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

本项目旨在研究从统计力学的角度发展非平衡和（或）非热力学系统所表现出的相变概念的可能性。我们的研究有潜在的适用性，以调查合作现象观察到的系统组成的活性元素，如生物组织中的活细胞。为了进行系统而严密的分析，我们采用了具有平均场相互作用的无限多粒子系统的随机模型。系统由受外部噪声影响的无穷多个耦合非线性振子组成，用无穷多个耦合朗格万方程的集合来描述。注意到随机系统简化为非线性的Fokker-Planck方程（NFPE），能够显示与系统相变相关的分岔，我们充分利用NFPE来深入了解与分岔相关的系统的动力学特性。NFPE分为两类。第一类以势条件为特征，与热力学系统的情况非常相似。在这种情况下，与遍历分量相对应的NFPE的吸引子是不动点型的。第二类是指在没有势条件的情况下，允许吸引子为极限环型或混沌型。对于第一类情况，我们通过证明NFPE上的h定理，阐明了系统的渐近平衡点，并将h泛函作为Lyapunov泛函进行了非线性稳定性分析。利用波动耗散定理研究了波动的动力学行为，证明了在分岔点附近出现临界慢化现象，并得到了波动的功率谱和相关函数。对于第二种情况，我们得到了一种新的相变类型，即随着外部噪声功率的降低，系统发生hopf分岔，NFPE解给出的概率分布开始以一定频率旋转。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