Mackey-Glass type delay differential equations near the boundary of absolute stability

Liz, E.; Trofimchuk, E.; Trofimchuk, S.

Mackey-Glass type delay differential equations near the boundary of absolute stability

Authors

Liz, E.

Trofimchuk, E.

Trofimchuk, S.

Description

For equations $ x'(t) = -x(t) + ζf(x(t-h)), x \in \R, f'(0)= -1, ζ> 0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and satisfies $xf(x) < 0$ for $x\not=0$, we prove convergence of all solutions to zero when both $ζ-1 >0$ and $h(ζ-1)^{1/8}$ are less than some constant (independent on $h,ζ$). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey-Glass type delay differential equations.
16 pages, 1 figure, accepted for publication in the Journal of Mathematical Analysis and Applications