A global stability criterion for scalar functional differential equations

2001/12/05 by E. Liz, Liz, E., V. Tkachenko +3
Mathematics · #34K20 #92D25 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:34K20 #msc:92D25

paper · pdf · doi:10.48550/arxiv.math/0112047

28 pages, 2 figures. Final version, to appear in the SIAM Journal on Mathematical Analysis

arxiv created 2003/04/10 · arxiv updated 2009/11/30

Abstract

We consider scalar delay differential equations x'(t) = -δx(t) + f(t,x_t) (*) with nonlinear f satisfying a sort of negative feedback condition combined with a boundedness condition. The well known Mackey-Glass type equations, equations satisfying the Yorke condition, equations with maxima are kept within our considerations. Here, we establish a criterion for the global asymptotical stability of a unique steady state to (*). As an example, we study Nicholson's blowflies equation, where our computations support Smith's conjecture about the equivalence of global and local asymptotical stability in this population model.

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