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 & Continuous Dynamical Systems - A, 2013, 33 (10) : 4595-4611. doi: 10.3934/dcds.2013.33.4595
References:
[1]

J. M. Aguirregabiria, "Dynamics Solver v. 1.91,", 2012, (). Google Scholar

[2]

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

[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. doi: 10.1137/070709542. Google Scholar

[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). doi: 10.1063/1.3339819. Google Scholar

[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, (). doi: 10.1016/j.na.2012.05.027. Google Scholar

[6]

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

[7]

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

[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. doi: 10.1098/rspa.2011.0053. Google Scholar

[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). doi: 10.1063/1.2799471. Google Scholar

[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. doi: 10.1137/070708810. Google Scholar

[11]

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

[12]

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

[13]

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

[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. doi: 10.1142/S0218127405012934. Google Scholar

[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. doi: 10.1098/rspb.1984.0024. Google Scholar

[16]

E. M. Izhikevich, "Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting,", MIT Press, (2007). Google Scholar

[17]

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

[18]

C. K. R. T. Jones, "Geometric Singular Perturbation Theory,", Dynamical Systems, (1609), 44. doi: 10.1007/BFb0095239. Google Scholar

[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. doi: 10.1137/S0036141099360919. Google Scholar

[20]

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

[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). doi: 10.1103/PhysRevLett.98.074104. Google Scholar

[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. doi: 10.1080/00036811.2010.511193. Google Scholar

[23]

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

[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). doi: 10.1103/PhysRevE.71.056214. Google Scholar

[25]

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

[26]

P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, J. Differential Equations, 177 (2001), 419. doi: 10.1006/jdeq.2001.4001. Google Scholar

[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. doi: 10.1137/030601995. Google Scholar

[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. doi: 10.3934/dcdss.2009.2.829. Google Scholar

show all references

References:
[1]

J. M. Aguirregabiria, "Dynamics Solver v. 1.91,", 2012, (). Google Scholar

[2]

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

[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. doi: 10.1137/070709542. Google Scholar

[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). doi: 10.1063/1.3339819. Google Scholar

[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, (). doi: 10.1016/j.na.2012.05.027. Google Scholar

[6]

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

[7]

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

[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. doi: 10.1098/rspa.2011.0053. Google Scholar

[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). doi: 10.1063/1.2799471. Google Scholar

[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. doi: 10.1137/070708810. Google Scholar

[11]

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

[12]

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

[13]

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

[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. doi: 10.1142/S0218127405012934. Google Scholar

[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. doi: 10.1098/rspb.1984.0024. Google Scholar

[16]

E. M. Izhikevich, "Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting,", MIT Press, (2007). Google Scholar

[17]

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

[18]

C. K. R. T. Jones, "Geometric Singular Perturbation Theory,", Dynamical Systems, (1609), 44. doi: 10.1007/BFb0095239. Google Scholar

[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. doi: 10.1137/S0036141099360919. Google Scholar

[20]

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

[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). doi: 10.1103/PhysRevLett.98.074104. Google Scholar

[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. doi: 10.1080/00036811.2010.511193. Google Scholar

[23]

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

[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). doi: 10.1103/PhysRevE.71.056214. Google Scholar

[25]

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

[26]

P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, J. Differential Equations, 177 (2001), 419. doi: 10.1006/jdeq.2001.4001. Google Scholar

[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. doi: 10.1137/030601995. Google Scholar

[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. doi: 10.3934/dcdss.2009.2.829. Google Scholar

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