Global geometry of the stable regions for two delay differential equations
The stable region for the two delay differential equation \[ \dot x(t) +ax(t)+bx(t-r)+cx(t-\sigma)=0,\quad r \geq 0,\;\sigma \geq 0, t \geq 0 \] is investigated. The previously obtained numerical results show that the boundary of a stable region may be a very complicated curve which has infinitely many kinks, therefore some interesting questions arise: What is the asymptotical behavior of the boundary of a stable region? Can it become chaotic? In the paper a complete geometrical description of the stable region in the \(r-\sigma\) plane is given. In particular it is proved that: (i) If a stable region is unstable, then its boundary will approach a straight line parallel to the \(r\)-axis or \(\sigma\)-axis as \(r+\sigma \to \infty\). (ii) If a half line in the first quadrant of the \(r-\sigma\) plane contains an unstable point, then the intersection of this line and the boundary of the stable region contains at most finitely many points and eventually the half line leaves the stable region.
global geometry for delay differential equations
scientific article; zbMATH DE number 484022