隨機藕合系統的動態研究

Abstract

這個三年的計畫，我們將研究(隨機的藕合)系統，主要的研究課題為(i) 同步的流形上的動態和 (ii)此系統的同步和非同步現象。隨機藕合系統的 動態是定義成系統的狀態X (k)和X t (k)相乘的期望值E(X (k)X t (k))；也即 autocorrelation matrix。 第一年我們將研究一個具體的例子但不是隨機藕合，這例子是 Stake and Iwasa [2,4] and Isagi [1]所提的森林的成長模型，其樹和樹的藕合是非 線性且不連續。 第二年我們將研究同一模型，但藕合方式是隨機的。 第三年我們將研究一個抽象且一般的模型。此模型可以含上述森林模 型和在[3, 5]中所討論的模型。預期這模型的理論獲得可以在物理、工程、 腦神經、生物和生態學中找得應用。

In this project, we are to study the dynamics of (stochastically) coupled maps. The dynamics of such coupled networks is defined in terms of their auto correlation matrix. Among other things, we are to investigate (i) the dynamics of synchronous manifold, and (ii) synchronization and desynchronization of the coupled networks. In the first year, we are to study a deterministic tree models proposed by Satake and Iwasa [2,4] and Isagi [1]. The coupling for the forest is nonlinear and discontinuous. For the secong year, we are to study the same model with the coupling matrix M(k) ' s being independent and identically distributed matrices with common random variable M . Finally, we are to formulate abstract networks that are stochastically coupled. Such formulation encompassed those tree models as well as abstract models discussed in [3, 5]. It is expected that our formulation may find applications in physics, engineering, neuron science, biology as well as ecology.