Except the toy problem \dot x = x(t-1) is really do not know an example of this kind.

I would recommend to see the book "Alfredo Bellen, Marino Zennaro: Numerical Methods for Delay Differential Equations, Oxford University Press, 10.01.2013" and the refs. cited therein. In my old paper togehter with Oberle "Numerical Treatment of Delay Differential Equations by Hermite Interpolation``, Numer. Math. 37 (1981), 235-255. (DOI 10.1007/BF01398255), there is now example of this kind.

Another source could be the optimal control problems from the recent papers of Maurer and  Goellmann et. al.

Hopefully these hint will help you to find appropriate examples.

Dear Prof. Pesch,

Thank you for your suggestion. I have seen several examples in the case of scalar equation. However, for higher dimensional system then not yet.

Regards, Phi Ha.

Well, there exist quite enough problems which can be modelled using the so-called pure systems of delay differential equations -- the simple artificial neural networks/communications networks,  the dynamics of species populations, driver's behaviour in the platoon of vehicles,  etc. All of such models are based on the common assumption of the delayed transfer of "something".

I once looked at the system \dot{x} = sin(t-tau) which has interesting chaotic dynamics: http://sprott.physics.wisc.edu/pubs/paper304.pdf

Dear Phi,

different DDEs of the form $ \dot{x}(t)=F(t,x(t),x(t-\tau)) $ may be transformed to the form $\dot{y}(t)=G(t,y(t-\tau))$ by change of variables (Examples: "Wright's equation" $ \dot{x}(t)=-\alpha x(t-1)  [1+x(t)] $ with real $\alpha>0 $ via $ y=log(1+x) $, or the "Hutchinson equation" $\dot{x}(t)=b x(t) [(K-x(t-1))/K]$ with reals $ b,K>0 $ via $y=log(x)$, or the equation $\dot{x}(t)=-\mu x(t)+f(x(t-1))$ via $y(t)=\exp(\mu t) x(t)$.) However, if you are looking for an application where the form $\dot{y}(t)=G(t,y(t-\tau))$ directly follows from the model then compare, for instance, the article "Synchronized systems with time delay in loop" (Proc. Inst. Radio Eng. 41 (1957)) of W.J. Cunningham and P.J. Wangersky, or the article "Existence of periodic solutions of one-dimensional differential-delay equations" (Tohoku math. J. 30 (1978)) of T. Furumochi. For further references, it could be also interesting to consult the survey article "Topics in Delay Differential Equations" of H.-O. Walther in DMV Jahresbericht vol. 116 (2014) or the monograph  "Applied Delay Differential Equations" of T. Erneux.

I hope I could help you with that.

Best regards,

Eugen Stumpf

Wang H., Nagy J. D., Gilg O., Kuang Y. The roles of predator maturation delay and functional response in determining the periodicity of predator–prey cycles // Mathematical Biosciences. 2009. V. 221. P. 1–10

What about difference (instead differetial) eq.?

Old, but nice classicall text from Maynard Smith & Slatkin 1973.

There are a few references at the end of the Maple worksheet "Constant Delay Differential Equations and the Method of Steps" at the Web site: http://www.maplesoft.com/applications/view.aspx?SID=33093.  In particular, R.D. Driver in his book "Ordinary and Delay Differential Equations" (Springer-Verlag, New York, 1977) lists some on references on page 239.  I have not looked at them but perhaps you can find something there.

Thank you all for many answer and suggestions. I am quite convinced now.

Dear Phi.

I'm work on many example of DDE,like that optical feedback,optical injection,optoelectronics feedback and many applications of these phenomenon on chaos communications.