arXiv subjects

Yu Ichida

Publications and source records attributed to Yu Ichida.

5 recordsLinked to original sources

arXiv2026

Dynamics and structure of pull-in and touchdown behavior in parallel-plate electrostatic MEMS actuators via geometric approaches

This paper focuses on parallel-plate electrostatic actuators, which are fundamental structures found in Micro-Electro-Mechanical Systems (MEMS) for numerous modern devices. It considers the equations used to model these actuators. In particular, we investigate the behavior of the solutions to a MEMS model that is represented by a second-order ordinary differential equation, where the spring is characterized as soft, linear, or hard. We present the mathematical structure underlying the pull-in and touchdown phenomena that characterize the behavior of systems incorporating soft and hard spring, based on the behavior of linear springs. The pull-in phenomenon corresponds to the degeneration of the solutions to the model equation. The touchdown phenomenon corresponds to the finite-time singularity of the solutions. These global dynamics and structures are systematically revealed through geometric approaches based on Poincar\'e-type compactification, the center manifold theorem, and blow-up technique. These methods successfully induce and resolve the dynamics at infinity.

math.DS↗

arXiv2026

Minimal speed of unbounded traveling wave solutions for a 1D reaction-diffusion equation and their relationship with the dynamics at infinity

This paper presents results on the unboundedness and minimal speed of traveling wave solutions for a one-dimensional spatial reaction-diffusion equation with an asymptotically linear reaction term and a saturation parameter. By applying a Poincar\'e-type compactification, we reveal the full dynamics (including infinity) of the two-dimensional system of ordinary differential equations satisfied by traveling wave solutions. This yields essential information characterizing traveling wave solutions: the classification of trajectories in the phase plane, the positivity and unboundedness of front-type and sign-changing profiles, and the explicit form of the minimal speed. This paper examines a special equation with an asymptotically linear reaction term. While, our results differ from those of conventional linear determinacy. We claim that the minimal speed is derived from information at infinity within the traveling wave system.

math.DS↗

arXiv2023

Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium equation

This paper reports results on the classification of traveling wave solutions, including nonnegative weak sense, in the spatial 1D degenerate parabolic equation. These are obtained through dynamical systems theory and geometric approaches (in particular, Poincar\'e compactification). Classification of traveling wave solutions means enumerating those that exist and presenting properties of each solution, such as its profile and asymptotic behavior. The results examine a different range of parameters included in the equation, using the same techniques as discussed in the earlier work [Y. Ichida, Discrete Contin. Dyn. Syst., Ser. B, {\bf{28}} (2023), no. 2, 1116--1132]. In a clear departure from this previous work, the classification results obtained in this paper and the successful application of known transformation also yield results for the classification of (weak) nonnegative traveling wave solutions for spatial 1D porous medium equations with special nonlinear terms and the simplest porous medium equation. Finally, the bifurcations at infinity occur in the two-dimensional ordinary differential equations that characterize these traveling wave solutions are shown.

math.AP↗

arXiv2022

On global behavior of a some SIR epidemic model based on the Poincar\'e compactification

It is important to study the global behavior of solutions to systems of ordinary differential equations describing the transmission dynamics of infectious disease. In this paper, we present a different approach from the Lyapunov function used in most of them. This approach is based on the Poincar\'e compactification. We then apply the method to a SIR endemic model as a test case, and discuss its effectiveness and the potential applications of this approach. In addition, we refine the discussion of dynamics near the equilibrium, derive the asymptotic behavior, and mention its relation to the basic reproduction number.

math.DS↗

arXiv2020

A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations

In this paper, we consider the asymptotic behavior of traveling wave solutions of the degenerate nonlinear parabolic equation: $u_{t}=u^{p}(u_{xx}+u)-\delta u$ ($\delta = 0$ or $1$) for $\xi \equiv x - ct \to - \infty$ with $c>0$. We give a refined one of them, which was not obtain in the preceding work [Ichida-Sakamoto, 2020], by an appropriate asymptotic study and properties of the Lambert $W$ function.

math.DS↗