# American Institute of Mathematical Sciences

December  2016, 36(12): 6855-6871. doi: 10.3934/dcds.2016098

## Global stability of a price model with multiple delays

 1 MTA-SZTE Analysis and Stochastics Research Group, 1 Aradi vértanúk tere, Szeged, Hungary 2 Bolyai Intstitute, University of Szeged, 1 Aradi vértanúk tere, Szeged, Hungary 3 MTA-SZTE Analysis and Stochastics Research Group, Bolyai Intstitute, University of Szeged, 1 Aradi vértanúk tere, Szeged, Hungary

Received  March 2016 Revised  July 2016 Published  October 2016

Consider the delay differential equation \begin{equation*} \dot{x}(t)=a \Bigg(\sum_{i=1}^n b_i\big[x(t-s_i)- x(t-r_i)\big]\Bigg)-g(x(t)), \end{equation*} where $a>0$, $b_i>0$ and $0\leq s_i < r_i$ $(i\in \{1,\dots,n\})$ are parameters, $g\colon \mathbb{R} \to \mathbb{R}$ is an odd $C^1$ function with $g'(0)=0$, the map $(0,\infty)\ni \xi \mapsto g(\xi)/\xi\in\mathbb{R}$ is strictly increasing and $\sup_{\xi>0} g(\xi)/\xi>2a$. This equation can be interpreted as a price model, where $x(t)$ represents the price of an asset (e.g. price of share or commodity, currency exchange rate etc.) at time $t$. The first term on the right-hand side represents the positive response for the recent tendencies of the price and $-g(x(t))$ is responsible for the instantaneous negative feedback to the deviation from the equilibrium price.
We study the local and global stability of the unique, non-hyperbolic equilibrium point. The main result gives a sufficient condition for global asymptotic stability of the equilibrium. The region of attractivity is also estimated in case of local asymptotic stability.
Citation: Ábel Garab, Veronika Kovács, Tibor Krisztin. Global stability of a price model with multiple delays. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6855-6871. doi: 10.3934/dcds.2016098
##### References:
 [1] B. Bánhelyi, T. Csendes, T. Krisztin and A. Neumaier, Global attractivity of the zero solution for Wright's equation, SIAM J. Appl. Dyn. Syst., 13 (2014), 537-563. doi: 10.1137/120904226. [2] D. I. Barnea, A method and new results for stability and instability of autonomous functional differential equations, SIAM J. Appl. Math., 17 (1969), 681-697. doi: 10.1137/0117064. [3] M. Bartha, On stability properties for neutral differential equations with state-dependent delay, Differential Equations Dynam. Systems, 7 (1999), 197-220. [4] P. Brunovský, A. Erdélyi and H.-O. Walther, On a model of a currency exchange rate - local stability and periodic solutions, J. Dynam. Differential Equations, 16 (2004), 393-432. doi: 10.1007/s10884-004-4285-1. [5] P. Brunovský, A. Erdélyi and H.-O. Walther, Erratum to: "On a model of a currency exchange rate - local stability and periodic solutions'' [J. Dynam. Differential Equations 16 (2004), no. 2, 393-432; mr2105782], J. Dynam. Differential Equations, 20 (2008), 271-276, URL http://dx.doi.org/10.1007/s10884-006-9062-x. [6] R. D. Driver, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401-426. doi: 10.1007/BF00281203. [7] A. Erdélyi, A delay differential equation model of oscillations of exchange rates, 2003, Diploma thesis. [8] J. R. Haddock, T. Krisztin, J. Terjéki and J. H. Wu, An invariance principle of Lyapunov-Razumikhin type for neutral functional-differential equations, J. Differential Equations, 107 (1994), 395-417. doi: 10.1006/jdeq.1994.1019. [9] J. R. Haddock, T. Krisztin and J. H. Wu, Asymptotic equivalence of neutral and infinite retarded differential equations, Nonlinear Anal., 14 (1990), 369-377. doi: 10.1016/0362-546X(90)90171-C. [10] J. Hale, Theory of Functional Differential Equations, 2nd edition, Springer-Verlag, New York-Heidelberg, 1977, Applied Mathematical Sciences, Vol. 3. [11] A. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54. [12] F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations, 37 (1980), 141-183. doi: 10.1016/0022-0396(80)90093-5. [13] J. Kato, On Liapunov-Razumikhin type theorems for functional differential equations, Funkcial. Ekvac., 16 (1973), 225-239. [14] T. Krisztin, On stability properties for one-dimensional functional-differential equations, Funkcial. Ekvac., 34 (1991), 241-256, URL http://www.math.kobe-u.ac.jp/~fe/xml/mr1130462.xml. [15] J. C. Lillo, Oscillatory solutions of the equation $y'(x)=m(x)y(x-n(x))$, J. Differential Equations, 6 (1969), 1-35. doi: 10.1016/0022-0396(69)90114-4. [16] E. Liz, V. Tkachenko and S. Trofimchuk, Yorke and Wright 3/2-stability theorems from a unified point of view, Discrete Contin. Dyn. Syst., (2003), 580-589, Dynamical systems and differential equations (Wilmington, NC, 2002). [17] J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. [18] A. D. Myškis, Lineĭnye Differencial’nye Uravneniya s Zapazdyvayuščim Argumentom (in Russian) [Linear Differential Equations with Retarded Argument], Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1951. [19] O. J. Staffans, A neutral FDE with stable $D$-operator is retarded, J. Differential Equations, 49 (1983), 208-217. doi: 10.1016/0022-0396(83)90012-8. [20] E. Stumpf, On a differential equation with state-dependent delay: A center-unstable manifold connecting an equilibrium and a periodic orbit, J. Dynam. Differential Equations, 24 (2012), 197-248. doi: 10.1007/s10884-012-9245-6. [21] H.-O. Walther, Convergence to square waves for a price model with delay, Discrete Contin. Dyn. Syst., 13 (2005), 1325-1342. doi: 10.3934/dcds.2005.13.1325. [22] H.-O. Walther, Bifurcation of periodic solutions with large periods for a delay differential equation, Ann. Mat. Pura Appl. (4), 185 (2006), 577-611. doi: 10.1007/s10231-005-0170-8. [23] E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 194 (1955), 66-87. [24] J. A. Yorke, Asymptotic stability for one dimensional differential-delay equations, J. Differential Equations, 7 (1970), 189-202. doi: 10.1016/0022-0396(70)90132-4.

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##### References:
 [1] B. Bánhelyi, T. Csendes, T. Krisztin and A. Neumaier, Global attractivity of the zero solution for Wright's equation, SIAM J. Appl. Dyn. Syst., 13 (2014), 537-563. doi: 10.1137/120904226. [2] D. I. Barnea, A method and new results for stability and instability of autonomous functional differential equations, SIAM J. Appl. Math., 17 (1969), 681-697. doi: 10.1137/0117064. [3] M. Bartha, On stability properties for neutral differential equations with state-dependent delay, Differential Equations Dynam. Systems, 7 (1999), 197-220. [4] P. Brunovský, A. Erdélyi and H.-O. Walther, On a model of a currency exchange rate - local stability and periodic solutions, J. Dynam. Differential Equations, 16 (2004), 393-432. doi: 10.1007/s10884-004-4285-1. [5] P. Brunovský, A. Erdélyi and H.-O. Walther, Erratum to: "On a model of a currency exchange rate - local stability and periodic solutions'' [J. Dynam. Differential Equations 16 (2004), no. 2, 393-432; mr2105782], J. Dynam. Differential Equations, 20 (2008), 271-276, URL http://dx.doi.org/10.1007/s10884-006-9062-x. [6] R. D. Driver, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401-426. doi: 10.1007/BF00281203. [7] A. Erdélyi, A delay differential equation model of oscillations of exchange rates, 2003, Diploma thesis. [8] J. R. Haddock, T. Krisztin, J. Terjéki and J. H. Wu, An invariance principle of Lyapunov-Razumikhin type for neutral functional-differential equations, J. Differential Equations, 107 (1994), 395-417. doi: 10.1006/jdeq.1994.1019. [9] J. R. Haddock, T. Krisztin and J. H. Wu, Asymptotic equivalence of neutral and infinite retarded differential equations, Nonlinear Anal., 14 (1990), 369-377. doi: 10.1016/0362-546X(90)90171-C. [10] J. Hale, Theory of Functional Differential Equations, 2nd edition, Springer-Verlag, New York-Heidelberg, 1977, Applied Mathematical Sciences, Vol. 3. [11] A. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54. [12] F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations, 37 (1980), 141-183. doi: 10.1016/0022-0396(80)90093-5. [13] J. Kato, On Liapunov-Razumikhin type theorems for functional differential equations, Funkcial. Ekvac., 16 (1973), 225-239. [14] T. Krisztin, On stability properties for one-dimensional functional-differential equations, Funkcial. Ekvac., 34 (1991), 241-256, URL http://www.math.kobe-u.ac.jp/~fe/xml/mr1130462.xml. [15] J. C. Lillo, Oscillatory solutions of the equation $y'(x)=m(x)y(x-n(x))$, J. Differential Equations, 6 (1969), 1-35. doi: 10.1016/0022-0396(69)90114-4. [16] E. Liz, V. Tkachenko and S. Trofimchuk, Yorke and Wright 3/2-stability theorems from a unified point of view, Discrete Contin. Dyn. Syst., (2003), 580-589, Dynamical systems and differential equations (Wilmington, NC, 2002). [17] J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. [18] A. D. Myškis, Lineĭnye Differencial’nye Uravneniya s Zapazdyvayuščim Argumentom (in Russian) [Linear Differential Equations with Retarded Argument], Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1951. [19] O. J. Staffans, A neutral FDE with stable $D$-operator is retarded, J. Differential Equations, 49 (1983), 208-217. doi: 10.1016/0022-0396(83)90012-8. [20] E. Stumpf, On a differential equation with state-dependent delay: A center-unstable manifold connecting an equilibrium and a periodic orbit, J. Dynam. Differential Equations, 24 (2012), 197-248. doi: 10.1007/s10884-012-9245-6. [21] H.-O. Walther, Convergence to square waves for a price model with delay, Discrete Contin. Dyn. Syst., 13 (2005), 1325-1342. doi: 10.3934/dcds.2005.13.1325. [22] H.-O. Walther, Bifurcation of periodic solutions with large periods for a delay differential equation, Ann. Mat. Pura Appl. (4), 185 (2006), 577-611. doi: 10.1007/s10231-005-0170-8. [23] E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 194 (1955), 66-87. [24] J. A. Yorke, Asymptotic stability for one dimensional differential-delay equations, J. Differential Equations, 7 (1970), 189-202. doi: 10.1016/0022-0396(70)90132-4.
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