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September  2014, 34(9): 3803-3830. doi: 10.3934/dcds.2014.34.3803

On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

1. 

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

Received  April 2013 Revised  December 2013 Published  March 2014

A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. However, arbitrarily small perturbations of the vector field can make forward evolution well-defined, and from an applied perspective, such perturbations may represent additional model features that enhance the realism of a piecewise-smooth mathematical model. Three physically motivated forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. The purpose of this paper is to characterise the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. In each case the limit leads to a novel probabilistic notion of forward evolution from the two-fold.
Citation: David J. W. Simpson. On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3803-3830. doi: 10.3934/dcds.2014.34.3803
References:
[1]

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

[2]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Kluwer Academic Publishers., (1988). Google Scholar

[3]

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

[4]

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). Google Scholar

[5]

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

[6]

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

[7]

M. Oestreich, N. Hinrichs and K. Popp, Bifurcation and stability analysis for a non-smooth friction oscillator,, Arch. Appl. Mech., 66 (1996), 301. doi: 10.1007/BF00795247. Google Scholar

[8]

B. Feeny and F. C. Moon, Chaos in a forced dry-friction oscillator: Experiments and numerical modelling,, J. Sound Vib., 170 (1994), 303. doi: 10.1006/jsvi.1994.1065. Google Scholar

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M. Johansson, Piecewise Linear Control Systems,, volume 284 of Lecture Notes in Control and Information Sciences. Springer-Verlag, (2003). Google Scholar

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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

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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

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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

[25]

S. Fernández-García, D. A. García, G. O. Tost, M. di Bernardo and M. R. Jeffrey, Structural stability of the two-fold singularity,, SIAM J. Appl. Dyn. Syst., 11 (2012), 1215. doi: 10.1137/120869134. Google Scholar

[26]

A. Colombo and M. R. Jeffrey, The two-fold singularity of non-smooth flows: Leading order dynamics in $n$-dimensions,, Phys. D, 263 (2013), 1. doi: 10.1016/j.physd.2013.07.015. Google Scholar

[27]

M. di Bernardo, A. Colombo and E. Fossas, Two-fold singularity in nonsmooth electrical systems,, In IEEE International Symposium on Circuits and Systems., (2011), 2713. Google Scholar

[28]

M. Desroches and M. R. Jeffrey, Canards and curvature: Nonsmooth approximation by pinching,, Nonlinearity, 24 (2011), 1655. doi: 10.1088/0951-7715/24/5/014. Google Scholar

[29]

M. Desroches and M. R. Jeffrey, Pinching of canards and folded nodes: Nonsmooth approximation of slow-fast dynamics,, Unpublished, (2012). Google Scholar

[30]

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

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A. M. Davie, Uniqueness of solutions of stochastic differential equations,, Int. Math. Res. Not., (2007). doi: 10.1093/imrn/rnm124. Google Scholar

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F. Flandoli and J. A. Langa, Markov attractors: A probabilistic approach to multivalued flows,, Stoch. Dyn., 8 (2008), 59. doi: 10.1142/S0219493708002202. Google Scholar

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V. S. Borkar and K. Suresh Kumar, A new Markov selection procedure for degenerate diffusions,, J. Theor. Probab., 23 (2010), 729. doi: 10.1007/s10959-009-0242-6. Google Scholar

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F. Flandoli, Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations,, Metrika, 69 (2009), 101. doi: 10.1007/s00184-008-0210-7. Google Scholar

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S. Attanasio and F. Flandoli, Zero-noise solutions of linear transport equations without uniqueness: An example,, C. R. Acad. Sci. Paris, 347 (2009), 753. doi: 10.1016/j.crma.2009.04.027. Google Scholar

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A. Yu. Veretennikov, Approximation of ordinary differential equations by stochastic differential equations,, Mat. Zametki, 33 (1983), 929. Google Scholar

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R. Bafico and P. Baldi, Small random perturbations of Peano phenomena,, Stochastics, 6 (): 279. doi: 10.1080/17442508208833208. Google Scholar

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

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Y. Bakhtin, Noisy heteroclinic networks,, Probab. Theory Relat. Fields, 150 (2011), 1. doi: 10.1007/s00440-010-0264-0. Google Scholar

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D. Armbruster, E. Stone and V. Kirk, Noisy heteroclinic networks,, Chaos, 13 (2003), 71. doi: 10.1063/1.1539951. Google Scholar

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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

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J. M. Gonçalves, A. Megretski and M. A. Dahleh, Global stability of relay feedback systems,, IEEE Trans. Automat. Contr., 46 (2001), 550. doi: 10.1109/9.917657. Google Scholar

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T. Kalmár-Nagy, P. Wahi and A Halder, Dynamics of a hysteretic relay oscillator with periodic forcing,, SIAM J. Appl. Dyn. Syst., 10 (2011), 403. doi: 10.1137/100784606. Google Scholar

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S. Varigonda and T. T. Georgiou, Dynamics of relay relaxation oscillators,, IEEE Trans. Automat. Contr., 46 (2001), 65. doi: 10.1109/9.898696. Google Scholar

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J. Sieber, Dynamics of delayed relay systems,, Nonlinearity, 19 (2006), 2489. doi: 10.1088/0951-7715/19/11/001. Google Scholar

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J. Sieber, P. Kowalczyk, S. J. Hogan and M. di Bernardo, Dynamics of symmetric dynamical systems with delayed switching,, J. Vib. Control, 16 (2010), 1111. doi: 10.1177/1077546309341124. Google Scholar

[50]

A. Colombo, M. di Bernardo, S. J. Hogan and P. Kowalczyk, Complex dynamics in a hysteretic relay feedback system with delay,, J. Nonlinear Sci., 17 (2007), 85. doi: 10.1007/s00332-005-0745-y. Google Scholar

[51]

Yu. V. Prokhorov and A. N. Shiryaev, editors, Probability Theory III: Stochastic Calculus,, Springer, (1998). Google Scholar

[52]

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

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

[54]

C. W. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Sciences,, Springer, (2009). Google Scholar

[55]

L. Zhang, Random perturbation of some multi-dimensional non-Lipschitz ordinary differential equations,, Unpublished, (2013). Google Scholar

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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

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D. J. W. Simpson and R. Kuske, The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise,, To appear: Stoch. Dyn., (2014). doi: 10.1142/S0219493714500105. Google Scholar

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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

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

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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

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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

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

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

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C. Knessl, Exact and asymptotic solutions to a PDE that arises in time-dependent queues,, Adv. Appl. Prob., 32 (2000), 256. doi: 10.1239/aap/1013540033. Google Scholar

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show all references

References:
[1]

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

[2]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Kluwer Academic Publishers., (1988). Google Scholar

[3]

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

[4]

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). Google Scholar

[5]

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

[6]

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

[7]

M. Oestreich, N. Hinrichs and K. Popp, Bifurcation and stability analysis for a non-smooth friction oscillator,, Arch. Appl. Mech., 66 (1996), 301. doi: 10.1007/BF00795247. Google Scholar

[8]

B. Feeny and F. C. Moon, Chaos in a forced dry-friction oscillator: Experiments and numerical modelling,, J. Sound Vib., 170 (1994), 303. doi: 10.1006/jsvi.1994.1065. Google Scholar

[9]

M. Johansson, Piecewise Linear Control Systems,, volume 284 of Lecture Notes in Control and Information Sciences. Springer-Verlag, (2003). Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

K. Deimling, Multivalued Differential Equations,, W. de Gruyter, (1992). doi: 10.1515/9783110874228. Google Scholar

[17]

G. V. Smirnov, Introduction to the Theory of Differential Inclusions,, American Mathematical Society, (2002). Google Scholar

[18]

J. Cortés, Discontinuous dynamical systems. A tutorial on solutions, nonsmooth analysis, and stability,, IEEE Contr. Sys. Magazine, 28 (2008), 36. doi: 10.1109/MCS.2008.919306. Google Scholar

[19]

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

[20]

M. A. Teixeira, Generic bifurcation of sliding vector fields,, J. Math. Anal. Appl., 176 (1993), 436. doi: 10.1006/jmaa.1993.1226. Google Scholar

[21]

Y. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems,, Int. J. Bifurcation Chaos, 13 (2003), 2157. doi: 10.1142/S0218127403007874. Google Scholar

[22]

M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems,, J. Differential Equations, 250 (2011), 1967. doi: 10.1016/j.jde.2010.11.016. Google Scholar

[23]

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

[24]

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

[25]

S. Fernández-García, D. A. García, G. O. Tost, M. di Bernardo and M. R. Jeffrey, Structural stability of the two-fold singularity,, SIAM J. Appl. Dyn. Syst., 11 (2012), 1215. doi: 10.1137/120869134. Google Scholar

[26]

A. Colombo and M. R. Jeffrey, The two-fold singularity of non-smooth flows: Leading order dynamics in $n$-dimensions,, Phys. D, 263 (2013), 1. doi: 10.1016/j.physd.2013.07.015. Google Scholar

[27]

M. di Bernardo, A. Colombo and E. Fossas, Two-fold singularity in nonsmooth electrical systems,, In IEEE International Symposium on Circuits and Systems., (2011), 2713. Google Scholar

[28]

M. Desroches and M. R. Jeffrey, Canards and curvature: Nonsmooth approximation by pinching,, Nonlinearity, 24 (2011), 1655. doi: 10.1088/0951-7715/24/5/014. Google Scholar

[29]

M. Desroches and M. R. Jeffrey, Pinching of canards and folded nodes: Nonsmooth approximation of slow-fast dynamics,, Unpublished, (2012). Google Scholar

[30]

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

[31]

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

[32]

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

[33]

A. M. Davie, Uniqueness of solutions of stochastic differential equations,, Int. Math. Res. Not., (2007). doi: 10.1093/imrn/rnm124. Google Scholar

[34]

F. Flandoli and J. A. Langa, Markov attractors: A probabilistic approach to multivalued flows,, Stoch. Dyn., 8 (2008), 59. doi: 10.1142/S0219493708002202. Google Scholar

[35]

V. S. Borkar and K. Suresh Kumar, A new Markov selection procedure for degenerate diffusions,, J. Theor. Probab., 23 (2010), 729. doi: 10.1007/s10959-009-0242-6. Google Scholar

[36]

F. Flandoli, Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations,, Metrika, 69 (2009), 101. doi: 10.1007/s00184-008-0210-7. Google Scholar

[37]

S. Attanasio and F. Flandoli, Zero-noise solutions of linear transport equations without uniqueness: An example,, C. R. Acad. Sci. Paris, 347 (2009), 753. doi: 10.1016/j.crma.2009.04.027. Google Scholar

[38]

S. S. Sastry, The effects of small noise on implicitly defined nonlinear dynamical systems,, IEEE Trans. Circuits Syst., 30 (1983), 651. doi: 10.1109/TCS.1983.1085404. Google Scholar

[39]

A. Yu. Veretennikov, Approximation of ordinary differential equations by stochastic differential equations,, Mat. Zametki, 33 (1983), 929. Google Scholar

[40]

R. Bafico and P. Baldi, Small random perturbations of Peano phenomena,, Stochastics, 6 (): 279. doi: 10.1080/17442508208833208. Google Scholar

[41]

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

[42]

Y. Bakhtin, Noisy heteroclinic networks,, Probab. Theory Relat. Fields, 150 (2011), 1. doi: 10.1007/s00440-010-0264-0. Google Scholar

[43]

D. Armbruster, E. Stone and V. Kirk, Noisy heteroclinic networks,, Chaos, 13 (2003), 71. doi: 10.1063/1.1539951. Google Scholar

[44]

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

[45]

J. M. Gonçalves, A. Megretski and M. A. Dahleh, Global stability of relay feedback systems,, IEEE Trans. Automat. Contr., 46 (2001), 550. doi: 10.1109/9.917657. Google Scholar

[46]

T. Kalmár-Nagy, P. Wahi and A Halder, Dynamics of a hysteretic relay oscillator with periodic forcing,, SIAM J. Appl. Dyn. Syst., 10 (2011), 403. doi: 10.1137/100784606. Google Scholar

[47]

S. Varigonda and T. T. Georgiou, Dynamics of relay relaxation oscillators,, IEEE Trans. Automat. Contr., 46 (2001), 65. doi: 10.1109/9.898696. Google Scholar

[48]

J. Sieber, Dynamics of delayed relay systems,, Nonlinearity, 19 (2006), 2489. doi: 10.1088/0951-7715/19/11/001. Google Scholar

[49]

J. Sieber, P. Kowalczyk, S. J. Hogan and M. di Bernardo, Dynamics of symmetric dynamical systems with delayed switching,, J. Vib. Control, 16 (2010), 1111. doi: 10.1177/1077546309341124. Google Scholar

[50]

A. Colombo, M. di Bernardo, S. J. Hogan and P. Kowalczyk, Complex dynamics in a hysteretic relay feedback system with delay,, J. Nonlinear Sci., 17 (2007), 85. doi: 10.1007/s00332-005-0745-y. Google Scholar

[51]

Yu. V. Prokhorov and A. N. Shiryaev, editors, Probability Theory III: Stochastic Calculus,, Springer, (1998). Google Scholar

[52]

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

[53]

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

[54]

C. W. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Sciences,, Springer, (2009). Google Scholar

[55]

L. Zhang, Random perturbation of some multi-dimensional non-Lipschitz ordinary differential equations,, Unpublished, (2013). Google Scholar

[56]

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

[57]

D. J. W. Simpson and R. Kuske, The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise,, To appear: Stoch. Dyn., (2014). doi: 10.1142/S0219493714500105. Google Scholar

[58]

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

[59]

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

[60]

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

[61]

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

[62]

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

[63]

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

[64]

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