• Previous Article
    Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables
  • DCDS Home
  • This Issue
  • Next Article
    Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms
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 and Continuous Dynamical Systems, 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), 99-128.

[2]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers., Norwell, 1988.

[3]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications. Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008.

[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, Berlin, 2004.

[5]

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

[6]

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

[7]

M. Oestreich, N. Hinrichs and K. Popp, Bifurcation and stability analysis for a non-smooth friction oscillator, Arch. Appl. Mech., 66 (1996), 301-314. doi: 10.1007/BF00795247.

[8]

B. Feeny and F. C. Moon, Chaos in a forced dry-friction oscillator: Experiments and numerical modelling, J. Sound Vib., 170 (1994), 303-323. doi: 10.1006/jsvi.1994.1065.

[9]

M. Johansson, Piecewise Linear Control Systems, volume 284 of Lecture Notes in Control and Information Sciences. Springer-Verlag, New York, 2003.

[10]

K. H. Johansson, A. Rantzer and K. J. Åström, Fast switches in relay feedback systems, Automatica, 35 (1999), 539-552. doi: 10.1016/S0005-1098(98)00160-5.

[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), 1121-1140. doi: 10.1142/S0218127401002584.

[12]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theor. Popul. Biol., 72 (2007), 197-213. doi: 10.1016/j.tpb.2007.06.003.

[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), 983-990. doi: 10.1098/rspb.2005.3398.

[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), 173-184. doi: 10.1007/s12591-012-0138-2.

[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), 1061-1080. doi: 10.1137/110847020.

[16]

K. Deimling, Multivalued Differential Equations, W. de Gruyter, New York, 1992. doi: 10.1515/9783110874228.

[17]

G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, RI, 2002.

[18]

J. Cortés, Discontinuous dynamical systems. A tutorial on solutions, nonsmooth analysis, and stability, IEEE Contr. Sys. Magazine, 28 (2008), 36-73. doi: 10.1109/MCS.2008.919306.

[19]

M. A. Teixeira, Stability conditions for discontinuous vector fields, J. Differential Equations, 88 (1990), 15-29. doi: 10.1016/0022-0396(90)90106-Y.

[20]

M. A. Teixeira, Generic bifurcation of sliding vector fields, J. Math. Anal. Appl., 176 (1993), 436-457. doi: 10.1006/jmaa.1993.1226.

[21]

Y. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurcation Chaos, 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874.

[22]

M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differential Equations, 250 (2011), 1967-2023. doi: 10.1016/j.jde.2010.11.016.

[23]

M. R. Jeffrey and A. Colombo, The two-fold singularity of discontinuous vector fields, SIAM J. Appl. Dyn. Sys., 8 (2009), 624-640. doi: 10.1137/08073113X.

[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), 423-451. doi: 10.1137/100801846.

[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), 1215-1230. doi: 10.1137/120869134.

[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), 1-10. doi: 10.1016/j.physd.2013.07.015.

[27]

M. di Bernardo, A. Colombo and E. Fossas, Two-fold singularity in nonsmooth electrical systems, In IEEE International Symposium on Circuits and Systems., pages 2713-2716, 2011.

[28]

M. Desroches and M. R. Jeffrey, Canards and curvature: Nonsmooth approximation by pinching, Nonlinearity, 24 (2011), 1655-1682. doi: 10.1088/0951-7715/24/5/014.

[29]

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

[30]

Y. Z. Tsypkin, Relay Control Systems, Cambridge University Press, New York, 1984.

[31]

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

[32]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Relat. Fields, 131 (2005), 154-196. doi: 10.1007/s00440-004-0361-z.

[33]

A. M. Davie, Uniqueness of solutions of stochastic differential equations, Int. Math. Res. Not., IMRN 2007, no. 24, Art. ID rnm124, 26 pp. doi: 10.1093/imrn/rnm124.

[34]

F. Flandoli and J. A. Langa, Markov attractors: A probabilistic approach to multivalued flows, Stoch. Dyn., 8 (2008), 59-75. doi: 10.1142/S0219493708002202.

[35]

V. S. Borkar and K. Suresh Kumar, A new Markov selection procedure for degenerate diffusions, J. Theor. Probab., 23 (2010), 729-747. doi: 10.1007/s10959-009-0242-6.

[36]

F. Flandoli, Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations, Metrika, 69 (2009), 101-123. doi: 10.1007/s00184-008-0210-7.

[37]

S. Attanasio and F. Flandoli, Zero-noise solutions of linear transport equations without uniqueness: An example, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 753-756. doi: 10.1016/j.crma.2009.04.027.

[38]

S. S. Sastry, The effects of small noise on implicitly defined nonlinear dynamical systems, IEEE Trans. Circuits Syst., 30 (1983), 651-663. doi: 10.1109/TCS.1983.1085404.

[39]

A. Yu. Veretennikov, Approximation of ordinary differential equations by stochastic differential equations, Mat. Zametki, 33 (1983), 929-932.

[40]

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

[41]

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

[42]

Y. Bakhtin, Noisy heteroclinic networks, Probab. Theory Relat. Fields, 150 (2011), 1-42. doi: 10.1007/s00440-010-0264-0.

[43]

D. Armbruster, E. Stone and V. Kirk, Noisy heteroclinic networks, Chaos, 13 (2003), 71-86. doi: 10.1063/1.1539951.

[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, Barcelona, Spain, 2002.

[45]

J. M. Gonçalves, A. Megretski and M. A. Dahleh, Global stability of relay feedback systems, IEEE Trans. Automat. Contr., 46 (2001), 550-562. doi: 10.1109/9.917657.

[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), 403-422. doi: 10.1137/100784606.

[47]

S. Varigonda and T. T. Georgiou, Dynamics of relay relaxation oscillators, IEEE Trans. Automat. Contr., 46 (2001), 65-77. doi: 10.1109/9.898696.

[48]

J. Sieber, Dynamics of delayed relay systems, Nonlinearity, 19 (2006), 2489-2527. doi: 10.1088/0951-7715/19/11/001.

[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), 1111-1140. doi: 10.1177/1077546309341124.

[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), 85-108. doi: 10.1007/s00332-005-0745-y.

[51]

Yu. V. Prokhorov and A. N. Shiryaev, editors, Probability Theory III: Stochastic Calculus, Springer, New York, 1998.

[52]

D. Stroock and S. R. S Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math., 22 (1969), 345-400. doi: 10.1002/cpa.3160220304.

[53]

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

[54]

C. W. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Sciences, Springer, New York, 2009.

[55]

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

[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), 229-237. doi: 10.1016/j.bulsci.2008.12.005.

[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.

[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), 819-828. doi: 10.1214/aop/1176993230.

[59]

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

[60]

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

[61]

M. Gradinaru, S. Herrmann and B. Roynette, A singular large deviations phenomenon, Ann. Inst. Henri Poincaré, 37 (2001), 555-580. doi: 10.1016/S0246-0203(01)01075-5.

[62]

Z. Schuss, Theory and Applications of Stochastic Processes, Springer, New York, 2010. doi: 10.1007/978-1-4419-1605-1.

[63]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, New York, 2003. doi: 10.1007/978-3-642-14394-6.

[64]

C. Knessl, Exact and asymptotic solutions to a PDE that arises in time-dependent queues, Adv. Appl. Prob., 32 (2000), 256-283. doi: 10.1239/aap/1013540033.

[65]

O. Vallée and M. Soares, Airy Functions and Applications to Physics, Second edition. Imperial College Press, London, 2010.

[66]

M. J. Ablowitz and A. S. Fokas, Complex Variables. Introduction and Applications, Cambridge University Press, New York, 2003. doi: 10.1017/CBO9780511791246.

show all references

References:
[1]

A. F. Filippov, Differential equations with discontinuous right-hand side, Mat. Sb., 51 (1960), 99-128.

[2]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers., Norwell, 1988.

[3]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications. Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008.

[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, Berlin, 2004.

[5]

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

[6]

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

[7]

M. Oestreich, N. Hinrichs and K. Popp, Bifurcation and stability analysis for a non-smooth friction oscillator, Arch. Appl. Mech., 66 (1996), 301-314. doi: 10.1007/BF00795247.

[8]

B. Feeny and F. C. Moon, Chaos in a forced dry-friction oscillator: Experiments and numerical modelling, J. Sound Vib., 170 (1994), 303-323. doi: 10.1006/jsvi.1994.1065.

[9]

M. Johansson, Piecewise Linear Control Systems, volume 284 of Lecture Notes in Control and Information Sciences. Springer-Verlag, New York, 2003.

[10]

K. H. Johansson, A. Rantzer and K. J. Åström, Fast switches in relay feedback systems, Automatica, 35 (1999), 539-552. doi: 10.1016/S0005-1098(98)00160-5.

[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), 1121-1140. doi: 10.1142/S0218127401002584.

[12]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theor. Popul. Biol., 72 (2007), 197-213. doi: 10.1016/j.tpb.2007.06.003.

[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), 983-990. doi: 10.1098/rspb.2005.3398.

[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), 173-184. doi: 10.1007/s12591-012-0138-2.

[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), 1061-1080. doi: 10.1137/110847020.

[16]

K. Deimling, Multivalued Differential Equations, W. de Gruyter, New York, 1992. doi: 10.1515/9783110874228.

[17]

G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, RI, 2002.

[18]

J. Cortés, Discontinuous dynamical systems. A tutorial on solutions, nonsmooth analysis, and stability, IEEE Contr. Sys. Magazine, 28 (2008), 36-73. doi: 10.1109/MCS.2008.919306.

[19]

M. A. Teixeira, Stability conditions for discontinuous vector fields, J. Differential Equations, 88 (1990), 15-29. doi: 10.1016/0022-0396(90)90106-Y.

[20]

M. A. Teixeira, Generic bifurcation of sliding vector fields, J. Math. Anal. Appl., 176 (1993), 436-457. doi: 10.1006/jmaa.1993.1226.

[21]

Y. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurcation Chaos, 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874.

[22]

M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differential Equations, 250 (2011), 1967-2023. doi: 10.1016/j.jde.2010.11.016.

[23]

M. R. Jeffrey and A. Colombo, The two-fold singularity of discontinuous vector fields, SIAM J. Appl. Dyn. Sys., 8 (2009), 624-640. doi: 10.1137/08073113X.

[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), 423-451. doi: 10.1137/100801846.

[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), 1215-1230. doi: 10.1137/120869134.

[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), 1-10. doi: 10.1016/j.physd.2013.07.015.

[27]

M. di Bernardo, A. Colombo and E. Fossas, Two-fold singularity in nonsmooth electrical systems, In IEEE International Symposium on Circuits and Systems., pages 2713-2716, 2011.

[28]

M. Desroches and M. R. Jeffrey, Canards and curvature: Nonsmooth approximation by pinching, Nonlinearity, 24 (2011), 1655-1682. doi: 10.1088/0951-7715/24/5/014.

[29]

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

[30]

Y. Z. Tsypkin, Relay Control Systems, Cambridge University Press, New York, 1984.

[31]

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

[32]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Relat. Fields, 131 (2005), 154-196. doi: 10.1007/s00440-004-0361-z.

[33]

A. M. Davie, Uniqueness of solutions of stochastic differential equations, Int. Math. Res. Not., IMRN 2007, no. 24, Art. ID rnm124, 26 pp. doi: 10.1093/imrn/rnm124.

[34]

F. Flandoli and J. A. Langa, Markov attractors: A probabilistic approach to multivalued flows, Stoch. Dyn., 8 (2008), 59-75. doi: 10.1142/S0219493708002202.

[35]

V. S. Borkar and K. Suresh Kumar, A new Markov selection procedure for degenerate diffusions, J. Theor. Probab., 23 (2010), 729-747. doi: 10.1007/s10959-009-0242-6.

[36]

F. Flandoli, Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations, Metrika, 69 (2009), 101-123. doi: 10.1007/s00184-008-0210-7.

[37]

S. Attanasio and F. Flandoli, Zero-noise solutions of linear transport equations without uniqueness: An example, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 753-756. doi: 10.1016/j.crma.2009.04.027.

[38]

S. S. Sastry, The effects of small noise on implicitly defined nonlinear dynamical systems, IEEE Trans. Circuits Syst., 30 (1983), 651-663. doi: 10.1109/TCS.1983.1085404.

[39]

A. Yu. Veretennikov, Approximation of ordinary differential equations by stochastic differential equations, Mat. Zametki, 33 (1983), 929-932.

[40]

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

[41]

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

[42]

Y. Bakhtin, Noisy heteroclinic networks, Probab. Theory Relat. Fields, 150 (2011), 1-42. doi: 10.1007/s00440-010-0264-0.

[43]

D. Armbruster, E. Stone and V. Kirk, Noisy heteroclinic networks, Chaos, 13 (2003), 71-86. doi: 10.1063/1.1539951.

[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, Barcelona, Spain, 2002.

[45]

J. M. Gonçalves, A. Megretski and M. A. Dahleh, Global stability of relay feedback systems, IEEE Trans. Automat. Contr., 46 (2001), 550-562. doi: 10.1109/9.917657.

[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), 403-422. doi: 10.1137/100784606.

[47]

S. Varigonda and T. T. Georgiou, Dynamics of relay relaxation oscillators, IEEE Trans. Automat. Contr., 46 (2001), 65-77. doi: 10.1109/9.898696.

[48]

J. Sieber, Dynamics of delayed relay systems, Nonlinearity, 19 (2006), 2489-2527. doi: 10.1088/0951-7715/19/11/001.

[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), 1111-1140. doi: 10.1177/1077546309341124.

[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), 85-108. doi: 10.1007/s00332-005-0745-y.

[51]

Yu. V. Prokhorov and A. N. Shiryaev, editors, Probability Theory III: Stochastic Calculus, Springer, New York, 1998.

[52]

D. Stroock and S. R. S Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math., 22 (1969), 345-400. doi: 10.1002/cpa.3160220304.

[53]

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

[54]

C. W. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Sciences, Springer, New York, 2009.

[55]

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

[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), 229-237. doi: 10.1016/j.bulsci.2008.12.005.

[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.

[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), 819-828. doi: 10.1214/aop/1176993230.

[59]

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

[60]

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

[61]

M. Gradinaru, S. Herrmann and B. Roynette, A singular large deviations phenomenon, Ann. Inst. Henri Poincaré, 37 (2001), 555-580. doi: 10.1016/S0246-0203(01)01075-5.

[62]

Z. Schuss, Theory and Applications of Stochastic Processes, Springer, New York, 2010. doi: 10.1007/978-1-4419-1605-1.

[63]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, New York, 2003. doi: 10.1007/978-3-642-14394-6.

[64]

C. Knessl, Exact and asymptotic solutions to a PDE that arises in time-dependent queues, Adv. Appl. Prob., 32 (2000), 256-283. doi: 10.1239/aap/1013540033.

[65]

O. Vallée and M. Soares, Airy Functions and Applications to Physics, Second edition. Imperial College Press, London, 2010.

[66]

M. J. Ablowitz and A. S. Fokas, Complex Variables. Introduction and Applications, Cambridge University Press, New York, 2003. doi: 10.1017/CBO9780511791246.

[1]

Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058

[2]

Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251

[3]

Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021

[4]

Richard H. Rand, Asok K. Sen. A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Communications on Pure and Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567

[5]

Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks and Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297

[6]

Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223

[7]

Linna Li, Changjun Yu, Ning Zhang, Yanqin Bai, Zhiyuan Gao. A time-scaling technique for time-delay switched systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1825-1843. doi: 10.3934/dcdss.2020108

[8]

B. Cantó, C. Coll, A. Herrero, E. Sánchez, N. Thome. Pole-assignment of discrete time-delay systems with symmetries. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 641-649. doi: 10.3934/dcdsb.2006.6.641

[9]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial and Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113

[10]

Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021076

[11]

Ming He, Xiaoyun Ma, Weijiang Zhang. Oscillation death in systems of oscillators with transferable coupling and time-delay. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 737-745. doi: 10.3934/dcds.2001.7.737

[12]

Qinqin Chai, Ryan Loxton, Kok Lay Teo, Chunhua Yang. A unified parameter identification method for nonlinear time-delay systems. Journal of Industrial and Management Optimization, 2013, 9 (2) : 471-486. doi: 10.3934/jimo.2013.9.471

[13]

Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, 2021, 29 (5) : 2973-2985. doi: 10.3934/era.2021022

[14]

Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339

[15]

Nasim Ullah, Ahmad Aziz Al-Ahmadi. A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertainty. Mathematical Foundations of Computing, 2020, 3 (2) : 81-99. doi: 10.3934/mfc.2020007

[16]

Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329

[17]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072

[18]

Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158

[19]

Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial and Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061

[20]

J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (118)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]