Shin-itiro Goto

Legendre submanifolds in contact manifolds as attractors and geometric nonequilibrium thermodynamics

It has been proposed that equilibrium thermodynamics is described on Legendre submanifolds in contact geometry. It is shown in this paper that Legendre submanifolds embedded in a contact manifold can be expressed as attractors in phase space for a certain class of contact Hamiltonian vector fields. By giving a physical interpretation that points outside the Legendre submanifold can represent nonequilibrium states of thermodynamic variables, in addition to that points of a given Legendre submanifold can represent equilibrium states of the variables, this class of contact Hamiltonian vector fields is physically interpreted as a class of relaxation processes, in which thermodynamic variables achieve an equilibrium state from a nonequilibrium state through a time evolution, a typical nonequilibrium phenomenon. Geometric properties of such vector fields on contact manifolds are characterized after introducing a metric tensor field on a contact manifold. It is also shown that a contact manifold and a strictly c...

Renormalization Reductions for Systems with Delay

The renormalization method which is a type of perturbation method is extended to a tool to study weakly nonlinear time-delay systems. For systems with order-one delay, we show that the renormalization method leads to reduced systems without delay. For systems with order-one and large-delay, we propose an extended renormalization method which leads to reduced systems with delay. In some examples, the validities of our perturbative results are confirmed analytically and numerically. We also compare our reduced equations with reduced ones obtained by another perturbation method.

Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics

Contact geometry has been applied to various mathematical sciences, and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry. Here, such a dually flat space is related to a Legendre submanifold in a contact manifold. In this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact Hamiltonian vector fields. Based on these descriptions, two ways of lifting vector fields on Legendre submanifolds to contact manifolds are given. For some classes of these lifted vector fields, invariant measures in contact manifolds and stability analysis around Legendre submanifolds are explicitly given. Throughout this paper, Legendre duality is explicitly stated. In addition, to show how to apply these general methodologies to applied mathematical disciplines, electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed.

Regularized Renormalization Group Reduction of Symplectic Maps

By means of the perturbative renormalization group method, we study a long-time behaviour of some symplectic discrete maps near elliptic and hyperbolic fixed points. It is shown that a naive renormalization group (RG) map breaks the symplectic symmetry and fails to describe a long-time behaviour. In order to preserve the symplectic symmetry, we present a regularization procedure, which gives a regularized symplectic RG map describing an approximate long-time behaviour successfully.

Renormalization Analysis of Resonance Structure in a 2-D Symplectic Map

A symplecticity-preserving RG analysis is carried out to study the resonance structure near an elliptic fixed point of a prototype symplectic map in two dimensions. Through analysis of fixed points of a reduced RG map, the topology of the resonance structure, such as a chain of resonant islands, can be determined analytically. The application of this analysis to the Henon map is also presented.

Random Wandering around Homoclinic-Like Manifolds in a Symplectic Map Chain

We present a method to construct a symplecticity preserving renormalization group map of a chain of weakly nonlinear symplectic maps and obtain a general reduced symplectic map describing its long-time behavior. It is found that the modulational instability in the reduced map triggers random wandering of orbits around some homoclinic-like manifolds. This behavior is understood as Bernoulli shifts.

Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems

By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative Henon map, both of which exhibit strong homoclinic chaos. In terms of the asymptotic expansion, a simple formulation is presented to give the first homoclinic point in the double- well map. Even a truncated expansion of the unstable manifold is shown to reproduce the well-known many-leaved (fractal) structure of the strange attractor in the Henon map.

Analytical expression for low-dimensional resonance islands in a 4-dimensional symplectic map

We study 2- and 4-dimensional nearly integrable symplectic maps using a singular perturbation method. Resonance island structures in the 2- and 4-dimensional maps are obtained. The validity of these perturbative results are confirmed numerically.

Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Maps on a parameter space for expressing distribution functions are exactly derived from the Perron-Frobenius equations for a generalized Boole transform family. Here the generalized Boole transform family is a one-parameter family of maps where it is defined on a subset of the real line and its probability distribution function is the Cauchy distribution with some parameters. With this reduction, some relations between the statistical picture and the orbital one are shown. From the viewpoint of information geometry, the parameter space can be identified with a statistical manifold, and then it is shown that the derived maps can be characterized. Also, with an induced symplectic structure from a statistical structure, symplectic and information geometric aspects of the derived maps are discussed.

From an Unstable Periodic Orbit to the Lyapunov Exponent and a Macroscopic Variable in a Hamiltonian Lattice Periodic Orbit Dependencies

We study the problem of determining which periodic orbits in phase space can predict the largest Lyapunov exponent and the expectation values of macroscopic variables in a Hamiltonian system with many degrees of freedom. We also attempt to elucidate the manner in which these orbits yield such predictions. The model which we use in this paper is a discrete