American Institute of Mathematical Sciences

Advanced
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

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 & 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

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.  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

[1]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[2]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[3]

C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603
[4]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[5]

A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701
[6]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[7]

Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678
[8]

Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139
[9]

Azmy S. Ackleh, Keng Deng. Stability of a delay equation arising from a juvenile-adult model. Mathematical Biosciences & Engineering, 2010, 7 (4) : 729-737. doi: 10.3934/mbe.2010.7.729
[10]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[11]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020392
[12]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220
[13]

Zhihua Liu, Pierre Magal. Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2271-2292. doi: 10.3934/dcdsb.2019227
[14]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020066
[15]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065
[16]

Hans-Otto Walther. Convergence to square waves for a price model with delay. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1325-1342. doi: 10.3934/dcds.2005.13.1325
[17]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[18]

Luciano Abadías, Carlos Lizama, Marina Murillo-Arcila. Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay. Communications on Pure & Applied Analysis, 2018, 17 (1) : 243-265. doi: 10.3934/cpaa.2018015
[19]

Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028
[20]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2020074

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences