Multiple Attractors and Itinerant Dynamics in High-dimensional Chaos

(1)对于全局耦合映象的多吸引子状态，用扰动后的返回率来研究吸引子的强度。研究发现，脆性(Milnor)吸引子在偏序相处具有较大的盆体积。脆弱吸引子的这种优势可以通过相空间中全局吸引子的鲁棒性来理解。我们将混沌巡游重新解释为Milnor吸引子网络上的动力学，发现噪声诱导的脆弱吸引子的选择强烈依赖于噪声幅度。(2)研究了全局耦合映射的集体动力学。首先，我们用自洽的Perron-Frobenius方程阐明了胶体运动的机制。其次，通过引入集体李亚普诺夫指数，证明了集体混沌的存在性。第三，研究了微观外部噪声对集体运动的影响，发现随着外部…的增大，集体运动中的自由度数随着-logsiguma^2的增大而减小说明了微观外部噪声如何在宏观层面上抑制自由度数。(3)基于内部动力学和繁殖相互作用单元的发展过程的数值研究，提出了一种生物组织方案。多样化、离散和递归类型的形成以及差异化规则被发现是这种系统的自然结果。规则是通过周围单位的内部表征形成的，既取决于内部状态，也取决于相互作用。发展过程的宏观稳健性被证明是这样一个系统的自然结果。通过引入空间定域力-化学相互作用，证明了具有生活史的多细胞有机体的出现，并证明了某一噪声动力系统具有阈值输入依赖性，这可能与信号转导有关。在此基础上，提出了一种利用混沌和受限热浴来存储能量的哈密顿系统的动力系统机制。(4)我们还研究了繁殖单位起源的瓦片自动机模型、动力系统的不可判断性、生成符号的函数动力学和规则。较少

(1) Strength of attractor is studied by the return rate to itself after perturbations, for a multi-attractor state of a globally coupled map. It is found that fragile (Milnor) attractors have a large basin volume at the partially ordered phase. Such dominance of fragile attractors is understood by robustness of global attraction in the phase space. We reinterpret chaotic itinerancy as the dynamics over Milnor attractor network, Noise-induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude.(2) Collective dynamics of globally coupled maps is studied. First, we have clarified the mechanism of collevtie motion in terms of self-consistent Perron-Frobenius equation. Second, existence of collective chaos is shown by introducing the collective Lyapunov exponent. Third, effect of microscopic external noise on the collective motion is studied, to find that the number of degrees of freedom in the collective motion decreases as-logsiguma^2 with the exter … More nal noise variance siguma^2. We show how the microscopic external noise can suppress the number of degrees of freedom in macroscopic level.(3) A scenario for biological organization is proposed, based on numerical studies of the developmental process of interacting units with internal dynamics and reproduction. Diversification, formation of discrete and recursive types, and rules for differentiation are found as a natural consequence of such a system. The rule is formed through the internal representation of the surrounding units, and depends both on internal state and on interaction. The macroscopic robustness of the developmental process is shown to be a natural consequence of such a system. By introducing spatially localized mechano-chemical interactions, the emergence of a multi-cellular organism with a life history is demonstrated.We have also shown that a certain noisy dynamical system shows threshold-type inpute dependence, in possible connection with signal transduction. Fur thermore, a dynamical systems mechanism to store energy in a Hamiltonian system is proposed with the use of chaos and restricted heat-bath.(4) We have also studied tile automaton model for the origin of reproduction unit, undecidability in terms of dynamical systems, functional dynamics to generate symbols and rules. Less