Sensitive boundary condition dependence (BCD) is reported in a convectively unstable system with noise, where the amplitude of generated oscillatory dynamics in the downstream depends sensitively on the boundary value. This BCD is explained in terms of the manner in which the co-moving Lyapunov exponent (characterizing the convective instability) decreases from upstream to downstream. It is shown that a fractal BCD appears if the dynamics that represent the spatial change of the fixed point includes transient chaotic dynamics. By considering as an example a one-way-coupled map lattice, this theory for BCD is demonstrated.
To understand the mechanism allowing for long-term storage of excess energy in proteins, we study a Hamiltonian system consisting of several coupled pendula in partial contact with a heat bath. It is found that energy absorption and storage are possible when the motion of each pendulum switches between oscillatory (vibrational) and rotational modes. The relevance of our mechanism to protein motors is discussed.
We consider the effect of microscopic external noise on the collective motion of a globally coupled map in fully desynchronized states. Without the external noise a macroscopic variable shows high-dimensional chaos distinguishable from random motions. With the increase of external noise intensity, the collective motion is successively simplified. The number of effective degrees of freedom in the collective motion is found to decrease as~$-\log{\sigma^2}$ with the external noise variance~$\sigma^2$. It is shown how the microscopic noise can suppress the number of degrees of freedom at a macroscopic level.
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. Finally the consequence of our intra-inter dynamics to evolution is discussed, which leads to the genetic fixation of interaction-induced phenotypic diversification.
To understand an architecture of a living system, ``Tile Automaton'' is introduced as an abstract model of chemical reaction of molecules scattered over a space. The model consists of tiles of various shapes which stand for molecules. The chemical reaction, induced by the collisions of tiles, is represented by the change of the tile shapes. Although the rules for reaction are deterministic, it is found that the evolution of the system strongly depends on mutual relationship among tiles. The evolution often leads to self-organization of a `factory', a set of tiles that produces tiles continuously and keeps its structure. Several functions to form such factory are extracted, while reproduction of the factory itself is also observed. It is proposed that the formation of the factory is due to the interference between different aspects of tiles -- the shape and the motion. A concept, ``tangling of contexts'' is introduced as a mechanism of living systems.
Classical model of helium atom is numerically investigated. by choosing several types of configurations. While most of orbits cause autoionization via chaotic transients, three types of tori are found, two of which are formed due to strong corrleation between two electrons. Chaotic motion, on the other hand, is always unbounded and leads to autoionization. Complex structure of phase space with tori and unstable autoionizing orbits is studied. One type of torus is quantized semiclassically with the EBK scheme, to estimate the energy level of the excited state.
Unidirectionally coupled dynamical system is studied by focusing on the input (or boundary) dependence. Due to convective instability, noise at an up-flow is spatially amplified to form an oscillation. The response, given by the down-flow dynamics, shows both analogue and digital changes, where the former is represented by oscillation frequency and the latter by different type of dynamics. The underlying universal mechanism for these changes is clarified by the spatial change of the co-moving Lyapunov exponent, with which the condition for the input dependence is formulated. The mechanism has a remarkable dependence on the noise strength, and works only within its medium range. Relevance of our mechanism to intra-cellular signal dynamics is discussed, by making our dynamics correspond to the auto-catalytic biochemical reaction for the chemical concentration, and the input to the external signal, and the noise to the concentration fluctuation of chemicals.
An algorithm to characterize collective motion is presented, with the introduction of ``collective Lyapunov exponent'', as the orbital instability at a macroscopic level. By applying the algorithm to a globally coupled map, existence of low-dimensional collective chaos is confirmed, where the scale of (high-dimensional) microscopic chaos is separated from the macroscopic motion, and the scale approaches zero in the thermodynamic limit.
Collective behavior is studied in globally coupled maps. Several coherent motions exist, even in fully desynchronized state. To characterize the collective behavior, we introduce scaling transformation of parameters, and detect in parameter space a tongue-like structure in which collective motions is possible. Such collective motion is supported by the separation of time scales, given by the self-consistent relationship between the collective motion and chaotic dynamics of each element. It is shown that the change of collective motion is related with the window structure of a single one-dimensional map. Formation and collapse of regular collective motion are understood as the internal bifurcation structure. Coexistence of multiple attractors with different collective behaviors is also found in fully desynchronized state.
Basic problems in complex systems are surveyed in connection with Life. As a key issue for complex systems, complementarity between syntax/rule/parts and semantics/behavior/whole is stressed. To address the issue, a constructive approach for a biological system is proposed. As a construction in a computer, intra-inter dynamics is presented for cell biology, where the following five general features are drawn from our model experiments; intrinsic diversification, recursive type formation, rule generation, formation of internal representation, and macroscopic robustness. Significance of the constructed logic to the biology of existing organisms is also discussed.
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. Change of the attractor strength and basin volume against the parameter and size are studied. In the partially ordered phase, the dynamics is often described as Milnor attractor network, which leads to a new interpretation of chaotic itinerancy. Noise-induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude. Relevance of the observed results to neural dynamics and cell differentiation is also discussed.
The origin of multicellular organism and mechanism of development in cell society are studied, by choosing a model with intracellular biochemical dynamics allowing for osicllations, cell-to-cell interaction through diffusive chemicals on a 2-dimensional plane, and state-dependent cell adhesion. Cells differentiate based on dynamical instability, following the isologous diversification theory. A fixed spatial pattern of differentiated cells appears, where spatial information is sustained by cell-to-cell interaction. This pattern has robustness against perturbations. With an adequate cell adhesion force, active cells are released, from which a new generation of multicellular organisms is created, accompanied by a death of the original multicellular unit as a halting state. It is shown that the emergence of multicellular organisms with the differentiation, regulation, and life-cycle is not a chance, but a necessity.
Basic problems for the construction of a scenario for the Life are discussed. To study the problems in terms of dynamical systems theory, a scheme of intra-inter dynamics is presented. It consists of internal dynamics of a unit, interaction among the units, and the dynamics to change the dynamics itself, for example by replication (and death) of units according to their internal states. Applying the dynamics to cell differentiation, isologous diversification theory is proposed. According to it, orbital instability leads to diversified cell behaviors first. At the next stage, several cell types are formed, first triggered by clustering of oscillations, and then as attracting states of internal dynamics stabilized by the cell-to-cell interaction. At the third stage, the differentiation is determined as a recursive state by cell division. At the last stage, hierarchical differentiation proceeds, with the emergence of stochastic rule for the differentiation to sub-groups, where regulation of the probability for the differentiation provides the diversity and stability of cell society. Relevance of the theory to cell biology is discussed.
The isologous diversification is proposed for cell differentiation, to claim that the amplification of noise-induced slight difference between cells leads to a noise-tolerant society with differentiated cell types. It is a general consequence of interacting cells with biochemical networks and cell divisions, as is confirmed by several model simulations. According to the theory, the differentiation proceeds as follows: (1) Up to some number, divisions bring about almost identical cells, where intracellular chemical oscillations are synchronized. (2)As the number exceeds some number, the oscillations lose the synchrony, and cells split into the clusters of different phases of oscillations. (3)Chemical compositions of cells start to differ from one cluster to another. (4) The differentiated compositions are transmitted by divisions to the next generation, leading to the determination of the differentiated cell types. (5) Further differentiation proceeds within some clusters, leading to a hierarchical cell society. These five stages are explained as an inevitable scenario of dynamical systems. As a testable consequence of the theory, we discuss the interaction-dependent tumor formation and negative correlation between growth speed and chemical diversity.
Geometric properties of accepted languages of Turing machines are numerically investigated by mapping those into two-dimensional space. The geometric representation of universal language, i.e. the accepted language o f universal Turing machine, has a different fine structure on an arbitrarily small scale, and is constructed non-uniformly and slower than ordinary fractals. The structure of the set has a fractal boundary dimension converging to the space dimension, which gives a geometric characterization of the undecidability of t he halting problem of Turing machines.
An abstract model for cell differentiation is studied, where cells with internal chemical reaction dynamics interact with each other and replicate. It leads to spontaneous differentiation of cells and determination, as is discussed in the isologous diversification. Following features of the differentiation are obtained: (1)Hierarchical differentiation based on dynamical instability and the emergence of the interaction-dependent rule of differentiation; (2)Global stability of an ensemble of cells consisting of several cell types, that is sustained by the autonomous control of the rate of differentiation; (3)Existence of several cell colonies with different cell-type distributions. The results provide a novel viewpoint on the origin of complex cell society, while relevance to some biological problems, especially to the hemopoietic system, is also discussed.
In a multi-attractor state of a globally coupled dynamical system, stability of the attractors is studied by recording the return rates to themselves after perturbations. Besides the basin volume, attractors are characterized by the strength, defined as the threshold perturbation for the full return rate. It is observed that Milnor attractors with a vanishing strength are dominant in the partially ordered phase. Attractions to some attractors are found to be enhanced with the addition of a noise, selectively for its amplitude.
A phenomenological model for cloud dynamics is proposed, which consists of the successive operations of the physical processes; buoyancy, diffusion, viscosity, adiabatic expansion, fall of a droplet by gravity, descent flow dragged by the falling droplet, and advection. Through extensive simulations, the phases corresponding to stratus, cumulus, stratocumulus and cumulonimbus are found, with the change of the ground temperature and the moisture of the air. They are characterized by order parameters such as the cluster number, perimeter-to-area ratio of a cloud, and Kolmogorov-Sinai entropy.
An extension of coupled maps is given which allows for the growth of the number of elements, and is inspired by the cell differentiation problem. The growth of elements is made possible first by clustering the phases, and then by differentiating roles. The former leads to the time sharing of resources, while the latter leads to the separation of roles for the growth. The mechanism of the differentiation of elements is studied. An extension to a model with several internal phase variables is given, which shows differentiation of internal states. The relevance of interacting dynamics with internal states ("intra-inter" dynamics) to biological problems is discussed with an emphasis on heterogeneity by clustering, macroscopic robustness by partial synchronization and recursivity with the selection of initial conditions and digitalization.
Isologous diversification theory for cell differentiation is proposed, based on simulations of interacting cells with biochemical networks and cell division process following consumption of some chemicals. According to the simulations of the interaction-based dynamical systems model, the following scenario of the cell differentiation is proposed.(1) Up to some threshold number, divisions bring about almost identical cells with synchronized biochemical oscillations. (2)As the number is increased the oscillations lose the synchrony, leading to groups of cells with different phases of oscillations. (3)Amplitudes of oscillation and averaged chemical compositions start to differ by groups of cells. he differentiated behavior of states is transmitted to daughter cells. (4)Recursivity is formed so that the daughter cells keep the identical chemical character. This "memory" is made possible through the transfer of initial conditions. (5) Successive differentiation proceeds. Mechanism of tumor cell formation, origin of stem cells, anomalous differentiation by transplantations, apoptosis and other features of cell differentiation process are also discussed, with some novel predictions.
Studies on coupled map lattice, originally introduced for the study of spatiotemporal chaos, are surveyed, with the emphasis on the suppression of chaos, spatiotemporal intermittency, supertransients, and stability of fully-developed spatiotemporal chaos. Extensions of coupled maps to the global coupling and hyper-cubic lattices are also discussed. In these studies we note the emergence of the dynamics of a higher-level, as is called chaotic itinerancy. The chaotic itinerancy supports the succession of several quasi-stable states. Based on these studieds, the notion ``homeochaos" is presented as a mechanism on the dynamic stability supporting diversity. Last, some speculations on the diversity and collective stability are given in connection with the evolutionary process.
Fractalization of torus and its transition to chaos in a quasi-periodically forced logistic map is re-investigated in relation with a strange nonchaotic attractor, with the aid of functional equation for the invariant curve. Existence of fractal torus in an interval in parameter space is confirmed by the length and the number of extrema of the torus attractor, as well as the Fourier mode analysis. Mechanisms of the onset of fractal torus and the transition to chaos are studied in connection with the intermittency.
Fluctuations of the mean field of a globally coupled dynamical systems are discussed. The origin of hidden coherence is related with the instability of the fixed point solution of the self-consistent Perron-Frobenius equation. Collective dynamics in globally coupled tent maps are re-examined, both with the help of direct simulation and the Perron-Frobenius equation. Collective chaos in a single band state, and bifurcation against initial conditions in a two-band state are clarified with the return maps of the mean-field, Lyapunov spectra, and also the newly introduced Lyapunov exponent for the Perron-Frobenius equation.
A newly discovered cascade process of clusters is studied in a network of chaotic elements. It is shown that the splitting of clusters and the synchronization of elements are balanced in a class of partially ordered states, where marginal stability is sustained over an interval of the bifurcation parameters. The partition information creation in bit space shows an avalanche process of information, which leads to the anomalous behavior of power spectra, roughly fitted by a power law form of the wavenumber. Lyapunov spectra have accumulation at null exponents, analogous with those studied in fluid turbulence models.
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