Periodicity and Stability in a Price Model with Two Delays

• Thi Thuy Hoang ^[1] ; Tibor Krisztin ^[1]
  1. [1] University of Szeged

     University of Szeged

     Hungría

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
• Idioma: inglés
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• Resumen

  • We study the delay differential equation x (t) = a[x(t) − x(t − 1)] − g(x(t − τ)) where a > 0, τ > 0, and g : R u → u|u|κ ∈ R with κ > 0. This equation is a modification of the Brunovský–Erdélyi–Walther price model by incorporating a reaction delay τ > 0. For any a > 0 and 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 in (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), in contrast to the case τ = 0. For a ∈ (0, 1) and τ ∈ (0, 1], an R1 > 0 is constructed such that 0 attracts the ball of radius R1 with center at 0. Local asymptotic stability of the zero solution follows as well. It is also shown that, for any a ∈ (0, 1), the region of attraction of 0 tends to the whole phase space C([−1, 0],R) as τ → 0+ .