Satoshi Saitô

Butcher's simplifying assumption for symplectic integrators

Implicit Runge-Kutta methods with vanishingM matrix are discussed for preserving the symplectic structure of Hamiltonian systems. The number of the order conditions independent of the number of stages can be reduced considerably for the symplectic IRK method through the analysis utilizing the rooted tree and the corresponding elementary differentials. Butchers simplifying condition further reduces the number of independent order conditions.

Family of symplectic implicit Runge-Kutta formulae

A family of formulae for the sympletic IRK method is investigated. Specifically, focus is given to general solutions for formula parameters of IRK under the symplectic and the order conditions. Examples of such formulae are constructed for up to three stages.

Hopf bifurcation analysis for a dissipative system with asymmetric interaction: Analytical explanation of a specific property of highway traffic.

A dissipative system with asymmetric interaction, the optimal velocity model, shows a Hopf bifurcation concerned with the transition from a homogeneous motion to the formation of a moving cluster, such as the emergence of a traffic jam. We investigate the properties of Hopf bifurcation depending on the particle density, using the dynamical system for the traveling cluster solution of the continuum system derived from the original discrete system of particles. The Hopf bifurcation is revealed as a subcritical one, and the property explains well the specific phenomena in highway traffic: the metastability of jamming transition and the hysteresis effect in the relation of car density and flow rate.

Higher Order Symmetry and Bifurcation of the Period-2 Step-1 Accelerator Mode in the Standard Map

The bifurcation process of the period-2 step-1 accelerator mode in the standard map has been investigated throughly. The formation of dominant 5-cycles around the central island and the subsequent period doubling of the central island have been analyzed by studying the structure of higher order symmetry lines. The present analysis confirms that analysis of the higher order symmetry structure provides a systematic approach for investigating the higher order complex structure in the standard map.

Symmetry structure of accelerator modes in the standard map

Abstract The regular motions in nonlinear Hamiltonian dynamical systems have been best studied in terms of the symmetry analysis based on the involution decomposition. As for the standard map, because of its intrinsic periodicity, there appear various orders of the accelerator modes, a special type of regular motions. Extension of the involution decomposition taking account of the periodicity makes it possible to investigate the symmetry structure of the accelerator modes. Explicit analysis has been carried out for the period-2, the period-3 and the period-4 accelerator modes. The present approach proves to be useful for other mappings with proper periodicity.

ANOMALOUS DIFFUSION IN THE KICKED HARPER SYSTEM

Stochastic property of the Hamiltonian dynamical systems exhibits anomalous behavior owing to intermittent processes between regular orbits and chaotic motion. In particular, the accelerator modes give rise to the anomalous enhancement of the diffusion process, which could be attributed to temporal fractal property. In order to explore the effect of the accelerator modes on the stochastic diffusion, the kicked Harper system is investigated in detail. It is found that the numerical observation of the diffusion coefficient shows good agreement with the analytical expression obtained by the characteristic function method, except in the domains where the accelerator modes exist. It is interesting to note that, unlike the case of the standard map, the multi-period, multi-step accelerator modes are responsible for much of the profound effect of the enhancement of diffusion even in the domain of larger stochastic parameters.