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

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

2001-11-30

arxiv.org/abs/math/0111318v2

Mathematics

Dynamical Systems

16 pages, 1 figure, accepted for publication in the Journal of Mathematical Analysis and Applications

Scientific paper

For equations $ x'(t) = -x(t) + \zeta f(x(t-h)), x \in \R, f'(0)= -1, \zeta > 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 $\zeta -1 >0$ and $h(\zeta-1)^{1/8}$ are less than some constant (independent on $h,\zeta$). 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.

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