# コロキウム

Delay differential equations and Lambert W function

2015年7月16日(木) 13時30分

Ordinary differential equations (ODEs) represent the rule how an unknown quantity which has a single independent variable changes infinitesimally. In ODEs, the derivative $x’(t)$ of an unknown function $x(\cdot)$ at $t$ depends $(t, x(t))$ precisely. However, the finiteness of the propagation time, for example, indicates that differential equations in which $x’(t)$ depends the history of $x(\cdot)$ at $t$ arise. Such equations are called the delay differential equations (DDEs), and the dynamics of DDEs are very rich in the sense that DDEs determine semi-dynamical systems in the space of continuous functions. In this talk, we glance at the theory of DDEs and investigate the asymptotic stability of constant solutions by using the Lambert W function, which is defined as the multi-valued inverse of a complex function $z\mathrm{e}^z$.

J. K. Hale & S. M. Verduyn Lunel (1993): Introduction to Functional Differential Equations. (Springer)
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffery, & D. E. Knuth (1996): On the Lambert $W$ function. Adv. Comput. Math. 5, 329–359.