The researches on the stochastic dynamics based on the large deviation theory

ZHU Jinjie; CHEN Zhen; KONG Chen; LIU Xianbin

doi:10.6052/1000-0992-18-021

Volume 50 Issue 1

Oct. 2020

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ZHU Jinjie, CHEN Zhen, KONG Chen, LIU Xianbin. The researches on the stochastic dynamics based on the large deviation theory[J]. Advances in Mechanics, 2020, 50(1): 202010. doi: 10.6052/1000-0992-18-021

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ZHU Jinjie, CHEN Zhen, KONG Chen, LIU Xianbin. The researches on the stochastic dynamics based on the large deviation theory[J]. Advances in Mechanics, 2020, 50(1): 202010. doi: 10.6052/1000-0992-18-021

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The researches on the stochastic dynamics based on the large deviation theory

doi: 10.6052/1000-0992-18-021

ZHU Jinjie^{1,2,†
,
,},
CHEN Zhen^2,3,
KONG Chen^2,4,
LIU Xianbin^{2,*
,}

¹ School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
² State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
³ Department of Brain and Cognitive Sciences, University of Rochester, Rochester 14627, United States
⁴ Department Digital Method and Product, Envision Energy, Shanghai 200051, China

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Corresponding author: ZHU Jinjie; LIU Xianbin
Received Date: 2018-09-05
Publish Date: 2020-10-08

Abstract

Abstract

This paper introduces the basic ideas and concepts of large deviation theory and its application in the study of exit problems. Three critical indicators of exit problems are reviewed: mean first passage time, exit location distribution and most probable escape path. Among them, the characterization of the most probable escape path is a structural conundrum. For the mean first passage time, its relationship with quasi-potential is introduced, and it is applied to analyze the time matching mechanism in stochastic resonance and self-induced stochastic resonance. For the exit location distribution, the relevant algorithms to accelerate the Monte Carlo numerical simulation are discussed, and the probability evolution method is specially clarified with some interesting examples. For the study of the most probable escape path, several calculation methods are discussed, and the singularity of the topological structure and its dynamical implications of the Lagrangian manifold formed by the auxiliary Hamilton system trajectories are analyzed. Furthermore, the corrected action method under the condition of finite noise intensity is given. Finally, the prospect of some open problems for the application and development of large deviation theory is discussed.
- large deviation theory,
- exit problem,
- most probable escape path,
- mean first passage time,
- exit location distribution

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References(154)

References

[1]	陈朕 . 2018. 基于大偏差理论的几类非线性随机系统动力学行为研究. [博士论文]. 南京: 南京航空航天大学 (Chen Z . 2018. Dynamical behaviors of several nonlinear stochastic systems based on the large deviation theory. [PhD Thesis]. Nanjing: Nanjing University of Aeronautics and Astronautics).
[2]	孔琛 . 2018. 噪声扰动下非线性动力系统离出行为研究. [博士论文]. 南京: 南京航空航天大学 (Kong C . 2018. On the exit problems in nonlinear dynamical systems driven by random perturbations. [PhD Thesis]. Nanjing: Nanjing University of Aeronautics and Astronautics).
[3]	孔琛, 刘先斌 . 2014. 受周期和白噪声激励的分段线性系统的吸引域与离出问题研究. 力学学报, 46:447-456 (Kong C, Liu X B . 2014. Research for attracting region and exit problem of a piecewise linear system under periodic and white noise excitations. Chinese Journal of Theoretical and Applied Mechanics, 46: 447-456).
[4]	刘先斌, 陈虬, 陈大鹏 . 1996. 非线性随机动力系统的稳定性和分岔研究. 力学进展, 26:437-452 (Liu X B, Chen Q, Chen D P . 1996. The researches on the stability and bifurcation of nonlinear stochastic dynamical systems. Advances in Mechanics, 26: 437-452).
[5]	孙建桥, 熊夫睿 . 2017. 非线性动力学系统全局分析之外的胞映射方法新发展. 力学进展, 47:150-177 (Sun J Q, Xiong F R . 2017. Cell mapping methods-beyond global analysis of nonlinear dynamic systems. Advances in Mechanics, 47: 150-177).
[6]	徐伟, 孙春艳, 孙建桥, 贺群 . 2013. 胞映射方法的研究和进展. 力学进展, 43:91-100 (Xu W, Sun C Y, Sun J Q, He Q . 2013. Development and study on cell mapping methods. Advances in Mechanics, 43: 91-100).
[7]	徐伟, 岳晓乐, 韩群 . 2017. 胞映射方法及其在非线性随机动力学中的应用. 动力学与控制学报, 15:200-208 (Xu W, Yue X L, Han Q . 2017. Cell mapping method and its applications in nonlinear stochastic dynamical systems. Journal of Dynamics and Control, 15: 200-208).
[8]	许勇, 裴斌, 徐伟 . 2017. 随机平均原理研究若干进展. 动力学与控制学报, 15:193-199 (Xu Y, Pei B, Xu W . 2017. Some recent developments of stochastic averaging principle. Journal of Dynamics and Control, 15: 193-199).
[9]	朱位秋 . 1987. 随机平均法及其应用. 力学进展, 17:342-352 (Zhu W Q . 1987. Stochastic averaging methods and their applications. Advances in Mechanics, 17: 342-352).
[10]	朱位秋 . 1992. 随机振动. 北京: 科学出版社 (Zhu W Q. 1992. Stochastic Vibration. Beijing: Science Press).
[11]	朱位秋 . 2003. 非线性随机动力学与控制——Hamilton理论体系框架. 北京: 科学出版社 (Zhu W Q. 2003. Nonlinear Stochastic Dynamics and Control—the Framework of Hamilton Theory. Beijing: Science Press).
[12]	朱位秋, 蔡国强 . 2017. 随机动力学引论. 北京: 科学出版社 (Zhu W Q, Cai G Q. 2017. Introduction to Stochastic Dynamics. Beijing: Science Press).
[13]	朱金杰 . 2018. 神经元同步、共振及离出问题研究. [博士论文]. 南京: 南京航空航天大学 (Zhu J J . 2018. Synchronization, resonance and exit problem for neuronal dynamical systems. [PhD Thesis]. Nanjing: Nanjing University of Aeronautics and Astronautics).
[14]	Adams D A, Sander L M, Ziff R M. 2010. The barrier method: A technique for calculating very long transition times. Journal of Chemical Physics, 133:124103.
[15]	Agazzi A, Dembo A, Eckmann J P. 2017. Large deviations theory for Markov jump models of chemical reaction networks. Annals of Applied Probability, 28:1821-1855.
[16]	Agranov T, Meerson B. 2018. Narrow Escape of Interacting Diffusing Particles. Physical Review Letters, 120:120601.
[17]	Allen R J, Warren P B, Ten Wolde P R. 2005. Sampling rare switching events in biochemical networks. Physical Review Letters, 94:018104.
[18]	Arnold V I. 1984. Catastrophe Theory. Berlin: Heidelberg: Springer-Verlag.
[19]	Bandrivskyy A, Beri S, Luchinsky D G. 2003 a. Noise-induced shift of singularities in the pattern of optimal paths. Physics Letters A, 314:386-391.
[20]	Bandrivskyy A, Beri S, Luchinsky D G, Mannella R, McClintock P V E. 2003 b. Fast Monte Carlo simulations and singularities in the probability distributions of nonequilibrium systems. Physical Review Letters, 90:210201.
[21]	Ben-Jacob E, Bergman D J, Matkowsky B J, Schuss Z. 1982. Lifetime of oscillatory steady-states. Physical Review A, 26:2805-2816.
[22]	Benzi R, Parisi G, Sutera A, Vulpiani A. 1982. Stochastic resonance in climatic change. Tellus, 34:10-16.
[23]	Benzi R, Sutera A, Vulpiani A. 1981. The mechanism of stochastic resonance. Journal of Physics A: Mathematical and General, 14:L453.
[24]	Beri S, Mannella R, Luchinsky D G, Silchenko A N, McClintock P V E. 2005. Solution of the boundary value problem for optimal escape in continuous stochastic systems and maps. Physical Review E, 72:036131.
[25]	Beri S, Mannella R, McClintock P V E. 2004. Dynamic importance sampling for the escape problem in nonequilibrium systems: Observation of shifts in optimal paths. Physical Review Letters, 92:020601.
[26]	Bernt ?ksendal. 2010. Stochastic Differential Equations. Berlin, Heidelberg: Springer-Verlag.
[27]	Bobrovsky B Z, Schluss Z. 1982. A singular perturbation method for computation of the mean first passage time in a nonlinear filter. SIAM J. Appl. Math., 42:174-187.
[28]	Bouchet F, Laurie J, Zaboronski O. 2014. Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations. Journal of Statistical Physics, 156:1066-1092.
[29]	Bray A J, McKane A J. 1989. Instanton calculation of the escape rate for activation over a potential barrier driven by colored noise. Physical Review Letters, 62:493-496.
[30]	Bressloff P, Newby J. 2013. Stochastic models of intracellular transport. Reviews of Modern Physics, 85:135-196.
[31]	Cai R, Chen X, Duan J, Kurths J, Li X. 2017. Lévy noise-induced escape in an excitable system. Journal of Statistical Mechanics: Theory and Experiment, 2017: 063503.
[32]	Cameron M K. 2012. Finding the quasipotential for nongradient SDEs. Physica D: Nonlinear Phenomena, 241:1532-1550.
[33]	Chan H B, Dykman M I, Stambaugh C. 2008. Paths of fluctuation induced switching. Physical Review Letters, 100:130602.
[34]	Chatterjee M, Robert M E. 2001. Noise enhances modulation sensitivity in cochlear implant listeners: Stochastic resonance in a prosthetic sensory system? JARO - Journal of the Association for Research in Otolaryngology, 2:159-171.
[35]	Chen L C, Deng M L, Zhu W Q. 2009. First passage failure of quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations. Acta Mechanica, 206:133-148.
[36]	Chen Z, Li Y, Liu X. 2016. Noise induced escape from a nonhyperbolic chaotic attractor of a periodically driven nonlinear oscillator. Chaos, 26:935-992.
[37]	Chen L, Li Z, Zhuang Q, Zhu W. 2013. First-passage failure of single-degree-of-freedom nonlinear oscillators with fractional derivative. Journal of Vibration and Control, 19:2154-2163.
[38]	Chen Z, Liu X. 2016. Patterns and singular features of extreme fluctuational paths of a periodically driven system. Physics Letters A, 380:1953-1958.
[39]	Chen Z, Liu X. 2017 a. Noise induced transitions and topological study of a periodically driven system. Communications in Nonlinear Science and Numerical Simulation, 48:454-461.
[40]	Chen Z, Liu X. 2017 b. Subtle escaping modes and subset of patterns from a nonhyperbolic chaotic attractor. Physical Review E, 95:012208.
[41]	Chen L, Zhu W Q. 2010 a. First passage failure of quasi non-integrable generalized Hamiltonian systems. Archive of Applied Mechanics, 80:883-893.
[42]	Chen L C, Zhu W Q. 2010 b. Reliability of quasi integrable generalized Hamiltonian systems. Probabilistic Engineering Mechanics, 25:61-66.
[43]	Chen L, Zhu W. 2010 c. First passage failure of dynamical power systems under random perturbations. Science China Technological Sciences, 53:2495-2500.
[44]	Chen Z, Zhu J, Liu X. 2017. Crossing the quasi-threshold manifold of a noise-driven excitable system. Proceedings of the Royal Society A, 473:20170058.
[45]	Cohen J, Lewis R. 1967. A ray method for the asymptotic solution of the diffusion equation. IMA J Appl Math, 3:266-290.
[46]	Crandall M G, Evans L C, Lions P L. 1984. Some properties of viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society, 282:487-502.
[47]	Crandall M G, Lions P L. 1983. Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society, 277:1-42.
[48]	Crooks G E, Chandler D. 2001. Efficient transition path sampling for nonequilibrium stochastic dynamics. Physical Review E, 64:026109.
[49]	Dahiya D, Cameron M K. 2018. An ordered line integral method for computing the quasi-potential in the case of variable anisotropic diffusion. Physica D: Nonlinear Phenomena, 382-383:33-45.
[50]	Deng M, Zhu W. 2009. Some applications of stochastic averaging method for quasi Hamiltonian systems in physics. Science in China, Series G: Physics, Mechanics and Astronomy, 52:1213-1222.
[51]	Dykman M I. 2010. Poisson-noise-induced escape from a metastable state. Physical Review E, 81:051124.
[52]	Dykman M I, Luchinsky D G, McClintock P V E, Smelyanskiy V N. 1996. Corrals and critical behavior of the distribution of fluctuational paths. Physical Review Letters, 77:5229-5232.
[53]	Dykman M I, McClintock P V E, Smelyanski V N, Stein N D, Stocks N G. 1992. Optimal paths and the prehistory problem for large fluctuations in noise-driven systems. Physical Review Letters, 68:2718-2721.
[54]	Dykman M I, Millonas M M, Smelyanskiy V N. 1994. Observable and hidden singular features of large fluctuations in nonequilibrium systems. Physics Letters A, 195:53-58.
[55]	Einstein A. 1905. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der Physik, 322:549-560.
[56]	Ermentrout B. 1996. Type I membranes, phase resetting curves, and synchrony. Neural Computation, 8:979-1001.
[57]	Faradjian A K, Elber R. 2004. Computing time scales from reaction coordinates by milestoning. Journal of Chemical Physics, 120:10880-10889.
[58]	Fenichel N. 1979. Geometric singular perturbation theory for ordinary differential equations. Journal of Differential Equations, 31:53-98.
[59]	Freidlin M I, Wentzell A D. 2012. Random Perturbations of Dynamical Systems. Berlin Heidelberg: Springer.
[60]	Gammaitoni L, H?nggi P, Jung P, Marchesoni F. 1998. Stochastic resonance. Reviews of Modern Physics, 70:223-287.
[61]	Glowacki D R, Paci E, Shalashilin D V. 2009. Boxed molecular dynamics: A simple and general technique for accelerating rare event kinetics and mapping free energy in large molecular systems. Journal of Physical Chemistry B, 113:16603-16611.
[62]	Grafke T, Grauer R, Sch?fer T. 2015. The instanton method and its numerical implementation in fluid mechanics. Journal of Physics A: Mathematical and Theoretical, 48:333001.
[63]	Gu X, Zhu W. 2014. A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations. Journal of Sound and Vibration, 333:2632-2642.
[64]	Guardia M, Seara T M, Teixeira M A. 2011. Generic bifurcations of low codimension of planar Filippov systems. Journal of Differential Equations, 250:1967-2023.
[65]	Gutkin B S, Jost J, Tuckwell H C. 2009. Inhibition of rhythmic neural spiking by noise: The occurrence of a minimum in activity with increasing noise. Naturwissenschaften, 96:1091-1097.
[66]	Han Q, Xu W, Yue X. 2016. Exit location distribution in the stochastic exit problem by the generalized cell mapping method. Chaos, Solitons and Fractals, 87:302-306.
[67]	Han Q, Xu W, Yue X, Zhang Y. 2015. First-passage time statistics in a bistable system subject to Poisson white noise by the generalized cell mapping method. Communications in Nonlinear Science and Numerical Simulation, 23:220-228.
[68]	Heymann M, Vanden-Eijnden E. 2008. The geometric minimum action method: A least action principle on the space of curves. Communications on Pure and Applied Mathematics, 61:1052-1117.
[69]	Higham D J. 2001. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43:525-546.
[70]	Holcman D, Schuss Z. 2004. Escape through a small opening: Receptor trafficking in a synaptic membrane. Journal of Statistical Physics, 117:975-1014.
[71]	Horsthemke W, Lefever R. 1984. Noise-induced Transitions: Theory and Applications in Physics, Chemistry, and Biology. Berlin: Springer Verlag.
[72]	Huang Y, Liu X. 2012. Stochastic stability of viscoelastic system under non-Gaussian colored noise excitation. Science China: Physics, Mechanics and Astronomy, 55:483-492.
[73]	Huang D, Yang J, Zhang J, Liu H. 2018. An improved adaptive stochastic resonance method for improving the efficiency of bearing faults diagnosis. Journal of Mechanical Engineering Science, 232:2352-2368.
[74]	Ji P, Lu W, Kurths J. 2018. Stochastic basin stability in complex networks. EPL, 122:1-6.
[75]	Khovanov I A, Khovanova N A, McClintock P V E. 2003. Noise-induced failures of chaos stabilization: Large fluctuations and their control. Physical Review E, 67:051102.
[76]	Klosek-Dygas M M, Matkowsky B J, Schuss Z. 2006. Stochastic stability of nonlinear oscillators. SIAM Journal on Applied Mathematics, 48:1115-1127.
[77]	Knessl C, Matkowsky B, Schuss Z, Tier C. 1985. An asymptotic theory of large deviations for Markov jump processes. SIAM Journal on Applied Mathematics, 45:1006-1028.
[78]	Kong C, Gao X, Liu X. 2016. On the global analysis of a piecewise linear system that is excited by a Gaussian white noise. Journal of Computational and Nonlinear Dynamics, 11:051029.
[79]	Kong C, Liu X. 2017. Noise-induced chaos in a piecewise linear system. International Journal of Bifurcation and Chaos, 27:1750137.
[80]	Kougioumtzoglou I A, Spanos P D. 2013. Response and first-passage statistics of nonlinear oscillators via a numerical path integral approach. Journal of Engineering Mechanics, 139:1207-1217.
[81]	Kramers H A. 1940. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7:284-304.
[82]	Kraut S, Feudel U. 2003 a. Enhancement of noise-induced escape through the existence of a chaotic saddle. Physical Review E, 67:015204.
[83]	Kraut S, Feudel U. 2003 b. Noise-induced escape through a chaotic saddle: lowering of the activation energy. Physica D: Nonlinear Phenomena, 181:222-234.
[84]	Kubo R. 1966. The fluctuation-dissipation theorem. Reports on Progress in Physics, 29:255-284.
[85]	Kuznetsov Y A, Rinaldi S, Gragnani A. 2003. One-parameter bifurcations in planar Filippov systems. International Journal of Bifurcation and Chaos, 13:2157-2188.
[86]	Lee DeVille R E, Vanden-Eijnden E, Muratov C B. 2005. Two distinct mechanisms of coherence in randomly perturbed dynamical systems. Physical Review E, 72:31105.
[87]	Li W, Chen L, Trisovic N, Cvekovic A, Zhao J. 2015. First passage of stochastic fractional derivative systems with power-form restoring force. International Journal of Non-Linear Mechanics, 71:83-88.
[88]	Lin Y K, Cai G Q. 1995. Probabilistic Structural Dynamics: Advanced Theory and Applications. Boston: McGraw-Hill.
[89]	Longtin A. 1997. Autonomous stochastic resonance in bursting neurons. Physical Review E, 55:868-876.
[90]	Lu S, He Q, Zhang H, Kong F. 2017. Rotating machine fault diagnosis through enhanced stochastic resonance by full-wave signal construction. Mechanical Systems and Signal Processing, 85:82-97.
[91]	Luchinsky D G, Beri S, Mannella R, McClintock P V E, Khovanov I A. 2002. Optimal fluctuations and the control of chaos. International Journal of Bifurcation and Chaos, 12:583-604.
[92]	Luchinsky D G, Maier R S, Mannella R, McClintock P V E, Stein D L. 1997. Experiments on critical phenomena in a noisy exit problem. Physical Review Letters, 79:3109-3112.
[93]	Luchinsky D G, Maier R S, Mannella R, McClintock P V E, Stein D L. 1999. Observation of saddle-point avoidance in noise-induced escape. Physical Review Letters, 82:1806.
[94]	Luchinsky D G, McClintock P V E. 1997. Irreversibility of classical fluctuations studied in analogue electrical circuits. Nature, 389:463-466.
[95]	Luchinsky D G, McClintock P V E, Dykman M I. 1998. Analogue studies of nonlinear systems. Reports on Progress in Physics, 61:889-997.
[96]	Lücken L, Yanchuk S, Popovych O V, Tass P A. 2013. Desynchronization boost by non-uniform coordinated reset stimulation in ensembles of pulse-coupled neurons. Frontiers in Computational Neuroscience, 7:63.
[97]	Ludwig D. 1975. Persistence of dynamical systems under random perturbations. SIAM Review, 17:605-640.
[98]	Maier R S, Stein D L. 1993 a. Effect of focusing and caustics on exit phenomena in systems lacking detailed balance. Physical Review Letters, 71:1783-1786.
[99]	Maier R S, Stein D L. 1993 b. Escape problem for irreversible systems. Physical Review E, 48:931-938.
[100]	Maier R S, Stein D L. 1996. A scaling theory of bifurcations in the symmetric weak-noise escape problem. Journal of Statistical Physics, 83:291-357.
[101]	Maier R S, Stein D L. 1997. Limiting exit location distributions in the stochastic exit problem. SIAM Journal on Applied Mathematics, 57:752-790.
[102]	Marshall J S. 2016. Analytical solutions for an escape problem in a disc with an arbitrary distribution of exit holes along its boundary. Journal of Statistical Physics, 165:920-952.
[103]	Matkowsky B J, Schuss Z. 1977. The exit problem for randomly perturbed dynamical systems. SIAM Journal on Applied Mathematics, 33:365-382.
[104]	Matkowsky B, Schuss Z. 1982. Diffusion across characteristic boundaries. SIAM Journal on Applied Mathematics, 42:822-834.
[105]	Matkowsky B J, Schuss Z, Ben-Jacob E. 1982. A singular perturbation approach to Kramers' duffusion problem. SIAM Journal on Applied Mathematics, 42:835-849.
[106]	Matkowsky B J, Schuss Z, Tier C. 1984. Uniform expansion of the transition rate in Kramers' problem. Journal of Statistical Physics, 35:443-456.
[107]	Menck P J, Heitzig J, Kurths J, Schellnhuber H J. 2014. How dead ends undermine power grid stability. Nature Communications, 5:3969.
[108]	Menck P J, Heitzig J, Marwan N, Kurths J. 2013. How basin stability complements the linear-stability paradigm. Nature Physics, 9:89-92.
[109]	Muratov C B, Vanden-Eijnden E, E W. 2005. Self-induced stochastic resonance in excitable systems. Physica D: Nonlinear Phenomena, 210:227-240.
[110]	Naeh T, Klosek M M, Matkowsky B J, Schuss Z. 1990. A direct approach to the exit problem. SIAM Journal on Applied Mathematics, 50:595-627.
[111]	Newby J M, Bressloff P C, Keener J P. 2013. Breakdown of fast-slow analysis in an excitable system with channel noise. Physical Review Letters, 111:128101.
[112]	Nolting B C, Abbott K C. 2016. Balls, cups, and quasi-potentials: Quantifying stability in stochastic systems. Ecology, 97:850-864.
[113]	Pei B, Xu Y, Yin G. 2017. Stochastic averaging for a class of two-time-scale systems of stochastic partial differential equations. Nonlinear Analysis, Theory, Methods and Applications, 160:159-176.
[114]	Pikovsky A S, Kurths J. 1997. Coherence resonance in a noise-driven excitable system. Physical Review Letters, 78:775-778.
[115]	Qiao Z, Lei Y, Lin J, Jia F. 2017. An adaptive unsaturated bistable stochastic resonance method and its application in mechanical fault diagnosis. Mechanical Systems and Signal Processing, 84:731-746.
[116]	Reimann P, Schmid G J, H鋘ggi P. 1999. Universal equivalence of mean first-passage time and Kramers rate. Physical Review E, 60:R1-R4.
[117]	Rodrigo G, Stocks N G. 2018. Suprathreshold stochastic resonance behind cancer. Trends in Biochemical Sciences, 43:483-485.
[118]	Roy R V. 1993. Noise perturbations of nonlinear dynamical systems. Computational Stochastic Mechanics, 79:125-148.
[119]	Roy R V. 1994 a. Asymptotic analysis of first-passage problems. International Journal of Non-Linear Mechanics, 32:173-186.
[120]	Roy R V. 1994 b. Noise perturbations of a non-linear system with multiple steady states. International Journal of Non-Linear Mechanics, 29:755-773.
[121]	Roy R V. 1995. Noise-induced transitions in weakly nonlinear oscillators near resonance. Journal of Applied Mechanics, 62:496-504.
[122]	Schuecker J, Diesmann M, Helias M. 2015. Modulated escape from a metastable state driven by colored noise. Physical Review E, 92:052119.
[123]	Schultz P, Menck P J, Heitzig J, Kurths J. 2017. Potentials and limits to basin stability estimation. New Journal of Physics, 19:023005.
[124]	Schuss Z, Matkowsky B. 1979. The exit problem: A new approach to diffusion across potential barriers. SIAM Journal on Applied Mathematics, 36:604-623.
[125]	Schuss Z, Singer A, Holcman D. 2007. The narrow escape problem for diffusion in cellular microdomains. Proceedings of the National Academy of Sciences, 104:16098-16103.
[126]	Schuss Z, Spivak A. 1998. Where is the exit point? Chemical Physics, 235:227-242.
[127]	Sethian J A, Vladimirsky A. 2001. Ordered upwind methods for static Hamilton-Jacobi equations. Proceedings of the National Academy of Sciences, 98:11069-11074.
[128]	Sethian J A, Vladimirsky A. 2003. Ordered upwind methods for static Hamilton-Jacobi equations: Theory and algorithms. SIAM Journal on Numerical Analysis, 41:325-363.
[129]	Sidney Redner. 2001. A Guide to First-Passage Processes. Cambridge: Cambridge University Press.
[130]	Smelyanskiy V N, Dykman M I. 1997. Optimal control of large fluctuations. Physical Review E, 55:2516-2521.
[131]	Smelyanskiy V N, Dykman M I, Maier R S. 1997. Topological features of large fluctuations to the interior of a limit cycle. Physical Review E, 55:2369-2391.
[132]	Spivak A, Schuss Z. 2002 a. Analytical and numerical study of Kramers' exit problem I. Applied Mathematics E-Notes, 2:132-140.
[133]	Spivak A, Schuss Z. 2002 b. The exit distribution on the stochastic separatrix in Kramers' exit problem. SIAM Journal on Applied Mathematics, 62:1698-1711.
[134]	Spivak A, Schuss Z. 2003. Analytical and numerical study of Kramers' exit problem II. Applied Mathematics E-Notes, 3:147-155.
[135]	Stocks N G, Allingham D, Morse R P. 2002. The application of suprathreshold stochastic resonance to cochlear implant coding. Fluctuation and Noise Letters, 02:L169-L181.
[136]	Sun J Q, Hsu C S. 1988. First-passage time probability of non-linear stochastic systems by generalized cell mapping method. Journal of Sound and Vibration, 124:233-248.
[137]	Tél T, Lai Y C. 2010. Quasipotential approach to critical scaling in noise-induced chaos. Physical Review E, 81:56208.
[138]	Tél T, Lai Y C, Gruiz M. 2008. Noise-induced chaos: A consequence of long deterministic transients. International Journal of Bifurcation and Chaos, 18:509-520.
[139]	Touchette H. 2009. The large deviation approach to statistical mechanics. Physics Reports, 478:1-69.
[140]	Tuckwell H C, Jost J. 2012. Analysis of inverse stochastic resonance and the long-term firing of Hodgkin-Huxley neurons with Gaussian white noise. Physica A: Statistical Mechanics and Its Applications, 391:5311-5325.
[141]	Wang J. 2015. Landscape and flux theory of non-equilibrium dynamical systems with application to biology. Advances in Physics, 64:1-137.
[142]	Wang W, Yan Z, Liu X. 2017. The escape problem and stochastic resonance in a bistable system driven by fractional Gaussian noise. Physics Letters A, 381:2324-2336.
[143]	Weber J. 1956. Fluctuation dissipation theorem. Physical Review, 101:1620-1626.
[144]	Wechselberger M, Mitry J, Rinzel J. 2013. Canard theory and excitability. Lecture Notes in Mathematics, 2102: 89-132.
[145]	Wentzell A D, Freidlin M I. 1970. On small random perturbations of dynamical systems. Russian Mathematical Surveys, 25:1-55.
[146]	Whitney H. 1955. On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane. Annals of Mathematics, 62:374-410.
[147]	Wuehr M, Boerner J C, Pradhan C , et al. 2017. Stochastic resonance in the human vestibular system - Noise-induced facilitation of vestibulospinal reflexes. Brain Stimulation, 11:261-263.
[148]	Xu M. 2018. First-passage failure of linear oscillator with non-classical inelastic impact. Applied Mathematical Modelling, 54:284-297.
[149]	Xu Y, Guo R, Liu D, Zhang H, Duan J. 2013. Stochastic averaging principle for dynamical systems with fractional brownian motion. Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 19:1197-1212.
[150]	Yang S, Potter S F, Cameron M K. 2019. Computing the quasipotential for nongradient SDEs in 3D. Journal of Computational Physics, 379:325-350.
[151]	Young H P. 2015. The evolution of social norms. Annual Review of Economics, 7:359-387.
[152]	Zhu J, Chen Z, Liu X. 2018. Probability evolution method for exit location distribution. Physics Letters A, 382:771-775.
[153]	Zhu W Q, Huang Z L, Deng M L. 2002. Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems. International Journal of Non-Linear Mechanics, 37:1057-1071.
[154]	Zhu W Q, Wu Y J. 2003. First-passage time of Duffing oscillator under combined harmonic and white-noise excitations. Nonlinear Dynamics, 32:291-305.