Characterisitic equation of delay differential equation

Question

How do I derive the characteristic equation around a fixed point $x_0$, when the DDE is defined as $\tau\, dx(t)/dt=-x(t) + f(p-w\,x(t-d)) $, where p,w,d $\in R$ are constants and $f:R\rightarrow R$ is a nonlinear function with $f(x)=1+tan(x)$ ?

Answer 1

You first replace $f$ with the tangent at $x_0$ namely $y=f'(x_0)(x-x_0)+f(x_0)$. That gives you a linear, non-homogeneous DDE $$ \frac{d}{dt}x(t)=-x(t)-f'(x_0)wx(t-d)+f'(x_0)p-f'(x_0)x_0+f(x_0). $$ Take the homogeneous part $$ \frac{d}{dt}x(t)=-x(t)-f'(x_0)wx(t-d) $$ and substitute $x(t):=e^{\lambda t}$ to get $$ \lambda e^{\lambda t} = -e^{\lambda t}-f'(x_0)we^{\lambda t}e^{-\lambda d}. $$ Divide by $e^{\lambda t}$ and obtain the characteristic equation $$ \lambda+1+f'(x_0)we^{-\lambda d}=0. $$