a properly defined conditional forward velocity (CFV), we give a rigorous definition of the mean mobility  as the expectation of the CFV. Based on this, we mathematically demonstrate the equivalence between the mean mobility, the time-averaged velocity and the probability current in ratchet systems.

 

 

 

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