On characteristic roots and stability charts of delay differential equations

SUMMARY This work is devoted to the analytic study of the characteristic roots oftextitscalar autonomous delay differential equations with either real or complex coefficients. The focus is placed on the robust analysis of the position of the roots in the complex plane with respect to the variation of the coefficients, with the final aim of obtaining suitable representations for the relevant stability boundaries and charts. While the real case is almost standard (and known), the investigation of the complex case is not as immediate. Hence, a preliminary shift of the coefficients is proposed, which reduces the number of free parameters. This allows to extend the techniques used for the real case, also allowing for useful graphical visualization of the relevant stability charts. The present research is motivated on the basis of studying the stability of systems with delay.