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Quantitative logarithmic Sobolev inequalities and stability estimates
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 |
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.
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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