June  2018, 38(6): 3099-3138. doi: 10.3934/dcds.2018135

Exit time asymptotics for small noise stochastic delay differential equations

Faculty of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

* Corresponding author: David Lipshutz

Received  October 2017 Revised  January 2018 Published  April 2018

Fund Project: This research was supported in part by NSF grants DMS-1206772 and DMS-1148284

Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE). We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for other infinite-dimensional small noise stochastic equations.

Citation: David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
References:
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R. AzencottB. Geiger and W. Ott, Large deviations for Gaussian diffusions with delay, J. Stat. Phys., 170 (2018), 254-285. doi: 10.1007/s10955-017-1909-5. Google Scholar

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M. Boué and P. Dupuis, A variational representation for certain functions of Brownian motion, Ann. Probab., 26 (1998), 1641-1659. Google Scholar

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D. BratsunD. VolfsonL. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation, Proc. Natl. Acad. Sci. USA, 102 (2005), 14593-14598. doi: 10.1073/pnas.0503858102. Google Scholar

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A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61. Google Scholar

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A. BudhirajaP. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420. doi: 10.1214/07-AOP362. Google Scholar

[7]

A. BudhirajaP. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. H. Poincaré Probab. Statist., 47 (2011), 725-747. doi: 10.1214/10-AIHP382. Google Scholar

[8]

A. Budhiraja, P. Dupuis and M. Salins, Uniform large deviations principle for Banach space valued stochastic differential equations, preprint, arXiv: 1803.00648.Google Scholar

[9]

S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32 (2004), 1100-1139. doi: 10.1214/aop/1079021473. Google Scholar

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F. Chenal and M. Millet, Uniform large deviations for parabolic SPDEs and their applications, Stochastic Process. Appl., 72 (1997), 161-186. doi: 10.1016/S0304-4149(97)00091-4. Google Scholar

[11]

S.-N. Chow and H.-O. Walther, Characteristic multipliers and stability of symmetric periodic solutions of $\dot{x}(t) = g(x(t-1))$, Trans. Amer. Math. Soc., 307 (1988), 127-142. Google Scholar

[12]

S. Coombes and C. Laing, Delays in activity-based neural networks, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 1117-1129. doi: 10.1098/rsta.2008.0256. Google Scholar

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G. da Prato, Stochastic Equations in Infinite Dimensions, 2nd edition, Cambridge University Press, Cambridge, 2014. Google Scholar

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M. Day, Large deviations results for the exit problem with characteristic boundary, J. Math. Anal. Appl., 147 (1990), 134-153. doi: 10.1016/0022-247X(90)90389-W. Google Scholar

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M. Day, Some phenomena of the characteristic boundary exit problem, in Diffusion Processes and Related Problems in Analysis, Vol. 1 (Evanston, IL, 1989) (ed. R. Pinsky), Birkhäuser Boston, 22 (1990), 55–71. Google Scholar

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M. Day, Conditional exits for small noise diffusions with characteristic boundary, Ann. Probab., 20 (1992), 1385-1419. doi: 10.1214/aop/1176989696. Google Scholar

[17]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Berlin, 2010. Google Scholar

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P. Dupuis and H. J. Kushner, Large deviations for systems with small noise effects, and applications to stochastic systems theory, SIAM J. Control Optim., 24 (1986), 979-1008. doi: 10.1137/0324058. Google Scholar

[20]

P. Dupuis and H. J. Kushner, Stochastic systems with small noise, analysis and simulation; a phase locked loop example, SIAM J. Appl. Math., 47 (1987), 643-661. doi: 10.1137/0147043. Google Scholar

[21]

P. Dupuis and R. S. Ellis, A Weak Convergence Approach to Large Deviations, Wiley, New York, 1997. Google Scholar

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A. Eizenberg, The exit distribution for small random perturbations of dynamical systems with a repulsive type stationary point, Stochastics, 12 (1984), 251-275. doi: 10.1080/17442508408833304. Google Scholar

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A. Eizenberg and Y. Kifer, The asymptotic behavior of the principal eigenvalue in a singular perturbation problem with invariant boundaries, Probab. Theory Related Fields, 76 (1987), 439-476. doi: 10.1007/BF00960068. Google Scholar

[24]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, SpringerVerlag, New York, 1984. Google Scholar

[25]

J. K. Hale, Sufficient conditions for stability and instability of autonomous functional differential equations, J. Differential Equations, 1 (1965), 452-482. doi: 10.1016/0022-0396(65)90005-7. Google Scholar

[26]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. Google Scholar

[27]

W. Z. Huang, Generalization of Liapunov's theorem in a linear delay system, J. Math. Anal. Appl., 142 (1989), 83-94. doi: 10.1016/0022-247X(89)90166-2. Google Scholar

[28]

A. F. Ivanov and J. Losson, Stable rapidly oscillating solutions in delay differential equations with negative feedback, Differ. Integral Equ. Appl., 12 (1999), 811-832. Google Scholar

[29]

J. L. Kaplan and J. A. Yorke, On the stability of a periodic solution of a differential-delay equation, SIAM J. Math. Anal., 6 (1975), 268-282. doi: 10.1137/0506028. Google Scholar

[30]

J. L. Kaplan and J. A. Yorke, On the nonlinear differential delay equation $x'(t) = -f(x(t),x(t-1))$, J. Differential Equations, 23 (1977), 293-314. doi: 10.1016/0022-0396(77)90132-2. Google Scholar

[31]

Y. Kifer, The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point, Israel J. Math., 40 (1981), 74-96. doi: 10.1007/BF02761819. Google Scholar

[32]

A. N. Kolmogorov and S. F. Folmin, Elements of the Theory of Functions and Functional Analysis, Graylock, Rochester, 1957. Google Scholar

[33]

R. LangevinW. M. Oliva and J. C. F. de Oliveira, Retarded functional differential equations with white noise perturbations, Ann. Inst. H. Poincaré Phys. Théor, 55 (1991), 671-687. Google Scholar

[34]

J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036. Google Scholar

[35]

J. Mallet-Paret and G. 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. Google Scholar

[36]

J. Mallet-Paret and N. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Differential Equations, 250 (2011), 4037-4084. doi: 10.1016/j.jde.2010.10.024. Google Scholar

[37]

W. H. Mather, M. R. Bennet, J. Hasty and L. S. Tsimring, Delay-induced degrade-and-fire oscillations in small genetic circuits, Phys. Rev. Lett. , 102 (2009), 068105. doi: 10.1103/PhysRevLett.102.068105. Google Scholar

[38]

C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal., 80 (2013), 202-210. doi: 10.1016/j.na.2012.10.004. Google Scholar

[39]

S. -E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984. Google Scholar

[40]

S.-E. A. Mohammed and T. Zhang, Large deviations for stochastic systems with memory, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 881-893. doi: 10.3934/dcdsb.2006.6.881. Google Scholar

[41]

F. Paganini and Z. Wang, Global stability with time-delay in network congestion control, in Proceedings of the 41st IEEE Conference on Decision and Control, 2002, (2002), 3632–3637.Google Scholar

[42]

A. Papachristodoulou, Global stability analysis of TCP/AQM protocol for arbitrary networks with delay, in 43rd IEEE Conference on Decision and Control, 2004. CDC., (2004), 1029– 1034. doi: 10.1109/CDC.2004.1428823. Google Scholar

[43]

A. Papachristodoulou, J. C. Doyle and S. H. Low, Analysis of nonlinear delay differentiable equation models of TCP/AQM protocols using sums of squares, in 43rd IEEE Conference on Decision and Control, 2004. CDC. , (2004), 4684–4689.Google Scholar

[44]

M. Peet and S. Lall, Global stability analysis of a nonlinear model of internet congestion control with delay, IEEE Trans. Automat. Control, 52 (2007), 553-559. doi: 10.1109/TAC.2007.892379. Google Scholar

[45]

B. S. Razumikhin, On the stability of systems with a delay, Prikl. Mat. Mekh., 20 (1956), 500-512. Google Scholar

[46]

B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay, Automat. i Telemekh., 21 (1960), 740-749. Google Scholar

[47]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd edition, SpringerVerlag, New York, 1999. Google Scholar

[48]

L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus, 2nd edition, Cambridge University Press, Cambridge, 2000. Google Scholar

[49]

A. Roxin, N. Brunel and D. Hansel, Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks, Phys. Rev. Lett. , 94 (2005), 238103. doi: 10.1103/PhysRevLett.94.238103. Google Scholar

[50]

A. Roxin and E. Montbrió, How effective delays shape oscillatory dynamics in neuronal networks, Phys. D, 240 (2011), 323-345. doi: 10.1016/j.physd.2010.09.009. Google Scholar

[51]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. Google Scholar

[52]

R. Sowers, Large deviations for a reaction-diffusion equation with non-{G}aussian perturbations, Ann. Probab., 20 (1992), 504-537. doi: 10.1214/aop/1176989939. Google Scholar

[53]

D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dynam. Differential Equations, 20 (2008), 201-238. doi: 10.1007/s10884-006-9068-4. Google Scholar

[54]

D. Stoffer, Two results on stable rapidly oscillating solutions of delay differential equations, Dyn. Syst., 26 (2011), 169-188. doi: 10.1080/14689367.2011.553715. Google Scholar

[55]

S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math., 19 (1966), 261-286. doi: 10.1002/cpa.3160190303. Google Scholar

[56]

A. D. Ventcel and M. I. Freidlin, On small random perturbations of dynamical systems, Russian Math. Surveys, 25 (1970), 3-55. Google Scholar

[57]

A. D. Ventcel and M. I. Freidlin, Some problems concerning stability under small random perturbations, Theory Probab. Appl., 17 (1972), 269-283. Google Scholar

[58]

H. -O. Walther, The 2-dimensional attractor of $x'(t) = μ x(t)+f(x(t-1))$, Mem. Amer. Math. Soc., 113 (1995), ⅵ+76 pp. Google Scholar

[59]

H.-O. Walther, Contracting return maps for monotone delayed feedback, Discrete Contin. Dyn. Syst., 7 (2001), 259-274. doi: 10.3934/dcds.2001.7.259. Google Scholar

[60]

H. -O. Walther, Contracting returns maps for some delay differential equations, in Topics in Functional Differential and Difference Equations (Lisbon, 1999), (eds. T. Faria and P. Freitas), American Mathematical Society, 29 (2001), 349–360. Google Scholar

[61]

H.-O. Walther, Stable periodic motion of a delayed spring, Topol. Methods Nonlinear Anal., 21 (2003), 1-28. doi: 10.12775/TMNA.2003.001. Google Scholar

[62]

J. Wu, Stable phase-locked periodic solutions in a delay differential system, J. Differential Equations, 194 (2003), 237-286. doi: 10.1016/S0022-0396(03)00187-6. Google Scholar

[63]

X. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equations with bounded nonlinearity, J. Dynam. Differential Equations, 3 (1991), 515-540. doi: 10.1007/BF01049098. Google Scholar

[64]

X. Xie, The multiplier equation and its applications to S-solutions of a differential delay equation, J. Dynam. Differential Equations, 95 (1992), 259-280. doi: 10.1016/0022-0396(92)90032-I. Google Scholar

[65]

X. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity, J. Differential Equations, 103 (1993), 350-374. doi: 10.1006/jdeq.1993.1054. Google Scholar

show all references

References:
[1]

D. F. Anderson, A modified next reaction method for simulating chemical systems with time dependent propensities and delays, J. Chemical Phys. , 127 (2007), 214107. doi: 10.1063/1.2799998. Google Scholar

[2]

R. AzencottB. Geiger and W. Ott, Large deviations for Gaussian diffusions with delay, J. Stat. Phys., 170 (2018), 254-285. doi: 10.1007/s10955-017-1909-5. Google Scholar

[3]

M. Boué and P. Dupuis, A variational representation for certain functions of Brownian motion, Ann. Probab., 26 (1998), 1641-1659. Google Scholar

[4]

D. BratsunD. VolfsonL. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation, Proc. Natl. Acad. Sci. USA, 102 (2005), 14593-14598. doi: 10.1073/pnas.0503858102. Google Scholar

[5]

A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61. Google Scholar

[6]

A. BudhirajaP. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420. doi: 10.1214/07-AOP362. Google Scholar

[7]

A. BudhirajaP. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. H. Poincaré Probab. Statist., 47 (2011), 725-747. doi: 10.1214/10-AIHP382. Google Scholar

[8]

A. Budhiraja, P. Dupuis and M. Salins, Uniform large deviations principle for Banach space valued stochastic differential equations, preprint, arXiv: 1803.00648.Google Scholar

[9]

S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab., 32 (2004), 1100-1139. doi: 10.1214/aop/1079021473. Google Scholar

[10]

F. Chenal and M. Millet, Uniform large deviations for parabolic SPDEs and their applications, Stochastic Process. Appl., 72 (1997), 161-186. doi: 10.1016/S0304-4149(97)00091-4. Google Scholar

[11]

S.-N. Chow and H.-O. Walther, Characteristic multipliers and stability of symmetric periodic solutions of $\dot{x}(t) = g(x(t-1))$, Trans. Amer. Math. Soc., 307 (1988), 127-142. Google Scholar

[12]

S. Coombes and C. Laing, Delays in activity-based neural networks, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 1117-1129. doi: 10.1098/rsta.2008.0256. Google Scholar

[13]

G. da Prato, Stochastic Equations in Infinite Dimensions, 2nd edition, Cambridge University Press, Cambridge, 2014. Google Scholar

[14]

M. Day, Large deviations results for the exit problem with characteristic boundary, J. Math. Anal. Appl., 147 (1990), 134-153. doi: 10.1016/0022-247X(90)90389-W. Google Scholar

[15]

M. Day, Some phenomena of the characteristic boundary exit problem, in Diffusion Processes and Related Problems in Analysis, Vol. 1 (Evanston, IL, 1989) (ed. R. Pinsky), Birkhäuser Boston, 22 (1990), 55–71. Google Scholar

[16]

M. Day, Conditional exits for small noise diffusions with characteristic boundary, Ann. Probab., 20 (1992), 1385-1419. doi: 10.1214/aop/1176989696. Google Scholar

[17]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Berlin, 2010. Google Scholar

[18]

R. D. Driver, Existence and continuous dependence of solutions of a neutral functional differential equation, Arch. Ration. Mech. Anal., 19 (1965), 149-166. doi: 10.1007/BF00282279. Google Scholar

[19]

P. Dupuis and H. J. Kushner, Large deviations for systems with small noise effects, and applications to stochastic systems theory, SIAM J. Control Optim., 24 (1986), 979-1008. doi: 10.1137/0324058. Google Scholar

[20]

P. Dupuis and H. J. Kushner, Stochastic systems with small noise, analysis and simulation; a phase locked loop example, SIAM J. Appl. Math., 47 (1987), 643-661. doi: 10.1137/0147043. Google Scholar

[21]

P. Dupuis and R. S. Ellis, A Weak Convergence Approach to Large Deviations, Wiley, New York, 1997. Google Scholar

[22]

A. Eizenberg, The exit distribution for small random perturbations of dynamical systems with a repulsive type stationary point, Stochastics, 12 (1984), 251-275. doi: 10.1080/17442508408833304. Google Scholar

[23]

A. Eizenberg and Y. Kifer, The asymptotic behavior of the principal eigenvalue in a singular perturbation problem with invariant boundaries, Probab. Theory Related Fields, 76 (1987), 439-476. doi: 10.1007/BF00960068. Google Scholar

[24]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, SpringerVerlag, New York, 1984. Google Scholar

[25]

J. K. Hale, Sufficient conditions for stability and instability of autonomous functional differential equations, J. Differential Equations, 1 (1965), 452-482. doi: 10.1016/0022-0396(65)90005-7. Google Scholar

[26]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. Google Scholar

[27]

W. Z. Huang, Generalization of Liapunov's theorem in a linear delay system, J. Math. Anal. Appl., 142 (1989), 83-94. doi: 10.1016/0022-247X(89)90166-2. Google Scholar

[28]

A. F. Ivanov and J. Losson, Stable rapidly oscillating solutions in delay differential equations with negative feedback, Differ. Integral Equ. Appl., 12 (1999), 811-832. Google Scholar

[29]

J. L. Kaplan and J. A. Yorke, On the stability of a periodic solution of a differential-delay equation, SIAM J. Math. Anal., 6 (1975), 268-282. doi: 10.1137/0506028. Google Scholar

[30]

J. L. Kaplan and J. A. Yorke, On the nonlinear differential delay equation $x'(t) = -f(x(t),x(t-1))$, J. Differential Equations, 23 (1977), 293-314. doi: 10.1016/0022-0396(77)90132-2. Google Scholar

[31]

Y. Kifer, The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point, Israel J. Math., 40 (1981), 74-96. doi: 10.1007/BF02761819. Google Scholar

[32]

A. N. Kolmogorov and S. F. Folmin, Elements of the Theory of Functions and Functional Analysis, Graylock, Rochester, 1957. Google Scholar

[33]

R. LangevinW. M. Oliva and J. C. F. de Oliveira, Retarded functional differential equations with white noise perturbations, Ann. Inst. H. Poincaré Phys. Théor, 55 (1991), 671-687. Google Scholar

[34]

J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036. Google Scholar

[35]

J. Mallet-Paret and G. 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. Google Scholar

[36]

J. Mallet-Paret and N. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Differential Equations, 250 (2011), 4037-4084. doi: 10.1016/j.jde.2010.10.024. Google Scholar

[37]

W. H. Mather, M. R. Bennet, J. Hasty and L. S. Tsimring, Delay-induced degrade-and-fire oscillations in small genetic circuits, Phys. Rev. Lett. , 102 (2009), 068105. doi: 10.1103/PhysRevLett.102.068105. Google Scholar

[38]

C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal., 80 (2013), 202-210. doi: 10.1016/j.na.2012.10.004. Google Scholar

[39]

S. -E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984. Google Scholar

[40]

S.-E. A. Mohammed and T. Zhang, Large deviations for stochastic systems with memory, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 881-893. doi: 10.3934/dcdsb.2006.6.881. Google Scholar

[41]

F. Paganini and Z. Wang, Global stability with time-delay in network congestion control, in Proceedings of the 41st IEEE Conference on Decision and Control, 2002, (2002), 3632–3637.Google Scholar

[42]

A. Papachristodoulou, Global stability analysis of TCP/AQM protocol for arbitrary networks with delay, in 43rd IEEE Conference on Decision and Control, 2004. CDC., (2004), 1029– 1034. doi: 10.1109/CDC.2004.1428823. Google Scholar

[43]

A. Papachristodoulou, J. C. Doyle and S. H. Low, Analysis of nonlinear delay differentiable equation models of TCP/AQM protocols using sums of squares, in 43rd IEEE Conference on Decision and Control, 2004. CDC. , (2004), 4684–4689.Google Scholar

[44]

M. Peet and S. Lall, Global stability analysis of a nonlinear model of internet congestion control with delay, IEEE Trans. Automat. Control, 52 (2007), 553-559. doi: 10.1109/TAC.2007.892379. Google Scholar

[45]

B. S. Razumikhin, On the stability of systems with a delay, Prikl. Mat. Mekh., 20 (1956), 500-512. Google Scholar

[46]

B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay, Automat. i Telemekh., 21 (1960), 740-749. Google Scholar

[47]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd edition, SpringerVerlag, New York, 1999. Google Scholar

[48]

L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus, 2nd edition, Cambridge University Press, Cambridge, 2000. Google Scholar

[49]

A. Roxin, N. Brunel and D. Hansel, Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks, Phys. Rev. Lett. , 94 (2005), 238103. doi: 10.1103/PhysRevLett.94.238103. Google Scholar

[50]

A. Roxin and E. Montbrió, How effective delays shape oscillatory dynamics in neuronal networks, Phys. D, 240 (2011), 323-345. doi: 10.1016/j.physd.2010.09.009. Google Scholar

[51]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. Google Scholar

[52]

R. Sowers, Large deviations for a reaction-diffusion equation with non-{G}aussian perturbations, Ann. Probab., 20 (1992), 504-537. doi: 10.1214/aop/1176989939. Google Scholar

[53]

D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dynam. Differential Equations, 20 (2008), 201-238. doi: 10.1007/s10884-006-9068-4. Google Scholar

[54]

D. Stoffer, Two results on stable rapidly oscillating solutions of delay differential equations, Dyn. Syst., 26 (2011), 169-188. doi: 10.1080/14689367.2011.553715. Google Scholar

[55]

S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math., 19 (1966), 261-286. doi: 10.1002/cpa.3160190303. Google Scholar

[56]

A. D. Ventcel and M. I. Freidlin, On small random perturbations of dynamical systems, Russian Math. Surveys, 25 (1970), 3-55. Google Scholar

[57]

A. D. Ventcel and M. I. Freidlin, Some problems concerning stability under small random perturbations, Theory Probab. Appl., 17 (1972), 269-283. Google Scholar

[58]

H. -O. Walther, The 2-dimensional attractor of $x'(t) = μ x(t)+f(x(t-1))$, Mem. Amer. Math. Soc., 113 (1995), ⅵ+76 pp. Google Scholar

[59]

H.-O. Walther, Contracting return maps for monotone delayed feedback, Discrete Contin. Dyn. Syst., 7 (2001), 259-274. doi: 10.3934/dcds.2001.7.259. Google Scholar

[60]

H. -O. Walther, Contracting returns maps for some delay differential equations, in Topics in Functional Differential and Difference Equations (Lisbon, 1999), (eds. T. Faria and P. Freitas), American Mathematical Society, 29 (2001), 349–360. Google Scholar

[61]

H.-O. Walther, Stable periodic motion of a delayed spring, Topol. Methods Nonlinear Anal., 21 (2003), 1-28. doi: 10.12775/TMNA.2003.001. Google Scholar

[62]

J. Wu, Stable phase-locked periodic solutions in a delay differential system, J. Differential Equations, 194 (2003), 237-286. doi: 10.1016/S0022-0396(03)00187-6. Google Scholar

[63]

X. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equations with bounded nonlinearity, J. Dynam. Differential Equations, 3 (1991), 515-540. doi: 10.1007/BF01049098. Google Scholar

[64]

X. Xie, The multiplier equation and its applications to S-solutions of a differential delay equation, J. Dynam. Differential Equations, 95 (1992), 259-280. doi: 10.1016/0022-0396(92)90032-I. Google Scholar

[65]

X. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity, J. Differential Equations, 103 (1993), 350-374. doi: 10.1006/jdeq.1993.1054. Google Scholar

[1]

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