Periodicity and Stability in a Price Model with Two Delays

We study the delay differential equation x'(t) = a [x(t) - x(t - 1)] - g(x(t - τ )) x ′ ( t ) = a [ x ( t ) - x ( t - 1 ) ] - g ( x ( t - τ ) ) where $$a>0$$ a > 0 , τ >0 τ > 0 , and g:R u↦ u |u|^κ ∈ R g : R ∋ u ↦ u | u | κ ∈ R with κ >0 κ > 0 . This equation is a modification of the Brunovský–Erdélyi–Walther price model by incorporating a reaction delay τ >0 τ > 0 . For any $$a>0$$ a > 0 and κ >0 κ > 0 , by the Kaplan–Yorke method and the homogeneity of the nonlinear function g , we find a countable and dense set of delays τ τ in (0,∞ ) ( 0 , ∞ ) for which there exists a periodic solution. A consequence is that global asymptotic stability of the zero solution cannot be expected if a∈ (0,1) a ∈ ( 0 , 1 ) , in contrast to the case <jats:alternati