American Institute of Mathematical Sciences

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September  2016, 21(7): 2091-2107. doi: 10.3934/dcdsb.2016038

Singular fold with real noise

Peter W. Bates 1, , Ji Li 2, and  Mingji Zhang 3,

1. 

Department of Mathematics, Michigan State University, East Lansing, MI 48824
2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, United States

Received  November 2015 Revised  January 2016 Published  August 2016

We study the effect of small real noise on the jump behavior near a singular fold point, which is an important step in understanding the burst-spike behavior in many biological models. We show by the theory of center manifolds and random invariant manifolds that if the order of the noise is high enough, trajectories essentially pass the fold point in the manner as though there is no noise.
Keywords: random manifolds, singular fold, blow-up., Random dynamical systems, center manifold.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Peter W. Bates, Ji Li, Mingji Zhang. Singular fold with real noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2091-2107. doi: 10.3934/dcdsb.2016038
References:
[1]

L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-fast Dynamical Systems. A Sample-Paths Approach, Probab. Appl., Springer, London, 2006.  Google Scholar

[3]

P. Bates, J. Li and K. Lu, Normally hyperbolic invariant manifolds for random dynamical systems, Trans. Amer. Math. Soc., 365 (2013), 5933-5966. doi: 10.1090/S0002-9947-2013-05825-4.  Google Scholar

[4]

P. Bates, J. Li and K. Lu, Invariant foliations for random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 3639-3666. doi: 10.3934/dcds.2014.34.3639.  Google Scholar

[5]

P. Bates, J. Li and K. Lu, Geometric singular perturbation theory with real noise, J. Differential Equations, 259 (2015), 5137-5167. doi: 10.1016/j.jde.2015.06.023.  Google Scholar

[6]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100 pp. doi: 10.1090/memo/0577.  Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[8]

M. Krupa and P. Szmolyan, Extending singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Analysis, 33 (2001), 286-314. doi: 10.1137/S0036141099360919.  Google Scholar

[9]

K. Lu and Q.-D. Wang, Chaos in differential equations driven by a nonautonomous force, Nonlinearity, 23 (2010), 2935-2975. doi: 10.1088/0951-7715/23/11/012.  Google Scholar

[10]

E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, New York, 1980. doi: 10.1007/978-1-4615-9047-7.  Google Scholar

[11]

P. Szmolyan and M. Wechselberger, Relaxation oscillation in $R^3$, J. Differential Equations, 200 (2004), 69-104. doi: 10.1016/j.jde.2003.09.010.  Google Scholar

[12]

D. Terman, The transition from bursting to continuous spiking in an excitable membrane model, J. Nonlinear Sci., 2 (1992), 133-182. doi: 10.1007/BF02429854.  Google Scholar

[13]

D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, SIAM J. Appl. Math., 51 (1991), 1418-1450. doi: 10.1137/0151071.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-fast Dynamical Systems. A Sample-Paths Approach, Probab. Appl., Springer, London, 2006.  Google Scholar

[3]

P. Bates, J. Li and K. Lu, Normally hyperbolic invariant manifolds for random dynamical systems, Trans. Amer. Math. Soc., 365 (2013), 5933-5966. doi: 10.1090/S0002-9947-2013-05825-4.  Google Scholar

[4]

P. Bates, J. Li and K. Lu, Invariant foliations for random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 3639-3666. doi: 10.3934/dcds.2014.34.3639.  Google Scholar

[5]

P. Bates, J. Li and K. Lu, Geometric singular perturbation theory with real noise, J. Differential Equations, 259 (2015), 5137-5167. doi: 10.1016/j.jde.2015.06.023.  Google Scholar

[6]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100 pp. doi: 10.1090/memo/0577.  Google Scholar

[7]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[8]

M. Krupa and P. Szmolyan, Extending singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Analysis, 33 (2001), 286-314. doi: 10.1137/S0036141099360919.  Google Scholar

[9]

K. Lu and Q.-D. Wang, Chaos in differential equations driven by a nonautonomous force, Nonlinearity, 23 (2010), 2935-2975. doi: 10.1088/0951-7715/23/11/012.  Google Scholar

[10]

E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, New York, 1980. doi: 10.1007/978-1-4615-9047-7.  Google Scholar

[11]

P. Szmolyan and M. Wechselberger, Relaxation oscillation in $R^3$, J. Differential Equations, 200 (2004), 69-104. doi: 10.1016/j.jde.2003.09.010.  Google Scholar

[12]

D. Terman, The transition from bursting to continuous spiking in an excitable membrane model, J. Nonlinear Sci., 2 (1992), 133-182. doi: 10.1007/BF02429854.  Google Scholar

[13]

D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, SIAM J. Appl. Math., 51 (1991), 1418-1450. doi: 10.1137/0151071.  Google Scholar

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