Yes it is possible to have such situation but depends upon the nature of roots of the algebraic equation for the equilibrium states of the system (E).

Hi Iren,

 I have performed the solution of your ODE, you will find the details in the attached file. Best regards.

Thanks a lot for your answers :-). Dear Slim, as you have written, the attached solution corresponds to the ODE. However, I'm dealing with delay differential equations, which generally have infinitely many solutions.  Moreover, the known phenomena of the third order DDEs is that it always possesses oscillatory solutions and could have non oscillatory solutions. 

Dear Malay: since we are dealing with linear constant coefficient differential equations, does it make sense to assume another equilibrium as trivial-one?  In such a case, it is familiar that the trivial steady-state is asymptotically stable iff all characteristic roots lie in the open left-half plane. However, the proof of such theorem clearly does not allow to restrict us on behaviour of nonoscillatory solutions.

So I could summarize the problem as follows: for given parameters a,b,c,tau: we have information that all characteristic roots are either complex (responding to oscillatory solutions) or negative, would it be possible to find such a nonoscillatory solution, which is unstable (tends to infinity)? 

Thm 6.5 in Bellman, Cook says that a solution to the n-th order system can be written as

\sum_{r=1}^\infty e^{s_r t}p_r(t)       (*)

where s_r are the roots of the charact. eq-n and p_r(t) are the polynomials in t. If Re(s_r)

Hi Dmitry, 

the mentioned theorem is well known to me, but it says that the convergence of (*) is uniform in any finite interval. Only in case of all negative roots of the CHR,i.e. if there exists such a negative constant larger than any Re(s_r);  it doesn't need be bounded and therefore, we can formulate the strong theorem about  asymptotic behaviour of (E) (see the following chapter).

For my own part, I personally "hope" :-) that answer to my question (which seems unreasonably at first sight for sure) is negative, however, the theorem looks insufficient.

Hi Iren,

the problem of transition from finite to infinite intervals is pretty subtle an I'm affraid I won't be able tob help you any further. I can only give you the paper by Leontiev, which was cited by Bellman, Cooke. May be it will help you (though it is written in Rusian). 

I also wonder what this nonoscillatory untable component would look like -- exp(st)p(t) for some s or somthing else? 

Thank you very much for the attached article, my attempt to find it was unsuccessful :-) 

Iren, you are welcome. Please let us know what will be the result of your investigation.

Hi Iren, you can have a look on the following link :

http://www.mathworks.com/help/matlab/delay-differential-equations.html

Best regards,

May you can try the roots of the characteristic polynomial of your equation . If s is Laplace transform argument  then

p(s)=s**3+a(s**2)+bs+cexp(-tau *s)=0

Then replace s=iw ( i= imaginary unit), i**=-1. Then,

-i*w(**3)-a*(w**2)+b*w+c*cos(-tau*w)+i*c*sin(tau*w)=0

Then , look if there is some compatibility with a solution +-w ( with  nonzero w)  ( an elementary osciallation) by zeroing both the real and imaginary parts of the above equation.  However, if c=0 ( no delay influence) then there is a solution with w=0 and an imaginary one +-w ( cancelling that one) associated with the roots of s**2+a*s+b=0 as it is well known.