Yoji Aizawa

The Breakup Condition of Shearless KAM Curves in the Quadratic Map

We determined the exact location of the shearless KAM curve in the quadratic map and numerically investigate the breakup thresholds of those curves in the entire parameter space. The breakup diagram reveals many sharp singularities like fractals on the reconnection thresholds of the twin-chains with rational rotation numbers.

Comments on the non-stationary chaos

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 Karamatas theory. Finally, the significance of the stationary/non-stationary interface is emphasized in relation to the universality of 1/f fluctuations.

Clustering Motions in N-Body Systems Computer Experiments of Kinetic Laws

The dynamical process of cluster formation is numerically studied by carrying out with 2-dimensional N-body systems under short-range interactions. First, we give a theoretical definition of cluster boundaries by use of a scalar field of the Gauss-Riemann curvature. Based on this lucid definition, we can obtain much reliable information regarding statistical aspects of clustering motion. The energy dependence of the cluster size exhibits phasetransition-like behavior, as predicted by the cell model, and the velocity distribution function obeys the Maxwell-Boltzmann statistics not only in the gaseous phase but also in the cluster. However, it is pointed that the fluctuations of the cluster’s shape reveal very long time memories, even in the equilibrium state. Secondly, the kinetic aspects of each particle are analyzed from the residence time distribution. The residence time in the gaseous phase obeys a Poisson distribution, but in the droplet phase it obeys a Negative-Weibull distribution with the exponent α (� 1.7) within a certain scaling regime. Also, it is elucidated that the intrinsic long time behavior obeys the universal law of nearly integrable Hamiltonian dynamics, and that the symbolic dynamics of one particle display 1/f spectra very stably. Lastly, it is pointed out that these two regimes, i.e., the Negative-Weibull regime and the universal long time regime, correspond to different phases coexisting in a cluster, and the interdependence between both phases is discussed in relation to the stochastic theory of nucleation.

Clustering motions in N-body systems - free-fall motions in the Gaussian three-body problem

Abstract 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 time 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.

On the Diagram for the Onset of Global Chaos and Reconnection Phenomena in the Quadratic Nontwist Map

The quadratic nontwist map is studied by defining the “indicator points” for the phase space. The indicator points enable us to obtain useful information concerning the onset of global chaos and reconnection phenomena. The diagram obtained by using the indicator points clearly reveals the effects of the reconnection phenomena on the transition to global chaos. A discussion is given of the last KAM curve for the quadratic nontwist map. In this paper, we consider area-preserving maps that violate the twist condition. They are often called area-preserving “nontwist” maps. Violation of the twist condition is encountered in models from a variety of fields such as fluid dynamics, plasma physics, celestial mechanics, etc. Moreover, violation of the twist condition generically occurs in the neighborhood of tripling bifurcations of a Hamiltonian system. 5),11) Originally, the twist condition is assumed in order to prove important theorems concerning the persistence of KAM curves, e.g. the Moser twist theorem 1) and the Aubry-Mather theorem. 2) However, the precise effects of violation of the twist condition on the persistence of KAM curves have not yet been elucidated. Numerical studies of nontwist systems revealed that violation of the twist condition generally leads to complicated phase space phenomena, called reconnection phenomena. 3) - 11) The reconnection phenomena drastically change the phase space structure and thus strongly influence the breakup process of KAM curves. In order to study in detail the properties arising from violation of the twist condition, we focus here on the quadratic nontwist map (QNM), which is one of the simplest prototypes of an area-preserving nontwist map. Previously, we showed that the onset of global chaos and the reconnection phenomena in the QNM could be systematically studied by defining the “indicator points” for the phase space. After summarizing the properties of the indicator points, we present the diagram for the onset of global chaos and reconnection phenomena. The diagram clearly reveals how reconnection phenomena influence the transition to global chaos. On the basis of the numerical results obtained by using the indicator points, a discussion is given of the last KAM curve for the QNM.