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
• Enlaces
  • Texto completo
• 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+ .

• Referencias bibliográficas
  • 1. Balázs, I., Krisztin, T.: Global Stability for Price Models with Delay. J. Dyn. Diff. Eq. 31, 1327–1339 (2019). https://doi.org/10.1007/s10884-017-9583-5
  • 2. Brunovský, P., Erdélyi, A., Walther, H.-O.: On a model of a currency exchange rate–local stability and periodic solutions. (2004) J. Dynam....
  • 3. Diekmann, O., Verduyn Lunel, S.M., van Gils, S.A, Walther, H.-O.: Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag,...
  • 4. Erdélyi, A.: A Delay Differential EquationModel of Oscillations of Exchange Rates, (2003). (Diploma Thesis)
  • 5. Garab, Á., Kovács, V., Krisztin, T.: Global stability of a price model with multiple delays. Discrete Cont. Dyn. Sys. 36, 6855–6871 (2016)
  • 6. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)
  • 7. Herz, A.V.M.: Solutions of ˙ x(t) = −g(x(t −1)) Approach the Kaplan-Yorke Orbits for Odd Sigmoid g. J. Differential Equations 118,...
  • 8. Kaplan, J.L., Yorke, J.A.: Ordinary Differential Equations Which Yield Periodic Solutions of Differential Delay Equations. J. Math. Anal....
  • 9. Kolmanovskii, V.,Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Math. Appl., vol. 463,...
  • 10. Krisztin, T.,Wu, J.: Monotone semiflows generated by neutral equations with different delays in neutral and retarded parts. Acta Math....
  • 11. Stumpf, E.: On a differential equation with state-dependent delay: a center-unstable manifold connecting an equilibrium and a periodic...
  • 12. Walther, H.-O.: The solution manifold and C1-smoothness for differential equations with statedependent delay. J. Diff. Equations 195,...
  • 13. Walther, H.-O.: Convergence to square waves for a price model with delay. Discrete Cont. Dyn. Sys. 13, 1325–1342 (2005)
  • 14. Walther, H.-O.: Bifurcation of periodic solutions with large periods for a delay differential equation. Ann. Mat. Pura Appl. 4(185), 577–611...