I want to know the method that determines the stability of dde for all situations.
It depends of which kind of stability you want to investigate.
(i) Local stability => Search for singular points and then, linearize the system around these points
(ii) Global stability => Search for a Poincarré-Bendixon domain and/or a Lyapunov function
(iii) Lyapunov exponent is a quantity that characterizes the rate of separation of infinitesimally close trajectories, it is often use to ivestigate if the system is chaotic or not.
(iV) I recommend you the two book (The first chapter of the first book talk about your question).
A-
Mathematical Biology: I. An Introduction (Interdisciplinary Applied Mathematics) (Pt. 1) Hardcover
by James D. Murray (Author)
B-
Differential Equations and Mathematical Biology Hardcover – February 26, 2003
by D.S. Jones (Author), B.D. Sleeman (Author), D. S. Jones (Author)
A compact and straightforward description is found in the works of Gabor Stepan. He focuses on the milling process but the outlined method is applicable in general. I suggest to look right into his PhD thesis at http://galilei.mm.bme.hu/~inspi/
I highly recommend a book by Thomas Erneux, "Applied delay differential equations".
The fundamental example is to study the stability of the ODE
y'(t)=-y(t-tau)
here tau is the delay. You plug in y(t)=exp(lambda*t) and you get
lambda=-exp(-lambda*tau)
Next you look for a Hopf bifurcation, that is plug in lambda=i*w with w purely real. You get
i w = (sin(w tau)- i cos(w tau))
that is,
sin(w tau)=0
w = - cos(w tau)
The smallest solution (with tau positive) is tau = pi corresponding to the frequency w=1. The next solution is tau=3 pi etc...
That is, the system is stable for 0
Hi Faeze
Some delay-dependent criteria for various stability types of a delay differential equations are derived in these papers :
http://www.sciencedirect.com/science/article/pii/S0925231214013964
http://www.sciencedirect.com/science/article/pii/S0925231215003057
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