Global stability of a price model with multiple delays

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

Global stability of a price model with multiple delays

Ábel Garab ^1, , Veronika Kovács ^2, and Tibor Krisztin ^3,

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

Abstract
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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.

Keywords: multiple delay, infinite delay, price model, Delay differential equation, stable $D$ operator., global stability, neutral equation.

Mathematics Subject Classification: Primary: 34K20, 91B25; Secondary: 34K12, 34K40, 34K0.

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.

show all references

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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