Periodicity and Stability in a Price Model with Two Delays

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