International Journal of Modern Physics B

We show how to introduce a characterization the "complexity" of random dynamical systems. More precisely we propose a suitable indicator of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by these systems. This indicator of complexity, which can be extracted from real experimental data, turns out to be very natural in the context of information theory. For dynamical systems with random perturbations, it coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. In presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent λ_σ computed through the consideration of two nearby trajectories evolving under the same realization of the random perturbation. However, the former is much more relevant than the latter from a physical point of view as illustrated by some numerical examples of noisy and random maps.

If we pay serious and conscientious attention to the connotation of the word "complexity", study of complex systems must basically be biology. Basic questions are raised from this point of view.

A simple computational model is proposed that reproduces various aspects of the complex dynamics of dunes such as the barchan dunes formation process, the evolution process from a barchan dune to a seif dune, the network-dunes formation under time-dependent directional winds, etc. Although this model may be oversimplified in several respects, there is a hope that it helps us to sift relevant factors out the vast sea of numerous factors influencing the rich dynamics of desert dunes.

Recent progresses in understanding earthquake dynamics with the aid of a simple spring-block system is reviewed from a physicists' point of view. Dynamical instability due to negative dynamical friction amplifies any perturbation and leads to a chaotic behavior for almost any initial configurations. It is also pointed out that static friction gives another source of the complex behavior which is characteristic to a threshold element system.

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 subgroups, 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 possibilities for observing the emergence of collective behavior in large chaotic dynamical systems are discussed. Nontrivial collective behavior in extensively-chaotic situations is presented with the help of a few examples and argued to offer all the desirable properties of a truly generic phenomenon.

The Titius–Bode law governs the planet distribution in our Solar system. In this paper a possible explanation is proposed based on inelastic collision effects among planetessimals during the evolution of the Solar system. The main purpose of this paper is, however, to introduce a strategy to study phenomena driven by rare but drastic events such as colllisions in the planetary problem. Many complex systems evolve through rare but violent events, so that an efficient strategy to simulate such systems is desirable. An event-driven strategy is proposed in this article, and is used to produce many runs of 10⁸ year evolution history of planetary systems. I have found that the Titius–Bode law holds approximately, if the gravitational effect (scattering) and the collisions are taken into account. The result illustrates the importance of inelastic collisions, which are often neglected in the standard classical mechanics courses. Therefore, for completeness, other simpler particle systems under the effect of inelastc collisions, such as one-dimensional systems, are also included.