Nonlinear dynamics approach to cooperative phenomena in active element systems and its application to the study of biological rhythms

本项目旨在从统计力学的角度研究发展非平衡和(或)非热力学系统所表现出的相变概念的可能性。我们的研究对研究生物组织中由活细胞等活性成分组成的系统中观察到的合作现象具有潜在的适用性。为了进行系统和严谨的分析，我们采用了平均场相互作用的无限多粒子系统的随机模型。系统由无限多个受外部噪声影响的耦合非线性振子组成，由无穷多个耦合朗之万方程组描述。注意到随机系统归结为能够表现出与系统相变相关的分叉的非线性福克-普朗克方程(NFPE)，我们充分利用NFPE来深入了解系统Associa…的动力学性质NFPE被分为两类。第一种类型以势条件为特征，与热力学系统的情况非常相似。在这种情况下，对应于遍历分量的NFPE的吸引子是固定点类型的。第二类是指在没有势能条件的情况下，允许吸引子是极限环型或混沌型。对于第一类情形，我们通过证明NFPE上的一个H-定理来说明系统平衡点的渐近逼近，并以H-泛函为Lyapunov泛函进行了非线性稳定性分析。利用涨落耗散定理研究了涨落的动力学行为，揭示了在分叉点附近出现临界慢化现象，并得到了涨落的功率谱和关联函数。对于第二种情况，我们得到了一种新的相变类型，当外部噪声功率减小时，系统经历Hopf分叉，并且由NFPE的解给出的概率分布开始以一定的频率旋转。较少

The present project aims at studying the possibility of developing the concepts of phase transitions exhibited by nonequilibrium and (or) nonthermodynamic systems from the statistical mechanics point of view.Our research has a potential applicability to investigations of cooperative phenomena observed in systems consisting of active elements such as living cells in biological organizations. In order to make a systematic and rigorous analysis we have employed stochastic models of infinitely many particle systems with mean-field interaction. The systems consist of infinitely many coupled nonlinear oscilltors subjected to the influence of external noise and are described by sets of infinitely many coupled langevin equations. Noting that the stochastic systems reduce to nonlinear Fokker-Planck equations (NFPE) capable of exhibiting bifurcations associated with phase transitions of the systems,we have made full use of the NFPE to get insights into dynamical properties of the systems associa … More ted with the bifurcations.The NFPE are classified into two categories. The first type is characterized by the potential condition and has a close resemblance to the case of thermodynamic systems. In this case attractors of the NFPE corresponding to the ergodic components are of fixed point type. The second type refers to the case without potential condition in which attractors are allowed to be of limit-cycle or chaos type. For the first type case,we have elucidated asymptotic approach to equilibrium of the system by proving an H-theorem on the NFPE and conducted nonlinear stability analysis with the H-functional being taken as a Lyapunov functional. Dynamical behavior of fluctuations has also been investigated,by employing Fluctuation-Dissipation theorem,to show the appearance of critical slowing down in the vicinity of the bifurcation points as well as to obtain power spectra and correlation functions of the fluctuations. With regard to the second case we obtained a new type of phase transition,in which as external noise power is decreased the system undergoes Hopf-bifurcation and the probability distribution given by the solution to the NFPE begins to rotate with a certain frequency. Less