November  2014, 19(9): 2889-2913. doi: 10.3934/dcdsb.2014.19.2889

Stochastically perturbed sliding motion in piecewise-smooth systems

1. 

Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

2. 

Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Received  April 2013 Revised  February 2014 Published  September 2014

Sliding motion is evolution on a switching manifold of a discontinuous, piecewise-smooth system of ordinary differential equations. In this paper we quantitatively study the effects of small-amplitude, additive, white Gaussian noise on stable sliding motion. For equations that are static in directions parallel to the switching manifold, the distance of orbits from the switching manifold approaches a quasi-steady-state density. From this density we calculate the mean and variance for the near sliding solution. Numerical results of a relay control system reveal that the noise may significantly affect the period and amplitude of periodic solutions with sliding segments.
Citation: D. J. W. Simpson, R. Kuske. Stochastically perturbed sliding motion in piecewise-smooth systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2889-2913. doi: 10.3934/dcdsb.2014.19.2889
References:
[1]

A. J. Van der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems,, Springer-Verlag, (2000). Google Scholar

[2]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, volume 18 of Lecture Notes in Applied and Computational Mathematics,, Springer-Verlag, (2004). doi: 10.1007/978-3-540-44398-8. Google Scholar

[3]

S. Banerjee and G. C. Verghese, Nonlinear Phenomena in Power Electronics,, IEEE Press, (2001). doi: 10.1109/9780470545393. Google Scholar

[4]

Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems,, World Scientific, (2003). Google Scholar

[5]

T. Puu and I. Sushko, editors, Business Cycle Dynamics: Models and Tools,, Springer-Verlag, (2006). Google Scholar

[6]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications,, Springer-Verlag, (2008). Google Scholar

[7]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems,, Springer, (2006). Google Scholar

[8]

B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems,, Phys. Reports, 392 (2004), 321. doi: 10.1016/j.physrep.2003.10.015. Google Scholar

[9]

A. S. Pikovsky and J. Kurths, Coherence resonance in a noise-driven excitable system,, Phys. Rev. Lett., 78 (1997), 775. doi: 10.1103/PhysRevLett.78.775. Google Scholar

[10]

L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance,, Rev. Modern Phys., 70 (1998), 223. doi: 10.1103/RevModPhys.70.223. Google Scholar

[11]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Kluwer Academic Publishers., (1988). doi: 10.1007/978-94-015-7793-9. Google Scholar

[12]

M. Wiercigroch and B. De Kraker, Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities,, Singapore, (2000). doi: 10.1142/9789812796301. Google Scholar

[13]

B. Brogliato, Nonsmooth Mechanics: Models, Dynamics and Control,, Springer-Verlag, (1999). Google Scholar

[14]

B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak and J. Wojewoda, Chaotic Mechanics in Systems with Impacts and Friction,, World Scientific, (1999). doi: 10.1142/9789812798565. Google Scholar

[15]

J. Awrejcewicz and C. Lamarque, Bifurcation and Chaos in Nonsmooth Mechanical Systems,, World Scientific, (2003). doi: 10.1142/9789812564801. Google Scholar

[16]

R. A. Ibrahim, Vibro-Impact Dynamics., volume 43 of Lecture Notes in Applied and Computational Mechanics,, Springer, (2009). doi: 10.1007/978-3-642-00275-5. Google Scholar

[17]

C. K. Tse, Complex Behavior of Switching Power Converters,, CRC Press, (2003). doi: 10.1109/JPROC.2002.1015006. Google Scholar

[18]

A. F. Filippov, Differential equations with discontinuous right-hand side., Mat. Sb., 51 (1960), 99. Google Scholar

[19]

P. Casini, O. Giannini and F. Vestroni, Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics,, Nonlinear Dyn., 46 (2006), 259. doi: 10.1007/s11071-006-9041-0. Google Scholar

[20]

A. C. J. Luo and B. C. Gegg, Stick and non-stick periodic motions in periodically forced oscillators with dry friction,, J. Sound Vib., 291 (2006), 132. doi: 10.1016/j.jsv.2005.06.003. Google Scholar

[21]

M. Johansson, Piecewise Linear Control Systems., volume 284 of Lecture Notes in Control and Information Sciencesm, Springer-Verlag, (2003). doi: 10.1007/3-540-36801-9. Google Scholar

[22]

M. di Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations,, Int J. Bifurcation Chaos, 11 (2001), 1121. doi: 10.1142/S0218127401002584. Google Scholar

[23]

Z. Schuss, Theory and Applications of Stochastic Differential Equations,, Wiley, (1980). Google Scholar

[24]

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,, Springer-Verlag, (1985). Google Scholar

[25]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems,, Springer, (2012). doi: 10.1007/978-3-642-25847-3. Google Scholar

[26]

J. Grasman and O. A. van Herwaarden, Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications,, Springer, (1999). doi: 10.1007/978-3-662-03857-4. Google Scholar

[27]

, Special Issue on Nonsmooth Systems,, Phys. D, 241 (2012). Google Scholar

[28]

M. Dutta, H. E. Nusse, E. Ott, J. A. Yorke and G. Yuan, Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems,, Phys. Rev. Lett., 83 (1999), 4281. doi: 10.1103/PhysRevLett.83.4281. Google Scholar

[29]

T. C. L. Griffin, Dynamics of Stochastic Nonsmooth Systems,, PhD thesis, (2005). Google Scholar

[30]

T. Griffin and S. Hogan, Dynamics of discontinuous systems with imperfections and noise,, in IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, (2005), 275. doi: 10.1007/1-4020-3268-4_26. Google Scholar

[31]

P. Glendinning, The border collision normal form with stochastic switching surface,, SIAM J. Appl. Dyn. Sys., 13 (2014), 181. doi: 10.1137/130931643. Google Scholar

[32]

M. A. Hassouneh, E. H. Abed and H. E. Nusse, Robust dangerous border-collision bifurcations in piecewise smooth systems,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.070201. Google Scholar

[33]

A. Ganguli and S. Banerjee, Dangerous bifurcation at border collision: When does it occur?,, Phys. Rev. E., 71 (2005). doi: 10.1103/PhysRevE.71.057202. Google Scholar

[34]

Y. Do, A mechanism for dangerous border collision bifurcations,, Chaos Solitons Fractals, 32 (2007), 352. doi: 10.1016/j.chaos.2006.07.018. Google Scholar

[35]

Y. Do and H. K. Baek, Dangerous border-collision bifurcations of a piecewise-smooth map,, Comm. Pure Appl. Anal., 5 (2006), 493. doi: 10.3934/cpaa.2006.5.493. Google Scholar

[36]

R. Wackerbauer, Noise-induced stabilization of one-dimensional discontinuous maps,, Phys. Rev. E, 58 (1998), 3036. Google Scholar

[37]

L. Zhang, P. Shi, C. Wang and H. Gao, Robust $H_\infty$ filtering for switched linear discrete-time systems with polytopic uncertainties,, Int. J. Adapt. Control Signal Process, 20 (2006), 291. doi: 10.1002/acs.901. Google Scholar

[38]

W. Zhang, J. Hu and J. Lian, Quadratic optimal control of switched linear stochastic systems,, Syst. Contr. Lett., 59 (2010), 736. doi: 10.1016/j.sysconle.2010.08.010. Google Scholar

[39]

P. H. E. Tiesinga, Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.041913. Google Scholar

[40]

A. B. Nordmark, Non-periodic motion caused by grazing incidence in impact oscillators,, J. Sound Vib., 2 (1991), 279. doi: 10.1016/0022-460X(91)90592-8. Google Scholar

[41]

H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick-slip oscillations,, Phys. D, 136 (2000), 280. doi: 10.1016/S0167-2789(99)00161-X. Google Scholar

[42]

M. H. Fredriksson and A. B. Nordmark, On normal form calculation in impact oscillators,, Proc. R. Soc. A, 456 (2000), 315. doi: 10.1098/rspa.2000.0519. Google Scholar

[43]

M. di Bernardo, C. J. Budd and A. R. Champneys, Normal form maps for grazing bifurcations in $n$-dimensional piecewise-smooth dynamical systems,, Phys. D, 160 (2001), 222. doi: 10.1016/S0167-2789(01)00349-9. Google Scholar

[44]

D. J. W. Simpson, J. Hogan and R. Kuske, Stochastic regular grazing bifurcations,, SIAM J. Appl. Dyn. Sys., 12 (2013), 533. doi: 10.1137/120884286. Google Scholar

[45]

M. F. Dimentberg and D. V. Iourtchenko, Random vibrations with impacts: A review,, Nonlinear Dyn., 36 (2004), 229. doi: 10.1023/B:NODY.0000045510.93602.ca. Google Scholar

[46]

M. F. Dimentberg and A. I. Menyailov, Response of a single-mass vibroimpact system to white-noise random excitation,, Z. Angew. Math. Mech., 59 (1979), 709. doi: 10.1002/zamm.19790591205. Google Scholar

[47]

N. Sri Namachchivaya and J. H. Park, Stochastic dynamics of impact oscillators,, J. Appl. Mech. Trans. ASME, 72 (2005), 862. doi: 10.1115/1.2041660. Google Scholar

[48]

J. Feng, W. Xu, H. Rong and R. Wang, Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations,, Int. J. Non-Linear Mech., 44 (2009), 51. doi: 10.1016/j.ijnonlinmec.2008.08.013. Google Scholar

[49]

V. F. Zhuravlev, A method for analyzing vibration-impact systems by means of special functions,, Mech. Solids, 11 (1976), 23. Google Scholar

[50]

P. D. Christofides and N. H. El-Farra, Control of Nonlinear and Hybrid Process Systems. Designs for Uncertainty, Constraints and Time-Delays,, Springer, (2005). Google Scholar

[51]

J. Raouf and H. Michalska, Robust stabilization of switched linear systems with Wiener process noise,, in 49th IEEE Conference on Decision and Control, (2010), 6493. doi: 10.1109/CDC.2010.5717694. Google Scholar

[52]

W. Feng and J.-F. Zhang, Stability analysis and stabilization control of multi-variable switched stochastic systems,, Automatica, 42 (2006), 169. doi: 10.1016/j.automatica.2005.08.016. Google Scholar

[53]

D. Chatterjee and D. Liberzon, On stability of stochastic switched systems,, in Proceedings of the 43rd IEEE Conference on Decision and Control., (2004), 4125. doi: 10.1109/CDC.2004.1429398. Google Scholar

[54]

E. Skafidas, R. J. Evans, A. V. Savkin and I. R. Petersen, Stability results for switched controller systems,, Automatica, 35 (1999), 553. doi: 10.1016/S0005-1098(98)00167-8. Google Scholar

[55]

P. Mhaskar, N. H. El-Farra and P. D. Christofides, Robust hybrid predictive control of nonlinear systems,, Automatica, 41 (2005), 209. doi: 10.1016/j.automatica.2004.08.020. Google Scholar

[56]

B. Hu, X. Xu, P.J. Antsaklis and A. N. Michel, Robust stabilizing control laws for a class of second-order switched systems,, Syst. Control Lett., 38 (1999), 197. doi: 10.1016/S0167-6911(99)00065-1. Google Scholar

[57]

Z. Sun, A robust stabilizing law for switched linear systems,, Int. J. Control, 77 (2004), 389. doi: 10.1080/00207170410001667468. Google Scholar

[58]

D. Liberzon, Switching in Systems and Control,, Birkhauser, (2003). doi: 10.1007/978-1-4612-0017-8. Google Scholar

[59]

S.-C. Tan, Y.-M. Lai and C. K. Tse, Sliding Mode Control of Switching Power Converters,, CRC Press, (2012). Google Scholar

[60]

J.-Q. Sun, Stochastic Dynamics and Control., volume 4 of Nonlinear Science and Complexity,, Elsevier, (2006). Google Scholar

[61]

Y. Niu, D. W. C. Ho and J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay,, Automatica, 41 (2005), 873. doi: 10.1016/j.automatica.2004.11.035. Google Scholar

[62]

L. Wu, D. W. C. Ho and C. W. Li, Stabilisation and performance synthesis for switched stochastic systems,, IET Control Theory Appl., 4 (2010), 1877. doi: 10.1049/iet-cta.2009.0179. Google Scholar

[63]

L. Wu, D. W. C. Ho and C. W. Li, Sliding mode control of switching hybrid systems with stochastic perturbation,, Syst. Contr. Lett., 60 (2011), 531. doi: 10.1016/j.sysconle.2011.04.007. Google Scholar

[64]

Y. Niu, D. W. C. Ho and X. Wang, Robust $H_\infty$ control for nonlinear stochastic systems: A sliding-mode approach,, IEEE Trans. Automat. Contr., 53 (2008), 1695. doi: 10.1109/TAC.2008.929376. Google Scholar

[65]

D. E. Stewart, Rigid-body dynamics with friction and impact,, SIAM Rev., 42 (2000), 3. doi: 10.1137/S0036144599360110. Google Scholar

[66]

M. Abadie, Dynamical simulation of rigid bodies: Modelling of frictional contact,, in Impacts in Mechanical Systems: Analysis and Modelling, (2000). doi: 10.1007/3-540-45501-9_2. Google Scholar

[67]

P. S. Goohpattader, S. Mettu and M. K. Chaudhury, Experimental investigation of the drift and diffusion of small objects on a surface subjected to a bias and an external white noise: Roles of Coulombic friction and hysteresis,, Langmuir, 25 (2009), 9969. doi: 10.1021/la901111u. Google Scholar

[68]

P. S. Goohpattader and M. K. Chaudhury, Diffusive motion with nonlinear friction: Apparently Brownian,, J. Chem. Phys., 133 (2010). doi: 10.1063/1.3460530. Google Scholar

[69]

P.-G. de Gennes, Brownian motion with dry friction,, J. Stat. Phys., 119 (2005), 953. doi: 10.1007/s10955-005-4650-4. Google Scholar

[70]

H. Hayakawa, Langevin equation with Coulomb friction,, Phys. D, 205 (2005), 48. doi: 10.1016/j.physd.2004.12.011. Google Scholar

[71]

H. Touchette, E. Van der Straeten and W. Just, Brownian motion with dry friction: Fokker-Planck approach,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/44/445002. Google Scholar

[72]

A. Baule, E. G. D. Cohen and H. Touchette, A path integral approach to random motion with nonlinear friction,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/2/025003. Google Scholar

[73]

A. Baule, H. Touchette and E. G. D. Cohen, Stick-slip motion of solids with dry friction subject to random vibrations and an external field,, Nonlinearity, 24 (2011), 351. doi: 10.1088/0951-7715/24/2/001. Google Scholar

[74]

H. Touchette, T. Prellberg and W. Just, Exact power spectra of Brownian motion with solid friction,, J. Phys. A: Math. Theor., 45 (2012). doi: 10.1088/1751-8113/45/39/395002. Google Scholar

[75]

D. J. W. Simpson and R. Kuske, Stochastic perturbations of periodic orbits with sliding, Submitted to: J. Nonlin. Sci., (2014). Google Scholar

[76]

A. Colombo, M. di Bernardo, S. J. Hogan and M. R. Jeffrey, Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems,, Phys. D, 241 (2012), 1845. doi: 10.1016/j.physd.2011.09.017. Google Scholar

[77]

M. di Bernardo, P. Kowalczyk and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings,, Phys. D, 170 (2002), 175. doi: 10.1016/S0167-2789(02)00547-X. Google Scholar

[78]

A. Colombo and M. R. Jeffrey, Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows,, SIAM J. Appl. Dyn. Sys., 10 (2011), 423. doi: 10.1137/100801846. Google Scholar

[79]

M. R. Jeffrey and A. Colombo, The two-fold singularity of discontinuous vector fields,, SIAM J. Appl. Dyn. Sys., 8 (2009), 624. doi: 10.1137/08073113X. Google Scholar

[80]

M. A. Teixeira, Stability conditions for discontinuous vector fields,, J. Differential Equations, 88 (1990), 15. doi: 10.1016/0022-0396(90)90106-Y. Google Scholar

[81]

T.-S. Chiang and S.-J. Sheu, Large deviation of small perturbation of some unstable systems,, Stoch. Anal. Appl., 15 (1997), 31. doi: 10.1080/07362999708809462. Google Scholar

[82]

R. Kuske and G. Papanicolaou, The invariant density of a chaotic dynamical system with small noise,, Phys. D, 120 (1998), 255. doi: 10.1016/S0167-2789(98)00085-2. Google Scholar

[83]

P. Hitczenko and G. S. Medvedev, Bursting oscillations induced by small noise,, SIAM J. Appl. Math., 69 (2009), 1359. doi: 10.1137/070711803. Google Scholar

[84]

Ya. Z. Tsypkin, Relay Control Systems,, Cambridge University Press, (1984). Google Scholar

[85]

G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems,, Prentice Hall, (2002). Google Scholar

[86]

R. C. Dorf and R. H. Bishop, Modern Control Systems,, Prentice Hall, (2001). doi: 10.1109/TSMC.1981.4308749. Google Scholar

[87]

K. J. Åström and R. M. Murray, Feedback Systems. An Introduction for Scientists and Engineers,, Princeton University Press, (2008). Google Scholar

[88]

K. H. Johansson, A. Rantzer and K. J. Åström, Fast switches in relay feedback systems,, Automatica, 35 (1999), 539. doi: 10.1016/S0005-1098(98)00160-5. Google Scholar

[89]

K. H. Johansson, A. E. Barabanov and K. J. Åström, Limit cycles with chattering in relay feedback systems,, IEEE Trans. Automat. Contr., 47 (2002), 1414. doi: 10.1109/TAC.2002.802770. Google Scholar

[90]

Y. Zhao, J. Feng and C. K. Tse, Discrete-time modeling and stability analysis of periodic orbits with sliding for switched linear systems,, IEEE Trans. Circuits Systems I Fund. Theory Appl., 57 (2010), 2948. doi: 10.1109/TCSI.2010.2050230. Google Scholar

[91]

F. Dercole, R. Ferrière, A. Gragnani and S. Rinaldi, Coevolution of slow-fast populations: Evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics,, Proc. R. Soc. B, 273 (2006), 983. doi: 10.1098/rspb.2005.3398. Google Scholar

[92]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models,, Theor. Popul. Biol., 72 (2007), 197. doi: 10.1016/j.tpb.2007.06.003. Google Scholar

[93]

J. A. Amador, G. Olivar and F. Angulo, Smooth and Filippov models of sustainable development: Bifurcations and numerical computations,, Differ. Equ. Dyn. Syst., 21 (2013), 173. doi: 10.1007/s12591-012-0138-2. Google Scholar

[94]

S. Tang, J. Liang, Y. Xiao and R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds,, SIAM J. Appl. Math., 72 (2012), 1061. doi: 10.1137/110847020. Google Scholar

[95]

R. Szalai and H. M. Osinga, Invariant polygons in systems with grazing-sliding,, Chaos, 18 (2008). doi: 10.1063/1.2904774. Google Scholar

[96]

M. Tanelli, G. Osorio, M. di Bernardo, S. M. Savaresi and A. Astolfi, Existence, stability and robustness analysis of limit cycles in hybrid anti-lock braking systems,, Int. J. Contr., 82 (2009), 659. doi: 10.1080/00207170802203598. Google Scholar

[97]

M. di Bernardo, K. H. Johansson, U. Jönsson and F. Vasca, On the robustness of periodic solutions in relay feedback systems,, in 15th Triennial World Congress, (2002). Google Scholar

[98]

F. Flandoli, Random Perturbations of PDEs and Fluid Dynamic Models., volume 2015 of Lecture Notes in Mathematics,, Springer, (2011). doi: 10.1007/978-3-642-18231-0. Google Scholar

[99]

Yu. V. Prokhorov and A. N. Shiryaev, Probability Theory III: Stochastic Calculus,, Springer, (1998). doi: 10.1007/978-3-662-03640-2. Google Scholar

[100]

E. D. Conway, Stochastic equations with discontinuous drift,, Trans. Am. Math. Soc., 157 (1971), 235. doi: 10.1090/S0002-9947-1971-0275532-6. Google Scholar

[101]

D. Stroock and S. R. S Varadhan, Diffusion processes with continuous coefficients. I,, Comm. Pure Appl. Math., 22 (1969), 345. doi: 10.1002/cpa.3160220304. Google Scholar

[102]

A. Ju. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations,, Math. USSR Sb., 39 (1981), 387. Google Scholar

[103]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift,, Probab. Theory Relat. Fields, 131 (2005), 154. doi: 10.1007/s00440-004-0361-z. Google Scholar

[104]

S. E. A. Mohammed, T. Nilssen and F. Proske, Sobolev differentiable stochastic flows for SDE's with singular coefficients: Applications to the transport equation,, Unpublished, (2012). Google Scholar

[105]

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications,, Springer-Verlag, (1984). doi: 10.1007/978-3-642-96807-5. Google Scholar

[106]

Z. Schuss, Theory and Applications of Stochastic Processes,, Springer, (2010). doi: 10.1007/978-1-4419-1605-1. Google Scholar

[107]

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers,, International Series in Pure and Applied Mathematics. McGraw-Hill, (1978). Google Scholar

[108]

R. Buckdahn, Y. Ouknine and M. Quincampoix, On limiting values of stochastic differential equations with small noise intensity tending to zero,, Bull. Sci. Math., 133 (2009), 229. doi: 10.1016/j.bulsci.2008.12.005. Google Scholar

[109]

O. Menoukeu-Pamen, T. Meyer-Brandis and F. Proske, A Gel'fand triple approach to the small noise problem for discontinuous ODE's,, Unpublished, (2011). Google Scholar

[110]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization,, Springer, (2008). Google Scholar

[111]

I. Karatzas and S. E. Shreve, Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control,, Ann. Prob., 12 (1984), 819. doi: 10.1214/aop/1176993230. Google Scholar

[112]

I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures,, Theory Prob. Appl., 5 (1960), 285. Google Scholar

[113]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar

[114]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Springer, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar

[115]

Z. Qian and W. Zheng, Sharp bounds for transition probability densities of a class of diffusions,, C.R. Acad. Sci. Paris, 335 (2002), 953. doi: 10.1016/S1631-073X(02)02579-7. Google Scholar

[116]

Z. Qian, F. Russo and W. Zheng, Comparison theorem and estimates for transition probability densities of diffusion processes,, Probab. Theory Relat. Fields., 127 (2003), 388. doi: 10.1007/s00440-003-0291-1. Google Scholar

[117]

M. Gradinaru, S. Herrmann and B. Roynette, A singular large deviations phenomenon,, Ann. Inst. Henri Poincaré, 37 (2001), 555. doi: 10.1016/S0246-0203(01)01075-5. Google Scholar

[118]

W. Zhang, Transition density of one-dimensional diffusion with discontinuous drift,, IEEE Trans. Automat. Contr., 35 (1990), 980. doi: 10.1109/9.58517. Google Scholar

show all references

References:
[1]

A. J. Van der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems,, Springer-Verlag, (2000). Google Scholar

[2]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, volume 18 of Lecture Notes in Applied and Computational Mathematics,, Springer-Verlag, (2004). doi: 10.1007/978-3-540-44398-8. Google Scholar

[3]

S. Banerjee and G. C. Verghese, Nonlinear Phenomena in Power Electronics,, IEEE Press, (2001). doi: 10.1109/9780470545393. Google Scholar

[4]

Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems,, World Scientific, (2003). Google Scholar

[5]

T. Puu and I. Sushko, editors, Business Cycle Dynamics: Models and Tools,, Springer-Verlag, (2006). Google Scholar

[6]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications,, Springer-Verlag, (2008). Google Scholar

[7]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems,, Springer, (2006). Google Scholar

[8]

B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems,, Phys. Reports, 392 (2004), 321. doi: 10.1016/j.physrep.2003.10.015. Google Scholar

[9]

A. S. Pikovsky and J. Kurths, Coherence resonance in a noise-driven excitable system,, Phys. Rev. Lett., 78 (1997), 775. doi: 10.1103/PhysRevLett.78.775. Google Scholar

[10]

L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance,, Rev. Modern Phys., 70 (1998), 223. doi: 10.1103/RevModPhys.70.223. Google Scholar

[11]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Kluwer Academic Publishers., (1988). doi: 10.1007/978-94-015-7793-9. Google Scholar

[12]

M. Wiercigroch and B. De Kraker, Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities,, Singapore, (2000). doi: 10.1142/9789812796301. Google Scholar

[13]

B. Brogliato, Nonsmooth Mechanics: Models, Dynamics and Control,, Springer-Verlag, (1999). Google Scholar

[14]

B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak and J. Wojewoda, Chaotic Mechanics in Systems with Impacts and Friction,, World Scientific, (1999). doi: 10.1142/9789812798565. Google Scholar

[15]

J. Awrejcewicz and C. Lamarque, Bifurcation and Chaos in Nonsmooth Mechanical Systems,, World Scientific, (2003). doi: 10.1142/9789812564801. Google Scholar

[16]

R. A. Ibrahim, Vibro-Impact Dynamics., volume 43 of Lecture Notes in Applied and Computational Mechanics,, Springer, (2009). doi: 10.1007/978-3-642-00275-5. Google Scholar

[17]

C. K. Tse, Complex Behavior of Switching Power Converters,, CRC Press, (2003). doi: 10.1109/JPROC.2002.1015006. Google Scholar

[18]

A. F. Filippov, Differential equations with discontinuous right-hand side., Mat. Sb., 51 (1960), 99. Google Scholar

[19]

P. Casini, O. Giannini and F. Vestroni, Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics,, Nonlinear Dyn., 46 (2006), 259. doi: 10.1007/s11071-006-9041-0. Google Scholar

[20]

A. C. J. Luo and B. C. Gegg, Stick and non-stick periodic motions in periodically forced oscillators with dry friction,, J. Sound Vib., 291 (2006), 132. doi: 10.1016/j.jsv.2005.06.003. Google Scholar

[21]

M. Johansson, Piecewise Linear Control Systems., volume 284 of Lecture Notes in Control and Information Sciencesm, Springer-Verlag, (2003). doi: 10.1007/3-540-36801-9. Google Scholar

[22]

M. di Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations,, Int J. Bifurcation Chaos, 11 (2001), 1121. doi: 10.1142/S0218127401002584. Google Scholar

[23]

Z. Schuss, Theory and Applications of Stochastic Differential Equations,, Wiley, (1980). Google Scholar

[24]

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,, Springer-Verlag, (1985). Google Scholar

[25]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems,, Springer, (2012). doi: 10.1007/978-3-642-25847-3. Google Scholar

[26]

J. Grasman and O. A. van Herwaarden, Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications,, Springer, (1999). doi: 10.1007/978-3-662-03857-4. Google Scholar

[27]

, Special Issue on Nonsmooth Systems,, Phys. D, 241 (2012). Google Scholar

[28]

M. Dutta, H. E. Nusse, E. Ott, J. A. Yorke and G. Yuan, Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems,, Phys. Rev. Lett., 83 (1999), 4281. doi: 10.1103/PhysRevLett.83.4281. Google Scholar

[29]

T. C. L. Griffin, Dynamics of Stochastic Nonsmooth Systems,, PhD thesis, (2005). Google Scholar

[30]

T. Griffin and S. Hogan, Dynamics of discontinuous systems with imperfections and noise,, in IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, (2005), 275. doi: 10.1007/1-4020-3268-4_26. Google Scholar

[31]

P. Glendinning, The border collision normal form with stochastic switching surface,, SIAM J. Appl. Dyn. Sys., 13 (2014), 181. doi: 10.1137/130931643. Google Scholar

[32]

M. A. Hassouneh, E. H. Abed and H. E. Nusse, Robust dangerous border-collision bifurcations in piecewise smooth systems,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.070201. Google Scholar

[33]

A. Ganguli and S. Banerjee, Dangerous bifurcation at border collision: When does it occur?,, Phys. Rev. E., 71 (2005). doi: 10.1103/PhysRevE.71.057202. Google Scholar

[34]

Y. Do, A mechanism for dangerous border collision bifurcations,, Chaos Solitons Fractals, 32 (2007), 352. doi: 10.1016/j.chaos.2006.07.018. Google Scholar

[35]

Y. Do and H. K. Baek, Dangerous border-collision bifurcations of a piecewise-smooth map,, Comm. Pure Appl. Anal., 5 (2006), 493. doi: 10.3934/cpaa.2006.5.493. Google Scholar

[36]

R. Wackerbauer, Noise-induced stabilization of one-dimensional discontinuous maps,, Phys. Rev. E, 58 (1998), 3036. Google Scholar

[37]

L. Zhang, P. Shi, C. Wang and H. Gao, Robust $H_\infty$ filtering for switched linear discrete-time systems with polytopic uncertainties,, Int. J. Adapt. Control Signal Process, 20 (2006), 291. doi: 10.1002/acs.901. Google Scholar

[38]

W. Zhang, J. Hu and J. Lian, Quadratic optimal control of switched linear stochastic systems,, Syst. Contr. Lett., 59 (2010), 736. doi: 10.1016/j.sysconle.2010.08.010. Google Scholar

[39]

P. H. E. Tiesinga, Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.041913. Google Scholar

[40]

A. B. Nordmark, Non-periodic motion caused by grazing incidence in impact oscillators,, J. Sound Vib., 2 (1991), 279. doi: 10.1016/0022-460X(91)90592-8. Google Scholar

[41]

H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick-slip oscillations,, Phys. D, 136 (2000), 280. doi: 10.1016/S0167-2789(99)00161-X. Google Scholar

[42]

M. H. Fredriksson and A. B. Nordmark, On normal form calculation in impact oscillators,, Proc. R. Soc. A, 456 (2000), 315. doi: 10.1098/rspa.2000.0519. Google Scholar

[43]

M. di Bernardo, C. J. Budd and A. R. Champneys, Normal form maps for grazing bifurcations in $n$-dimensional piecewise-smooth dynamical systems,, Phys. D, 160 (2001), 222. doi: 10.1016/S0167-2789(01)00349-9. Google Scholar

[44]

D. J. W. Simpson, J. Hogan and R. Kuske, Stochastic regular grazing bifurcations,, SIAM J. Appl. Dyn. Sys., 12 (2013), 533. doi: 10.1137/120884286. Google Scholar

[45]

M. F. Dimentberg and D. V. Iourtchenko, Random vibrations with impacts: A review,, Nonlinear Dyn., 36 (2004), 229. doi: 10.1023/B:NODY.0000045510.93602.ca. Google Scholar

[46]

M. F. Dimentberg and A. I. Menyailov, Response of a single-mass vibroimpact system to white-noise random excitation,, Z. Angew. Math. Mech., 59 (1979), 709. doi: 10.1002/zamm.19790591205. Google Scholar

[47]

N. Sri Namachchivaya and J. H. Park, Stochastic dynamics of impact oscillators,, J. Appl. Mech. Trans. ASME, 72 (2005), 862. doi: 10.1115/1.2041660. Google Scholar

[48]

J. Feng, W. Xu, H. Rong and R. Wang, Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations,, Int. J. Non-Linear Mech., 44 (2009), 51. doi: 10.1016/j.ijnonlinmec.2008.08.013. Google Scholar

[49]

V. F. Zhuravlev, A method for analyzing vibration-impact systems by means of special functions,, Mech. Solids, 11 (1976), 23. Google Scholar

[50]

P. D. Christofides and N. H. El-Farra, Control of Nonlinear and Hybrid Process Systems. Designs for Uncertainty, Constraints and Time-Delays,, Springer, (2005). Google Scholar

[51]

J. Raouf and H. Michalska, Robust stabilization of switched linear systems with Wiener process noise,, in 49th IEEE Conference on Decision and Control, (2010), 6493. doi: 10.1109/CDC.2010.5717694. Google Scholar

[52]

W. Feng and J.-F. Zhang, Stability analysis and stabilization control of multi-variable switched stochastic systems,, Automatica, 42 (2006), 169. doi: 10.1016/j.automatica.2005.08.016. Google Scholar

[53]

D. Chatterjee and D. Liberzon, On stability of stochastic switched systems,, in Proceedings of the 43rd IEEE Conference on Decision and Control., (2004), 4125. doi: 10.1109/CDC.2004.1429398. Google Scholar

[54]

E. Skafidas, R. J. Evans, A. V. Savkin and I. R. Petersen, Stability results for switched controller systems,, Automatica, 35 (1999), 553. doi: 10.1016/S0005-1098(98)00167-8. Google Scholar

[55]

P. Mhaskar, N. H. El-Farra and P. D. Christofides, Robust hybrid predictive control of nonlinear systems,, Automatica, 41 (2005), 209. doi: 10.1016/j.automatica.2004.08.020. Google Scholar

[56]

B. Hu, X. Xu, P.J. Antsaklis and A. N. Michel, Robust stabilizing control laws for a class of second-order switched systems,, Syst. Control Lett., 38 (1999), 197. doi: 10.1016/S0167-6911(99)00065-1. Google Scholar

[57]

Z. Sun, A robust stabilizing law for switched linear systems,, Int. J. Control, 77 (2004), 389. doi: 10.1080/00207170410001667468. Google Scholar

[58]

D. Liberzon, Switching in Systems and Control,, Birkhauser, (2003). doi: 10.1007/978-1-4612-0017-8. Google Scholar

[59]

S.-C. Tan, Y.-M. Lai and C. K. Tse, Sliding Mode Control of Switching Power Converters,, CRC Press, (2012). Google Scholar

[60]

J.-Q. Sun, Stochastic Dynamics and Control., volume 4 of Nonlinear Science and Complexity,, Elsevier, (2006). Google Scholar

[61]

Y. Niu, D. W. C. Ho and J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay,, Automatica, 41 (2005), 873. doi: 10.1016/j.automatica.2004.11.035. Google Scholar

[62]

L. Wu, D. W. C. Ho and C. W. Li, Stabilisation and performance synthesis for switched stochastic systems,, IET Control Theory Appl., 4 (2010), 1877. doi: 10.1049/iet-cta.2009.0179. Google Scholar

[63]

L. Wu, D. W. C. Ho and C. W. Li, Sliding mode control of switching hybrid systems with stochastic perturbation,, Syst. Contr. Lett., 60 (2011), 531. doi: 10.1016/j.sysconle.2011.04.007. Google Scholar

[64]

Y. Niu, D. W. C. Ho and X. Wang, Robust $H_\infty$ control for nonlinear stochastic systems: A sliding-mode approach,, IEEE Trans. Automat. Contr., 53 (2008), 1695. doi: 10.1109/TAC.2008.929376. Google Scholar

[65]

D. E. Stewart, Rigid-body dynamics with friction and impact,, SIAM Rev., 42 (2000), 3. doi: 10.1137/S0036144599360110. Google Scholar

[66]

M. Abadie, Dynamical simulation of rigid bodies: Modelling of frictional contact,, in Impacts in Mechanical Systems: Analysis and Modelling, (2000). doi: 10.1007/3-540-45501-9_2. Google Scholar

[67]

P. S. Goohpattader, S. Mettu and M. K. Chaudhury, Experimental investigation of the drift and diffusion of small objects on a surface subjected to a bias and an external white noise: Roles of Coulombic friction and hysteresis,, Langmuir, 25 (2009), 9969. doi: 10.1021/la901111u. Google Scholar

[68]

P. S. Goohpattader and M. K. Chaudhury, Diffusive motion with nonlinear friction: Apparently Brownian,, J. Chem. Phys., 133 (2010). doi: 10.1063/1.3460530. Google Scholar

[69]

P.-G. de Gennes, Brownian motion with dry friction,, J. Stat. Phys., 119 (2005), 953. doi: 10.1007/s10955-005-4650-4. Google Scholar

[70]

H. Hayakawa, Langevin equation with Coulomb friction,, Phys. D, 205 (2005), 48. doi: 10.1016/j.physd.2004.12.011. Google Scholar

[71]

H. Touchette, E. Van der Straeten and W. Just, Brownian motion with dry friction: Fokker-Planck approach,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/44/445002. Google Scholar

[72]

A. Baule, E. G. D. Cohen and H. Touchette, A path integral approach to random motion with nonlinear friction,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/2/025003. Google Scholar

[73]

A. Baule, H. Touchette and E. G. D. Cohen, Stick-slip motion of solids with dry friction subject to random vibrations and an external field,, Nonlinearity, 24 (2011), 351. doi: 10.1088/0951-7715/24/2/001. Google Scholar

[74]

H. Touchette, T. Prellberg and W. Just, Exact power spectra of Brownian motion with solid friction,, J. Phys. A: Math. Theor., 45 (2012). doi: 10.1088/1751-8113/45/39/395002. Google Scholar

[75]

D. J. W. Simpson and R. Kuske, Stochastic perturbations of periodic orbits with sliding, Submitted to: J. Nonlin. Sci., (2014). Google Scholar

[76]

A. Colombo, M. di Bernardo, S. J. Hogan and M. R. Jeffrey, Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems,, Phys. D, 241 (2012), 1845. doi: 10.1016/j.physd.2011.09.017. Google Scholar

[77]

M. di Bernardo, P. Kowalczyk and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings,, Phys. D, 170 (2002), 175. doi: 10.1016/S0167-2789(02)00547-X. Google Scholar

[78]

A. Colombo and M. R. Jeffrey, Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows,, SIAM J. Appl. Dyn. Sys., 10 (2011), 423. doi: 10.1137/100801846. Google Scholar

[79]

M. R. Jeffrey and A. Colombo, The two-fold singularity of discontinuous vector fields,, SIAM J. Appl. Dyn. Sys., 8 (2009), 624. doi: 10.1137/08073113X. Google Scholar

[80]

M. A. Teixeira, Stability conditions for discontinuous vector fields,, J. Differential Equations, 88 (1990), 15. doi: 10.1016/0022-0396(90)90106-Y. Google Scholar

[81]

T.-S. Chiang and S.-J. Sheu, Large deviation of small perturbation of some unstable systems,, Stoch. Anal. Appl., 15 (1997), 31. doi: 10.1080/07362999708809462. Google Scholar

[82]

R. Kuske and G. Papanicolaou, The invariant density of a chaotic dynamical system with small noise,, Phys. D, 120 (1998), 255. doi: 10.1016/S0167-2789(98)00085-2. Google Scholar

[83]

P. Hitczenko and G. S. Medvedev, Bursting oscillations induced by small noise,, SIAM J. Appl. Math., 69 (2009), 1359. doi: 10.1137/070711803. Google Scholar

[84]

Ya. Z. Tsypkin, Relay Control Systems,, Cambridge University Press, (1984). Google Scholar

[85]

G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems,, Prentice Hall, (2002). Google Scholar

[86]

R. C. Dorf and R. H. Bishop, Modern Control Systems,, Prentice Hall, (2001). doi: 10.1109/TSMC.1981.4308749. Google Scholar

[87]

K. J. Åström and R. M. Murray, Feedback Systems. An Introduction for Scientists and Engineers,, Princeton University Press, (2008). Google Scholar

[88]

K. H. Johansson, A. Rantzer and K. J. Åström, Fast switches in relay feedback systems,, Automatica, 35 (1999), 539. doi: 10.1016/S0005-1098(98)00160-5. Google Scholar

[89]

K. H. Johansson, A. E. Barabanov and K. J. Åström, Limit cycles with chattering in relay feedback systems,, IEEE Trans. Automat. Contr., 47 (2002), 1414. doi: 10.1109/TAC.2002.802770. Google Scholar

[90]

Y. Zhao, J. Feng and C. K. Tse, Discrete-time modeling and stability analysis of periodic orbits with sliding for switched linear systems,, IEEE Trans. Circuits Systems I Fund. Theory Appl., 57 (2010), 2948. doi: 10.1109/TCSI.2010.2050230. Google Scholar

[91]

F. Dercole, R. Ferrière, A. Gragnani and S. Rinaldi, Coevolution of slow-fast populations: Evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics,, Proc. R. Soc. B, 273 (2006), 983. doi: 10.1098/rspb.2005.3398. Google Scholar

[92]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models,, Theor. Popul. Biol., 72 (2007), 197. doi: 10.1016/j.tpb.2007.06.003. Google Scholar

[93]

J. A. Amador, G. Olivar and F. Angulo, Smooth and Filippov models of sustainable development: Bifurcations and numerical computations,, Differ. Equ. Dyn. Syst., 21 (2013), 173. doi: 10.1007/s12591-012-0138-2. Google Scholar

[94]

S. Tang, J. Liang, Y. Xiao and R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds,, SIAM J. Appl. Math., 72 (2012), 1061. doi: 10.1137/110847020. Google Scholar

[95]

R. Szalai and H. M. Osinga, Invariant polygons in systems with grazing-sliding,, Chaos, 18 (2008). doi: 10.1063/1.2904774. Google Scholar

[96]

M. Tanelli, G. Osorio, M. di Bernardo, S. M. Savaresi and A. Astolfi, Existence, stability and robustness analysis of limit cycles in hybrid anti-lock braking systems,, Int. J. Contr., 82 (2009), 659. doi: 10.1080/00207170802203598. Google Scholar

[97]

M. di Bernardo, K. H. Johansson, U. Jönsson and F. Vasca, On the robustness of periodic solutions in relay feedback systems,, in 15th Triennial World Congress, (2002). Google Scholar

[98]

F. Flandoli, Random Perturbations of PDEs and Fluid Dynamic Models., volume 2015 of Lecture Notes in Mathematics,, Springer, (2011). doi: 10.1007/978-3-642-18231-0. Google Scholar

[99]

Yu. V. Prokhorov and A. N. Shiryaev, Probability Theory III: Stochastic Calculus,, Springer, (1998). doi: 10.1007/978-3-662-03640-2. Google Scholar

[100]

E. D. Conway, Stochastic equations with discontinuous drift,, Trans. Am. Math. Soc., 157 (1971), 235. doi: 10.1090/S0002-9947-1971-0275532-6. Google Scholar

[101]

D. Stroock and S. R. S Varadhan, Diffusion processes with continuous coefficients. I,, Comm. Pure Appl. Math., 22 (1969), 345. doi: 10.1002/cpa.3160220304. Google Scholar

[102]

A. Ju. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations,, Math. USSR Sb., 39 (1981), 387. Google Scholar

[103]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift,, Probab. Theory Relat. Fields, 131 (2005), 154. doi: 10.1007/s00440-004-0361-z. Google Scholar

[104]

S. E. A. Mohammed, T. Nilssen and F. Proske, Sobolev differentiable stochastic flows for SDE's with singular coefficients: Applications to the transport equation,, Unpublished, (2012). Google Scholar

[105]

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications,, Springer-Verlag, (1984). doi: 10.1007/978-3-642-96807-5. Google Scholar

[106]

Z. Schuss, Theory and Applications of Stochastic Processes,, Springer, (2010). doi: 10.1007/978-1-4419-1605-1. Google Scholar

[107]

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers,, International Series in Pure and Applied Mathematics. McGraw-Hill, (1978). Google Scholar

[108]

R. Buckdahn, Y. Ouknine and M. Quincampoix, On limiting values of stochastic differential equations with small noise intensity tending to zero,, Bull. Sci. Math., 133 (2009), 229. doi: 10.1016/j.bulsci.2008.12.005. Google Scholar

[109]

O. Menoukeu-Pamen, T. Meyer-Brandis and F. Proske, A Gel'fand triple approach to the small noise problem for discontinuous ODE's,, Unpublished, (2011). Google Scholar

[110]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization,, Springer, (2008). Google Scholar

[111]

I. Karatzas and S. E. Shreve, Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control,, Ann. Prob., 12 (1984), 819. doi: 10.1214/aop/1176993230. Google Scholar

[112]

I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures,, Theory Prob. Appl., 5 (1960), 285. Google Scholar

[113]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar

[114]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Springer, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar

[115]

Z. Qian and W. Zheng, Sharp bounds for transition probability densities of a class of diffusions,, C.R. Acad. Sci. Paris, 335 (2002), 953. doi: 10.1016/S1631-073X(02)02579-7. Google Scholar

[116]

Z. Qian, F. Russo and W. Zheng, Comparison theorem and estimates for transition probability densities of diffusion processes,, Probab. Theory Relat. Fields., 127 (2003), 388. doi: 10.1007/s00440-003-0291-1. Google Scholar

[117]

M. Gradinaru, S. Herrmann and B. Roynette, A singular large deviations phenomenon,, Ann. Inst. Henri Poincaré, 37 (2001), 555. doi: 10.1016/S0246-0203(01)01075-5. Google Scholar

[118]

W. Zhang, Transition density of one-dimensional diffusion with discontinuous drift,, IEEE Trans. Automat. Contr., 35 (1990), 980. doi: 10.1109/9.58517. Google Scholar

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