Global behavior in nonlinear systems with delayed feedback

The problem of global stability in scalar delay differential equations of the form x˙(t)=f(x(t-τ))-g(x(t)) is studied. Functions f and g are continuous and such that the equation assumes a unique equilibrium. Two types of the sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size τ of the delay. Delay independent stability conditions make use of the global stability in the limiting (as τ→∞) difference equation g(x_n+1)=f(x_n): the latter always implying the global stability in the differential equation for all values of the delay τ⩾0. The delay dependent conditions involve the global attractivity in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f and g, and the delay τ