Papers - Ei Shin-Ichiro
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APPLICATION OF A CENTER MANIFOLD THEORY TO A REACTION-DIFFUSION SYSTEM OF COLLECTIVE MOTION OF CAMPHOR DISKS AND BOATS Reviewed
S.-I. Ei, K. Ikeda, M. Nagayama, A. Tomoeda
MATHEMATICA BOHEMICA 139 ( 2 ) 363 - 371 2014
Language:English Publishing type:Research paper (scientific journal)
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Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems Reviewed
Shin-Ichiro Ei, Toshio Ishimoto
Networks and Heterogeneous Media 8 ( 1 ) 191 - 209 2013
Language:English Publishing type:Research paper (scientific journal)
We consider pulse-like localized solutions for reaction-diffusion sys- tems on a half line and impose various boundary conditions at one end of it. It is shown that the movement of a pulse solution with the homogeneous Neumann boundary condition is completely opposite from that with the Dirichlet boundary condition. As general cases, Robin type boundary conditions are also considered. Introducing one parameter connecting the Neumann and the Dirichlet boundary conditions, we clarify the transition of motions of solutions with respect to boundary conditions. © American Institute of Mathematical Sciences.
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Dynamics and interactions of spikes on smoothly curved boundaries for reaction-diffusion systems in 2D Reviewed
Shin-Ichiro Ei, Toshio Ishimoto
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 30 ( 1 ) 69 - 90 2013
Language:English Publishing type:Research paper (scientific journal) Publisher:SPRINGER JAPAN KK
It is known that for special types of reaction-diffusion Systems, such as the Gierer-Meinhardt model and the Gray-Scott model, stable stationary spike solutions exist on boundary points with maximal curvature. In this paper, we rigorously give the equation describing the motion of spike solutions along boundaries for general types of reaction-diffusion systems in R-2. We also apply the general results to the Gierer-Meinhardt model and show that a single spike solution moves toward a boundary point with locally maximal curvature. Moreover, by showing the repulsive interaction of spikes along boundaries for solutions of the Gierer-Meinhardt model, we have stable multispike stationary solutions in the neighborhood of a boundary point with locally maximal curvature.
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INFINITE DIMENSIONAL RELAXATION OSCILLATION IN AGGREGATION-GROWTH SYSTEMS Reviewed
Shin-Ichiro Ei, Hirofumi Izuhara, Masayasu Mimura
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B 17 ( 6 ) 1859 - 1887 2012.09
Language:English Publishing type:Research paper (scientific journal) Publisher:AMER INST MATHEMATICAL SCIENCES
Two types of aggregation systems with Fisher-KPP growth are proposed. One is described by a normal reaction-diffusion system, and the other is described by a cross-diffusion system. If the growth effect is dominant, a spatially constant equilibrium solution is stable. When the growth effect becomes weaker and the aggregation effect become dominant, the solution is destabilized so that spatially non-constant equilibrium solutions, which exhibit Turing's patterns, appear. When the growth effect weakens further, the spatially non-constant equilibrium solutions are destabilized through Hopf bifurcation, so that oscillatory Turing's patterns appear. Finally, when the growth effect is extremely weak, there appear spatio-temporal periodic solutions exhibiting infinite dimensional relaxation oscillation.
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DYNAMICS OF A BOUNDARY SPIKE FOR THE SHADOW GIERER-MEINHARDT SYSTEM Reviewed
Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS 11 ( 1 ) 115 - 145 2012.01
Language:English Publishing type:Research paper (scientific journal) Publisher:AMER INST MATHEMATICAL SCIENCES-AIMS
The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. The authors of [3] showed that if an initial value is close to a spiky pattern and its peak is far away from the boundary, the solution of the shadow Gierer-Meinhardt system, called a interior spike solution, moves towards a point on boundary which is the closest to the peak. However it has not been studied how a solution close to a spiky pattern with the peak on the boundary, called a boundary spike solution moves along the boundary. In this paper, we consider the shadow Gierer-Meinhardt system and dynamics of a boundary spike solution. Our results state that a boundary spike moves towards a critical point of the curvature of the boundary and approaches a stable stationary solution.
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Dynamics of pulses on a thin strip-like domain in R² Reviewed
EI Shin-Ichiro
RIMS Kokyuroku Bessatsu 31 ( B31 ) 195 - 210 2012
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Infinite dimensional relaxation oscillation in reaction-diffusion systems Reviewed
S.-I. Ei, H. Izuhara, M. Mimura
RIMS Kokyuroku Bessatsu 35 ( B35 ) 31 - 40 2012
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A NEW TREATMENT FOR PERIODIC SOLUTIONS AND COUPLED OSCILLATORS Reviewed
Shin-Ichiro Ei, Kunishige Ohgane
KYUSHU JOURNAL OF MATHEMATICS 65 ( 2 ) 197 - 217 2011.09
Language:English Publishing type:Research paper (scientific journal) Publisher:KYUSHU UNIV, FAC MATHEMATICS
We develop a systematic method for deriving the phase dynamics of perturbed periodic solutions. The method is to regard periodic solutions as slowly modulated traveling solutions on the circle. There, problems are reduced to the perturbed problems from stationary solutions on the circle. This makes the treatment of periodic solutions far easier and systematic. We also give the rigorous proofs for this method.
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Front dynamics in heterogeneous diffusive media Reviewed
Hideo Ikeda, Shin-Ichiro Ei
PHYSICA D-NONLINEAR PHENOMENA 239 ( 17 ) 1637 - 1649 2010.09
Language:English Publishing type:Research paper (scientific journal) Publisher:ELSEVIER SCIENCE BV
We herein consider two-component reaction-diffusion systems with a specific bistable and odd symmetric nonlinearity, which have the bifurcation structure of pitchfork type traveling front solutions with opposite velocities. We introduce a spatial heterogeneity, for example, a Heaviside-like abrupt change at the origin in the space, into diffusion coefficients. Numerically, the responses of traveling fronts via the heterogeneity can be classified into four types of behavior depending on the strength of the heterogeneity, which, in the present paper, is represented by the height of the jump: passage, stoppage, and two types of reflection. The goal of the present paper is to reduce the PDE dynamics to finite-dimensional ODE systems on a center manifold and show the mathematical mechanism for producing the four types of response in the PDE systems using finite-dimensional ODE systems. The reduced ODE systems include the terms (referred to as heterogeneous perturbations) originating from the interaction between traveling front solutions and the heterogeneity, which is very important for determining the dynamics of the ODE systems. In the present paper, we succeed in calculating these heterogeneous perturbations exactly and explicitly. (C) 2010 Elsevier B.V. All rights reserved.
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Neuron phase shift adaptive to time delay in locomotor control Reviewed
Kunishige Ohgane, Shin-Ichiro Ei, Hitoshi Mahara
APPLIED MATHEMATICAL MODELLING 33 ( 2 ) 797 - 811 2009.02
Language:English Publishing type:Research paper (scientific journal) Publisher:ELSEVIER SCIENCE INC
Based oil neurophysiological evidence, theoretical studies have shown that walking can be generated by mutual entrainment of oscillations of a central pattern generator (CPG) and a body. However, it has also been shown that the time delay in the sensorimotor loop destabilizes mutual entrainment, and results in the failure to walk. Recently, it has been reported that if (a) the neuron model used to construct the CPG is replaced by physiologically faithful neuron model (Bonhoeffer-Van der Pol type) and (b) the mechanical impedance of the body (muscle viscoelasticity) is controlled depending oil the angle between two legs, the phase relationship between CPG activity and body motion could be flexibly locked according to the loop delay and, therefore, mutual entrainment can be stabilized. That is, locomotor control adaptive to the loop delay can emerge from the coupling between CPG and body. Here, we call this mechanism flexible-phase locking. In this paper, we construct a system of coupled oscillators as a simplified model of a walking system to theoretically investigate the mechanism of flexible-phase locking, and to analyze the simplified model. The analysis suggests that the following are required as the essential mechanism: (i) an asymptotically stable limit cycle of the coupling system of CPG and body and (ii) a sign difference between afferent and efferent coupling coefficients. (C) 2007 Elsevier hic. All rights reserved.
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Eigenfunctions of the adjoint operator associated with a pulse solution of some reaction-diffusion systems Reviewed
S.-I. Ei, H. Ikeda, K. Ikeda, E. Yanagida
Bull. Inst. Math. Academia Sinica 3 603 - 666 2008
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'Initial state' coordinations reproduce the instant flexibility for human walking Reviewed
A Ohgane, K Ohgane, S Ei, H Mahara, T Ohtsuki
BIOLOGICAL CYBERNETICS 93 ( 6 ) 426 - 435 2005.12
Language:English Publishing type:Research paper (scientific journal) Publisher:SPRINGER
An important feature of human locomotor control is the instant adaptability to unpredictable changes of conditions surrounding the locomotion. Humans, for example, can seamlessly adapt their walking gait following a sudden ankle impairment (e.g., as a result of an injury). In this paper, we propose a theoretical study of the mechanisms underlying flexible locomotor control. We hypothesize that flexibility is achieved by modulating the posture at the beginning of the stance phase-the initial state. Using a walking model, we validate our hypothesis through computer simulations.
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A variational approach to singular perturbation problems in reaction-diffusion systems Reviewed
SI Ei, M Kuwamura, Y Morita
PHYSICA D-NONLINEAR PHENOMENA 207 ( 3-4 ) 171 - 219 2005.08
Language:English Publishing type:Research paper (scientific journal) Publisher:ELSEVIER SCIENCE BV
In this paper singular perturbation problems in reaction-diffusion systems are studied from a viewpoint of variational principle. The goal of the study is to provide an unified and transparent framework to understand existence, stability and dynamics of solutions with transition layers in contrast to previous works in many literatures on singular perturbation theory. (c) 2005 Elsevier B.V. All rights reserved.
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Emergence of adaptability to time delay in bipedal locomotion Reviewed
K Ohgane, S Ei, K Kazutoshi, T Ohtsuki
BIOLOGICAL CYBERNETICS 90 ( 2 ) 125 - 132 2004.02
Language:English Publishing type:Research paper (scientific journal) Publisher:SPRINGER-VERLAG
Based on neurophysiological evidence, theoretical studies have shown that locomotion is generated by mutual entrainment between the oscillatory activities of central pattern generators (CPGs) and body motion. However, it has also been shown that the time delay in the sensorimotor loop can destabilize mutual entrainment and result in the failure to walk. In this study, a new mechanism called flexible-phase locking is proposed to overcome the time delay. It is realized by employing the Bonhoeffer-Van der Pol formalism - well known as a physiologically faithful neuronal model - for neurons in the CPG. The formalism states that neurons modulate their phase according to the delay so that mutual entrainment is stabilized. Flexible-phase locking derives from the phase dynamics related to an asymptotically stable limit cycle of the neuron. The effectiveness of the mechanism is verified by computer simulations of a bipedal locomotion model.
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Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions Reviewed
SI Ei, JC Wei
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 19 ( 2 ) 181 - 226 2002.06
Language:English Publishing type:Research paper (scientific journal) Publisher:SPRINGER JAPAN KK
In this paper, the Gierer-Meinhardt model systems with finite diffusion constants in the whole space R-2 is considered. We give a regorous proof on the existence and the stability of a single spike solution, and by using such informations, the repulsive dynamics of the interacting multi single-spike solutions is also shown when distances between spike solutions are sufficiently large. This clarifies some part of the mechanism of the evolutional process of localized patterns appearing in the Gierer-Meinhardt model equations.
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2n-Splitting or Edge-Splitting? A Manner of Splitting in Dissipative Systems Reviewed
Shin-Ichiro Ei, Yasumasa Nishiura, Kei-Ichi Ueda
Japan Journal of Industrial and Applied Mathematics 18 ( 2 ) 181 - 205 2001
Language:English Publishing type:Research paper (scientific journal) Publisher:Kinokuniya Co. Ltd
Since early 90's, much attention has been paid to dynamic dissipative patterns in laboratories, especially, self-replicating pattern (SRP) is one of the most exotic phenomena. Employing model system such as the Gray-Scott model, it is confirmed also by numerics that SRP can be obtained via destabilization of standing or traveling spots. SRP is a typical example of transient dynamics, and hence it is not a priori clear that what kind of mathematical framework is appropriate to describe the dynamics. A framework in this direction is proposed by Nishiura-Ueyama [16], i.e., hierarchy structure of saddle-node points, which gives a basis for rigorous analysis. One of the interesting observation is that when there occurs self-replication, then only spots (or pulses) located at the boundary (or edge) are able to split. Internal ones do not duplicate at all. For 1D-case, this means that the number of newly born pulses increases like 2k after k-th splitting, not 2n-splitting where all pulses split simultaneously. The main objective in this article is two-fold: One is to construct a local invariant manifold near the onset of self-replication, and derive the nonlinear ODE on it. The other is to study the manner of splitting by analysing the resulting ODE, and answer the question "2n-splitting or edge-splitting?" starting from a single pulse. It turns out that only the edge-splitting occurs, which seems a natural consequence from a physical point of view, because the pulses at edge are easier to access fresh chemical resources than internal ones.
DOI: 10.1007/BF03168570
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Pattern formation in heterogeneous reaction-diffusion- advection systems with an application to population dynamics Reviewed
S.-I. Ei, M. Mimura
SIAM J. Math. Anal. 21 ( 346 ) 361 1990