Research

Introduction

The main purpose of our research is to construct qualitative theory of delay equations. In the classical ordinary differential equations, the rate of change of state depends only on present time. But, to model some phenomena more realistically, it is natural that the rate of change of state depends on past time as well as on present time in the field of engineering and mathematical biology. Thus, ordinary differential equations that take account of the effect of past state have been investigated by many researchers in the last half century.

Our research theme is to clarify the effect of delay on asymptotic properties of differential equations. For example, in a linear differential system with a delay in the diagonal terms 𝒙'(𝒕)=-3𝒙(𝒕-𝒓)-𝒚(𝒕), 𝒚'(𝒕)=-𝒙(𝒕)-0.98𝒚(𝒕-𝒓), if the delay 𝒓 changes 0.7, 0.8, 0.87, and 0.89, then the zero solution becomes asymptotically stable, unstable, asymptotically stable, and unstable, respectively (see the figures below). Such properties on stability are referred to as stability switches. For more detail on the above system, see Hata, Matsunaga, DCDSB 28, 4910-4936 (2023) .

We are also interested in delay difference equations as well as delay differential equations.

figure2
𝒓=0.7
figure3
𝒓=0.8
figure4
𝒓=0.87
figure5
𝒓=0.89

Keywords

  • Delay differential equations
  • Difference equations
  • Asymptotic stability
  • Stability switches

Research Grants (Leader)

  • 2024-2027, JSPS KAKENHI Grant-in-Aid for Scientific Research (C)
  • 2019-2023, JSPS KAKENHI Grant-in-Aid for Scientific Research (C)
  • 2014-2017, JSPS KAKENHI Grant-in-Aid for Scientific Research (C)
  • 2009-2011, JSPS KAKENHI Grant-in-Aid for Young Scientists (B)
  • 2007-2008, JSPS KAKENHI Grant-in-Aid for Young Scientists (B)
  • 2006, Basic Science Research Projects of the Sumitomo Foundation
  • 2003-2005, JSPS KAKENHI Grant-in-Aid for Young Scientists (B)
  • 2002, Sasakawa Grants for Science Fellows of the Japan Science Society