Multiple Attractors and Itinerant Dynamics in High-dimensional Chaos

(1)对于全局耦合映射的多吸引子状态，通过扰动后对自身的回归率研究了吸引子的强度。发现脆性（Milnor）吸引子在部分有序相具有较大的盆体积。脆弱吸引子的这种优势可以通过相空间中全局吸引的鲁棒性来理解。我们将混沌流动重新解释为米尔诺吸引子网络上的动力学，发现噪声诱导的脆弱吸引子选择与噪声幅值有很大的依赖性。(2)研究了全局耦合映射的集体动力学。首先，我们用自洽Perron-Frobenius方程阐明了集体运动的机理。其次，通过引入集体Lyapunov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，发现随着外部噪声的变化，集体运动的自由度以logsiguma^2的形式减小。我们展示了微观外部噪声如何抑制宏观水平上的自由度数。(3)通过对具有内部动力学和繁殖的相互作用单元的发育过程的数值研究，提出了生物组织的情景。多样化，离散和递归类型的形成，以及微分规则是这样一个系统的自然结果。规则是通过周围单位的内部表征形成的，既依赖于内部状态，也依赖于相互作用。发展过程的宏观稳健性被证明是这种系统的自然结果。通过引入空间局部的机械化学相互作用，展示了具有生命史的多细胞生物的出现。我们还表明，某些噪声动力系统表现出阈值型输入依赖，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在哈密顿系统中储存能量的动力系统机制。(4)研究了生殖单元起源的自动机模型、动力系统的不确定性、生成符号和规则的功能动力学。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