# Another proof of Hayes' and Sakata's results by critical delay and its comparison with the method of D-partitions in combination with delay sequence for purely imaginary roots

@inproceedings{Nishiguchi2021AnotherPO, title={Another proof of Hayes' and Sakata's results by critical delay and its comparison with the method of D-partitions in combination with delay sequence for purely imaginary roots}, author={Junya Nishiguchi}, year={2021} }

The location of roots of the characteristic equation of a linear delay differential equation (DDE) determines the stability of the linear DDE. However, by its transcendency, there is no general criterion on the contained parameters for the stability. Here we concentrate on the study of a simple transcendental equation (∗) z+a−we−zτ = 0 with coefficients of real a and complex w and a delay parameter τ > 0 to tackle this transcendency brought by delay. The consideration is twofold: (i) to give… Expand

#### References

SHOWING 1-10 OF 39 REFERENCES

The stability chart for the linearized Cushing equation with a discrete delay and with gamma-distributed delays

- Mathematics
- 1989

In a proper setting, the problem of the title amounts to the determination of all pairs (d, T) for which the function ƒ(z): = (z + 1)exp(zT) + d and the functions fn(z)≔(z+1)(1+zTn)n+d, n∈N, d, T⩾0,… Expand

On characteristic roots and stability charts of delay differential equations

- Mathematics
- 2012

SUMMARY
This work is devoted to the analytic study of the characteristic roots oftextitscalar autonomous delay differential equations with either real or complex coefficients. The focus is placed… Expand

Analysis of a System of Linear Delay Differential Equations

- Mathematics
- 2003

A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity with the concept of… Expand

Stability implications of delay distribution for first-order and second-order systems

- Mathematics
- 2009

In application areas, such as biology, physics and engineering, delays
arise naturally because of the time it takes for the system to react
to internal or external events. Often the associated… Expand

On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay

- Physics
- 2016

A transcendental equation $\lambda + \alpha - \beta\mathrm{e}^{-\lambda\tau} = 0$
with complex coefficients is investigated.
This equation can be obtained from the characteristic equation of a… Expand

Stability switches in a system of linear differential equations with diagonal delay

- Mathematics, Computer Science
- Appl. Math. Comput.
- 2009

As @t increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if 0<4a<-bc; and from instability to Stability to instability if --bc<2a<0. Expand

Sufficient conditions for stability of linear differential equations with distributed delay

- Mathematics
- 2001

We develop conditions for the stability of the constant (steady
state) solutions oflinear delay differential equations with distributed delay
when only information about the moments of the density… Expand

Robust stability analysis of linear time-delay systems by Lambert W function: Some extreme point results

- Mathematics, Computer Science
- Autom.
- 2006

It is proven that if uncertainties in the coefficients of the quasi-polynomial are set in appropriate regions in the complex plane, the authors can enjoy extreme point results: finite number of stability checks at some points of the boarder of the regions suffice. Expand

Discrete delay, distributed delay and stability switches

- Mathematics
- 1982

In modelling in the biological, physical and social sciences, it is sometimes necessary to take account of time delays inherent in the phenomena. The inclusion of delays explicitly in the equations… Expand

Stability of Runge-Kutta methods for linear delay differential equations

- Mathematics, Computer Science
- Numerische Mathematik
- 2000

This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear scalar delay differential equation and proves that implicit Euler method isstable. Expand