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Singular fold with real noise
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 |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-fast Dynamical Systems. A Sample-Paths Approach, Probab. Appl., Springer, London, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-fast Dynamical Systems. A Sample-Paths Approach, Probab. Appl., Springer, London, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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