'Rule-dynamics' is a framework for simple models of rule-changing systems using cellular automata (CA) rules. We realize the transition process from CA to rule-dynamics by adjusting one parameter. Through this process, we show that a correlation between the rule and the density is generated, and also show that this correlation increases although the entropies of the density and the rule increase respectively. This suggests that we can extract the density information from the local rule description.

Rule-dynamics' is a framework for simple models of rule-changing systems using cellular automata (CA) rules. It is known that the mechanism of some cluster formation can be explained in terms of the spatio-temporal action of only two local rules. Among the various self-organizing patterns which are induced by the relationships between the local rules, 'switching phenomena' are especially interesting. The pattern of one cluster can be exchanged with the pattern of another cluster due to boundary interactions between nearest neighboring clusters. That is, a non-local effect can be generated instantaneously beyond the direct interaction range assumed in the original system. The significance of the switching phenomena is briefly discussed in relation to information transfer in biological systems.

We study computer simulations of two financial market models, the second a simplified model of the first. The first is a model of the self-organized formation and breakup of crowds of traders, motivated by the dynamics of competitive evolving systems which shows interesting self-organized critical (SOC)-type behaviour without any fine tuning of control parameters. This SOC-type avalanching and stasis appear as realistic volatility clustering in the price returns time series. The market becomes highly ordered at 'crashes' but gradually loses this order through randomization during the intervening stasis periods. The second model is a model of stocks interacting through a competitive evolutionary dynamic in a common stock exchange. This model shows a self-organized 'market-confidence'. When this is high the market is stable but when it gets low the market may become highly volatile. Volatile bursts rapidly increase the market confidence again. This model shows a phase transition as temperature parameter is varied. The price returns time series in the transition region is very realistic power-law truncated Levy distribution with clustered volatility and volatility superdiffusion. This model also shows generally positive stock cross-correlations as is observed in real markets. This model may shed some light on why such phenomena are observed. (C) 2000 published by Elsevier Science B.V. All rights reserved.

Recent works on self-organised criticality (SOC) in pulse coupled relaxation oscillators have shown SOC to be related to frustrated attempts of the system to synchronise. Criticality has also been connected to punctuated equilibrium behaviour. We describe a model which shows a non-equilibrium phase transition and seems to link these two ideas. Near the transition, punctuated equilibrium behaviour is seen, with avalanches occurring on all scales. This scaling is described by an exponent very near 1. (C) 2000 Elsevier Science Ltd. All rights reserved.

A two-dimensional earthquake model which has two significant features is studied. In this model, the nucleation length is proportional to the final size of the earthquake and fault strength healing occurs at an exponential rate. The time evolution of the spatial average of the crust strength shows f(-v) fluctuations when the characteristic healing time tau is longer than a critical value tau(c). Furthermore we obtain a relation which is independent of the model parameters. Our results indicate that crust rigidity can be worked out from earthquake data statistics. (C) 2000 Elsevier Science Ltd. All rights reserved.

Clustering motions are typical and universal phenomena in N-body systems. Basic mechanisms leading to escaping and/or to trapping of particles are pursued in the analysis of a global structure for the three-body problem. The global structure of the three-body problem is numerically studied under the short range Gaussian interaction potential. As the Gaussian potential does not have any singularities at zero distance, we can avoid the computational errors in the long lime simulations. Main concerns are the analysis of the collinear three-body problem, and the result compared with the case of gravitational potential. The distributions of periodic orbits are precisely searched and their stability is determined by the linear stability analysis. The collapsing of quasi-periodic motions is correlated to the destabilization of the three-body cluster in the case of the free-fall motions, and that the boundary for the collapsing tori displays fractal curves. Finally the escape diagram for two-dimensional three-body problems are discussed in comparison with the case of gravitational potential, where the remarkable difference near the triple collision is pointed out. (C) 1999 Elsevier Science Ltd. All rights reserved.

Non-stationary chaos is a universal phenomenon in non-hyperbolic dynamical systems. Basic problems regarding the non-stationarity are discussed from ergodic-theoretical viewpoints. By use of a simple system, it is shown that "the law of large number" as well as "the law of small number" break down in the non-stationary regime. The non-stationarity in dynamical systems proposes a crucial problem underlying in the transitional region between chance and necessity, where non-observable processes behind reality interplay with observable ones. The incompleteness of statistical ensembles is discussed from the Karamata's theory. Finally, the significance of the stationary/non-stationary interface is emphasized in relation to the universality of 1/f fluctuations. (C) 1999 Elsevier Science Ltd. Ail rights reserved.

　アクトミオシン系における張力の発生起源は、それぞれのフィラメント上の活性部位およびミオシンヘッドによるＡＴＰの加水分解作用によって生じる化学エネルギーから力学エネルギーへの変換機構にある。　本年度の研究では、不応期と化学反応に起因する加速度、さらに粘性の効果を取り入れた力学モデルによってミオシンヘッドの運動の多様性を解析した。不応期の長さはもっとも敏感に運動の特性に影響を与えるパラメータとなることを確認し、精密な分岐構造を明らかにした。現在までに得られた結果を整理すると、(i) 　運動様式には周期振動とカオス的振動がある。(ii) 　カオスの発生ルートは、2n分岐、フィボナッチ分岐、サドルノード分岐を、不応期のそれぞれの領域で示す。(iii) 化学エネルギーから力学エネルギーへの変換効率は、周期的運動とカオス的運動では著しく異なる。本研究で精密な分岐構造が把握できたことにより、アクチン、ミオシンの両フィラメントの多体問題のモデル化に一定の指針を得ることができた。現在、すべり運動の理論モデルの作成に取りかかっている。

本研究では、ハミルトン力学系の相空間の微細構造を詳細に決定し、それらが輸送現象やクラスター化に与える影響を明らかにすることを目指している。ハミルトン系の摂動に対する相空間構造の変化は、コルモゴロフ-アーノルド-モーザー(KAM)の理論を基盤に研究が進められ、相空間が低次元の場合については、大域的なカオスへの遷移機構が詳しく調べられている。しかし、実際の物理系との対応を考える場合、相空間が高次元であったり、KAM条件を満たさない相空間構造が現れることがしばしばあり、そのような系のカオスへの遷移機構は、既存の理論では十分には説明出来ていない。そこで我々は、相空間が高次元であることによる効果や、KAM条件の破れが系の大域的性質に及ぼす影響を調べ、次のような成果を得た。　1. 4次元シンプレクティック写像において、KAMトーラス崩壊のメカニズムを、 摂動論的手法を用いて調べた。自由度 2の系では、KAMトーラスの摂動に対する強さは、回転数の数論的性質のみで決定されると考えられているが、多自由度系では、加える摂動によって より強いKAMトーラスは異なることが、摂動級数に現れる小さな分母に関する考察から明らかとなった。　2. KAM条件を満たさないノン・ツイスト系において、KAM条件の破れが作り出す複雑な相空間構造を詳しく調べた。KAM条件を破るトーラスは、摂動に対して非常に頑丈であることが分かった。また、ノン・ツイスト系では、摂動の加え方の僅かな違いによって、大域的なカオスへと遷移する臨界パラメータ値が、大きく変わってしまうことが分かった。