# 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 & Continuous Dynamical Systems - A, 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.  doi: 10.1137/120904226.  Google Scholar [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.  doi: 10.1137/0117064.  Google Scholar [3] M. Bartha, On stability properties for neutral differential equations with state-dependent delay,, Differential Equations Dynam. Systems, 7 (1999), 197.   Google Scholar [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.  doi: 10.1007/s10884-004-4285-1.  Google Scholar [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.   Google Scholar [6] R. D. Driver, Existence and stability of solutions of a delay-differential system,, Arch. Rational Mech. Anal., 10 (1962), 401.  doi: 10.1007/BF00281203.  Google Scholar [7] A. Erdélyi, A delay differential equation model of oscillations of exchange rates, 2003,, Diploma thesis., ().   Google Scholar [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.  doi: 10.1006/jdeq.1994.1019.  Google Scholar [9] J. R. Haddock, T. Krisztin and J. H. Wu, Asymptotic equivalence of neutral and infinite retarded differential equations,, Nonlinear Anal., 14 (1990), 369.  doi: 10.1016/0362-546X(90)90171-C.  Google Scholar [10] J. Hale, Theory of Functional Differential Equations,, 2nd edition, (1977).   Google Scholar [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.   Google Scholar [12] F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations,, J. Differential Equations, 37 (1980), 141.  doi: 10.1016/0022-0396(80)90093-5.  Google Scholar [13] J. Kato, On Liapunov-Razumikhin type theorems for functional differential equations,, Funkcial. Ekvac., 16 (1973), 225.   Google Scholar [14] T. Krisztin, On stability properties for one-dimensional functional-differential equations,, Funkcial. Ekvac., 34 (1991), 241.   Google Scholar [15] J. C. Lillo, Oscillatory solutions of the equation $y'(x)=m(x)y(x-n(x))$,, J. Differential Equations, 6 (1969), 1.  doi: 10.1016/0022-0396(69)90114-4.  Google Scholar [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.   Google Scholar [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.  doi: 10.1006/jdeq.1996.0037.  Google Scholar [18] A. D. Myškis, Lineĭnye Differencial’nye Uravneniya s Zapazdyvayuščim Argumentom (in Russian) [Linear Differential Equations with Retarded Argument],, Gosudarstv. Izdat. Tehn.-Teor. Lit., (1951).   Google Scholar [19] O. J. Staffans, A neutral FDE with stable $D$-operator is retarded,, J. Differential Equations, 49 (1983), 208.  doi: 10.1016/0022-0396(83)90012-8.  Google Scholar [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.  doi: 10.1007/s10884-012-9245-6.  Google Scholar [21] H.-O. Walther, Convergence to square waves for a price model with delay,, Discrete Contin. Dyn. Syst., 13 (2005), 1325.  doi: 10.3934/dcds.2005.13.1325.  Google Scholar [22] H.-O. Walther, Bifurcation of periodic solutions with large periods for a delay differential equation,, Ann. Mat. Pura Appl. (4), 185 (2006), 577.  doi: 10.1007/s10231-005-0170-8.  Google Scholar [23] E. M. Wright, A non-linear difference-differential equation,, J. Reine Angew. Math., 194 (1955), 66.   Google Scholar [24] J. A. Yorke, Asymptotic stability for one dimensional differential-delay equations,, J. Differential Equations, 7 (1970), 189.  doi: 10.1016/0022-0396(70)90132-4.  Google Scholar

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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.  doi: 10.1137/120904226.  Google Scholar [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.  doi: 10.1137/0117064.  Google Scholar [3] M. Bartha, On stability properties for neutral differential equations with state-dependent delay,, Differential Equations Dynam. Systems, 7 (1999), 197.   Google Scholar [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.  doi: 10.1007/s10884-004-4285-1.  Google Scholar [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.   Google Scholar [6] R. D. Driver, Existence and stability of solutions of a delay-differential system,, Arch. Rational Mech. Anal., 10 (1962), 401.  doi: 10.1007/BF00281203.  Google Scholar [7] A. Erdélyi, A delay differential equation model of oscillations of exchange rates, 2003,, Diploma thesis., ().   Google Scholar [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.  doi: 10.1006/jdeq.1994.1019.  Google Scholar [9] J. R. Haddock, T. Krisztin and J. H. Wu, Asymptotic equivalence of neutral and infinite retarded differential equations,, Nonlinear Anal., 14 (1990), 369.  doi: 10.1016/0362-546X(90)90171-C.  Google Scholar [10] J. Hale, Theory of Functional Differential Equations,, 2nd edition, (1977).   Google Scholar [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.   Google Scholar [12] F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations,, J. Differential Equations, 37 (1980), 141.  doi: 10.1016/0022-0396(80)90093-5.  Google Scholar [13] J. Kato, On Liapunov-Razumikhin type theorems for functional differential equations,, Funkcial. Ekvac., 16 (1973), 225.   Google Scholar [14] T. Krisztin, On stability properties for one-dimensional functional-differential equations,, Funkcial. Ekvac., 34 (1991), 241.   Google Scholar [15] J. C. Lillo, Oscillatory solutions of the equation $y'(x)=m(x)y(x-n(x))$,, J. Differential Equations, 6 (1969), 1.  doi: 10.1016/0022-0396(69)90114-4.  Google Scholar [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.   Google Scholar [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.  doi: 10.1006/jdeq.1996.0037.  Google Scholar [18] A. D. Myškis, Lineĭnye Differencial’nye Uravneniya s Zapazdyvayuščim Argumentom (in Russian) [Linear Differential Equations with Retarded Argument],, Gosudarstv. Izdat. Tehn.-Teor. Lit., (1951).   Google Scholar [19] O. J. Staffans, A neutral FDE with stable $D$-operator is retarded,, J. Differential Equations, 49 (1983), 208.  doi: 10.1016/0022-0396(83)90012-8.  Google Scholar [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.  doi: 10.1007/s10884-012-9245-6.  Google Scholar [21] H.-O. Walther, Convergence to square waves for a price model with delay,, Discrete Contin. Dyn. Syst., 13 (2005), 1325.  doi: 10.3934/dcds.2005.13.1325.  Google Scholar [22] H.-O. Walther, Bifurcation of periodic solutions with large periods for a delay differential equation,, Ann. Mat. Pura Appl. (4), 185 (2006), 577.  doi: 10.1007/s10231-005-0170-8.  Google Scholar [23] E. M. Wright, A non-linear difference-differential equation,, J. Reine Angew. Math., 194 (1955), 66.   Google Scholar [24] J. A. Yorke, Asymptotic stability for one dimensional differential-delay equations,, J. Differential Equations, 7 (1970), 189.  doi: 10.1016/0022-0396(70)90132-4.  Google Scholar
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