October  2013, 33(10): 4595-4611. doi: 10.3934/dcds.2013.33.4595

Canard trajectories in 3D piecewise linear systems

1. 

Dep. Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca, Illes Balears, Spain, Spain

Received  July 2012 Revised  January 2013 Published  April 2013

We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold ${\mathcal S}_{\varepsilon}$. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle.
Citation: Rafel Prohens, Antonio E. Teruel. Canard trajectories in 3D piecewise linear systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4595-4611. doi: 10.3934/dcds.2013.33.4595
References:
[1]

J. M. Aguirregabiria, "Dynamics Solver v. 1.91," 2012, http://tp.lc.ehu.es/jma.html.

[2]

E. Benoît, J. L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math., 32, (1981), 37-119.

[3]

V. Carmona, F. Fernández-Sanchez and A. E. Teruel, Existence of a reversible T-point heteroclinic cycle in a piecewise linear version of the Michelson system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1032-1048. doi: 10.1137/070709542.

[4]

V. Carmona, F. Fernández-Sanchez, E. García-Medina and A. E. Teruel, Existence of homoclinic connections in continuous piecewise linear systems, Chaos, 20 (2010), pp.8. doi: 10.1063/1.3339819.

[5]

V. Carmona, F. Fernández-Sanchez, E. García-Medina and A. E. Teruel, Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations, to appear in Nonlinear Analysis: Theory, Methods & Applications. doi: 10.1016/j.na.2012.05.027.

[6]

M. Desroches, J. M. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, (Preprint), to appear in SIAM Review. doi: 10.1137/100791233.

[7]

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

[8]

M. Desroches and M. Jeffrey, Canards and curvature: The smallness of epsilon in the slow-fast dynamics, Proc. R. Soc. A, 467 (2011), 2404-2421. doi: 10.1098/rspa.2011.0053.

[9]

M. Desroches, B. Krauskopf and H. M. Osinga, Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system, Chaos, 18 (2008), pp. 8 , 015107. doi: 10.1063/1.2799471.

[10]

M. Desroches, B. Krauskopf and H. M. Osinga, The geometry of slow manifolds near a folded node, SIAM J. Appl. Dyn. Syst., 7 (2008), 1131-1162. doi: 10.1137/070708810.

[11]

F. Dumortier and R. Roussarie, Canards cycles and center manifolds, Mem. Amer. Math. Soc., 557 (1996).

[12]

B. Ermentrout and M. Wechselberger, Canards, clusters, and synchronization in a weakly coupled interneuron model, SIAM J. Appl. Dyn. Syst., 8 (2009), 253-278. doi: 10.1137/080724010.

[13]

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.

[14]

J. M. Ginoux, B. Rossetto and J. L. Jamet, Chaos in a three-dimensional Volterra-Gause model of predator-prey type, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1689-1708. doi: 10.1142/S0218127405012934.

[15]

J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.

[16]

E. M. Izhikevich, "Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting," MIT Press, Computational Neuroscience. Cambridge, MA, 2007.

[17]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Kluwer Academic Publishers, The Netherlands, 1988.

[18]

C. K. R. T. Jones, "Geometric Singular Perturbation Theory," Dynamical Systems, Lecture Notes in Math. 1609, Springer Berlin/Heidelberg, 1995, 44-118. (electronic). doi: 10.1007/BFb0095239.

[19]

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

[20]

M. Krupa and P. Szmolyan, Relaxation oscillations and canard explosion, J. Differential Equations, 174 (2001), 312-368. doi: 10.1006/jdeq.2000.3929.

[21]

F. Marino, F. Marin, S. Balle and O. Piro, Chaotically spiking canards in an excitable system with 2D inertial fast manifolds, Phys. Rev. Lett., 98 (2007), pp.4, 074104. (electronic). doi: 10.1103/PhysRevLett.98.074104.

[22]

A. Pokrovskii, D. Rachinskii, V. Sobolev and A. Zhezherun, Topological degree in analysis of canard-type trajectories in 3-D systems, Applicable Analysis: An International Journal, 90 (2011), 1123-1139 (electronic). doi: 10.1080/00036811.2010.511193.

[23]

J. Rubin and M. Wechselberger, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model, Biological Cybernetics, 97 (2007), 5-32, (electronic). doi: 10.1007/s00422-007-0153-5.

[24]

A. Shilnikov, R. L. Calabrese and G. Cymbalyuk, Mechanism of bistability: Tonic spiking and bursting in a neuron model, Phys. Rev. E (3), 71 (2005), pp.9, 056214 (electronic). doi: 10.1103/PhysRevE.71.056214.

[25]

A. Shilnikov, R. L. Calabrese and G. Cymbalyuk, How a neuron model can demonstrate co-existence of tonic spiking and bursting, Neurocomputing, 65-66 (2005), 869-875, (electronic). doi: 10.1016/j.neucom.2004.10.107.

[26]

P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001.

[27]

M. Wechselberger, Existence and bifurcation of canards in $\mathbb R^3$ in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), 101-139, (electronic). doi: 10.1137/030601995.

[28]

M. Wechselberger and W. Weckesser, Homoclinic clusters and chaos associated with a folded node in a stellate cell model, Discrete Contin. Dyn. Syst. Ser. S, 4 (2009), 829-850, (electronic). doi: 10.3934/dcdss.2009.2.829.

show all references

References:
[1]

J. M. Aguirregabiria, "Dynamics Solver v. 1.91," 2012, http://tp.lc.ehu.es/jma.html.

[2]

E. Benoît, J. L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math., 32, (1981), 37-119.

[3]

V. Carmona, F. Fernández-Sanchez and A. E. Teruel, Existence of a reversible T-point heteroclinic cycle in a piecewise linear version of the Michelson system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1032-1048. doi: 10.1137/070709542.

[4]

V. Carmona, F. Fernández-Sanchez, E. García-Medina and A. E. Teruel, Existence of homoclinic connections in continuous piecewise linear systems, Chaos, 20 (2010), pp.8. doi: 10.1063/1.3339819.

[5]

V. Carmona, F. Fernández-Sanchez, E. García-Medina and A. E. Teruel, Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations, to appear in Nonlinear Analysis: Theory, Methods & Applications. doi: 10.1016/j.na.2012.05.027.

[6]

M. Desroches, J. M. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, (Preprint), to appear in SIAM Review. doi: 10.1137/100791233.

[7]

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

[8]

M. Desroches and M. Jeffrey, Canards and curvature: The smallness of epsilon in the slow-fast dynamics, Proc. R. Soc. A, 467 (2011), 2404-2421. doi: 10.1098/rspa.2011.0053.

[9]

M. Desroches, B. Krauskopf and H. M. Osinga, Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system, Chaos, 18 (2008), pp. 8 , 015107. doi: 10.1063/1.2799471.

[10]

M. Desroches, B. Krauskopf and H. M. Osinga, The geometry of slow manifolds near a folded node, SIAM J. Appl. Dyn. Syst., 7 (2008), 1131-1162. doi: 10.1137/070708810.

[11]

F. Dumortier and R. Roussarie, Canards cycles and center manifolds, Mem. Amer. Math. Soc., 557 (1996).

[12]

B. Ermentrout and M. Wechselberger, Canards, clusters, and synchronization in a weakly coupled interneuron model, SIAM J. Appl. Dyn. Syst., 8 (2009), 253-278. doi: 10.1137/080724010.

[13]

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.

[14]

J. M. Ginoux, B. Rossetto and J. L. Jamet, Chaos in a three-dimensional Volterra-Gause model of predator-prey type, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1689-1708. doi: 10.1142/S0218127405012934.

[15]

J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.

[16]

E. M. Izhikevich, "Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting," MIT Press, Computational Neuroscience. Cambridge, MA, 2007.

[17]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Kluwer Academic Publishers, The Netherlands, 1988.

[18]

C. K. R. T. Jones, "Geometric Singular Perturbation Theory," Dynamical Systems, Lecture Notes in Math. 1609, Springer Berlin/Heidelberg, 1995, 44-118. (electronic). doi: 10.1007/BFb0095239.

[19]

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

[20]

M. Krupa and P. Szmolyan, Relaxation oscillations and canard explosion, J. Differential Equations, 174 (2001), 312-368. doi: 10.1006/jdeq.2000.3929.

[21]

F. Marino, F. Marin, S. Balle and O. Piro, Chaotically spiking canards in an excitable system with 2D inertial fast manifolds, Phys. Rev. Lett., 98 (2007), pp.4, 074104. (electronic). doi: 10.1103/PhysRevLett.98.074104.

[22]

A. Pokrovskii, D. Rachinskii, V. Sobolev and A. Zhezherun, Topological degree in analysis of canard-type trajectories in 3-D systems, Applicable Analysis: An International Journal, 90 (2011), 1123-1139 (electronic). doi: 10.1080/00036811.2010.511193.

[23]

J. Rubin and M. Wechselberger, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model, Biological Cybernetics, 97 (2007), 5-32, (electronic). doi: 10.1007/s00422-007-0153-5.

[24]

A. Shilnikov, R. L. Calabrese and G. Cymbalyuk, Mechanism of bistability: Tonic spiking and bursting in a neuron model, Phys. Rev. E (3), 71 (2005), pp.9, 056214 (electronic). doi: 10.1103/PhysRevE.71.056214.

[25]

A. Shilnikov, R. L. Calabrese and G. Cymbalyuk, How a neuron model can demonstrate co-existence of tonic spiking and bursting, Neurocomputing, 65-66 (2005), 869-875, (electronic). doi: 10.1016/j.neucom.2004.10.107.

[26]

P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001.

[27]

M. Wechselberger, Existence and bifurcation of canards in $\mathbb R^3$ in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), 101-139, (electronic). doi: 10.1137/030601995.

[28]

M. Wechselberger and W. Weckesser, Homoclinic clusters and chaos associated with a folded node in a stellate cell model, Discrete Contin. Dyn. Syst. Ser. S, 4 (2009), 829-850, (electronic). doi: 10.3934/dcdss.2009.2.829.

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