Stability of delay differential equations

Question

I have encountered a 2-dimensional system of differential equations. One of them is a delay differential equation (DDE). Can anybody explain to me how to analyze the stability of a DDE?

Answer 1

In a nutshell, to analyse the stability of a DDE \begin{equation} \dot{x}=A_0 x(t)-\sum_{i=1}^m A_i x(t-\tau_i), \end{equation} with $A_0$ and $A_i$ are $n\times n$ matrices, you insert a solution of the form $x=e^{\lambda t} v$. This leads to a characteristic equation \begin{equation} \rm{det} \Delta(\lambda)=0 \end{equation} where \begin{equation} \Delta(\lambda)=\lambda I-A_0-\sum_{i=1}^m A_i e^{-\lambda_i \tau_i}. \end{equation} The characteristic equation is a transcendental equation with infinitely many characteristic roots. The stability of the DDE is set by the spectral abscissa, that is, the root with the largest real part. Once that root passes through the imaginary axes ($\rm{Re}(\lambda_i)>0$), the system is unstable. In general, the characteristic roots need to be determined numerically. There are standard tools for this, for instance the DDE-BIFTOOL.

Have a look at

• Michiels and Niculescu - Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach
• Smith - An Introduction to Delay Differential Equations with Applications to the Life Sciences
• Breda, Maset, Vermiglio - Stability of Linear Delay Differential Equations A Numerical Approach with MATLAB