June 2017, 4(1&2): 143-165. doi: 10.3934/jcd.2017005

A Lin's method approach for detecting all canard orbits arising from a folded node

Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

* Corresponding author: b.krauskopf@auckland.ac.nz

Published  March 2018

Canard orbits are relevant objects in slow-fast dynamical systems that organize the spiraling of orbits nearby. In three-dimensional vector fields with two slow and one fast variables, canard orbits arise from the intersection between an attracting and a repelling two-dimensional slow manifold. Special points called folded nodes generate such intersections: in a suitable transverse two-dimensional section Σ, the attracting and repelling slow manifolds are counter-rotating spirals that intersect in a finite number of points. We present an implementation of Lin's method that is able to detect all of these intersection points and, hence, all of the canard orbits arising from a folded node. With a boundary-value-problem setup we compute orbit segments on each slow manifold up to Σ, where we require that the corresponding end points in Σ lie in a one-dimensional subspace known as the Lin space Z. The Lin space Z must be transverse to the slow manifolds and it remains fixed during the detection of canard orbits as zeros of the signed distance along Z. During the computation, a tangency of Z with one of the intersection curves in Σ may arise. To overcome this, we update the Lin space at an intermediate continuation step to detect a double tangency of Z to both curves in Σ, after which the canard detection is able to continue. Our method is demonstrated with the examples of the normal form for a folded node and of the Koper model.

Citation: José Mujica, Bernd Krauskopf, Hinke M. Osinga. A Lin's method approach for detecting all canard orbits arising from a folded node. Journal of Computational Dynamics, 2017, 4 (1&2) : 143-165. doi: 10.3934/jcd.2017005
References:
[1]

V. I. Arnol'd, ed., Encyclopedia of Mathematical Sciences: Dynamical Systems V, Springer-Verlag, Berlin, New York, 1994.

[2]

E. Benoît, Systèmes lents-rapides dans $\mathbb R^3$ et leurs canards, Astérisque, 2 (1983), 159-191.

[3]

B. Braaksma, Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlinear Sci., 8 (1998), 457-490. doi: 10.1007/s003329900058.

[4]

M. Brøns and K. Bar-Eli, Canard explosion and excitation in a model of the Belousov-Zhabotinskii reaction, J. Phys. Chem, 95 (1991), 8706-8713.

[5]

M. BrønsM. Krupa and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon, Fields Institute Communications, 49 (2006), 39-63.

[6]

P. De Maesschalck and M. Wechselberger, Neural excitability and singular bifurcations, Journal of Mathematical Neuroscience, 5 (2015), 16.

[7]

M. DesrochesB. 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.

[8]

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

[9]

M. DesrochesJ. GuckenheimerB. KrauskopfC. KuehnH. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Review, 54 (2012), 211-288. doi: 10.1137/100791233.

[10]

E. J. Doedel and B. E. Oldeman, AUTO-07p: Continuation and bifurcation software for ordinary differential equations, with major contribution from A. R. Champneys, F. Dercole, T. F. Fairgrieve, Yu. A. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. Zhang, 2012. Available from: http://cmvl.cs.concordia.ca/auto/.

[11]

J. DroverJ. RubinJ. Su and B. Ermentrout, Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies, SIAM J. Appl. Math., 65 (2004), 69-92. doi: 10.1137/S0036139903431233.

[12]

F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Memoirs of the AMS, 121, 1996.

[13]

J. P. EnglandB. Krauskopf and H. M. Osinga, Computing two-dimensional global invariant manifolds in slow-fast systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 805-822. doi: 10.1142/S0218127407017562.

[14]

I. R. Epstein and K. Showalter, Nonlinear chemical dynamics: Oscillations, patterns, and chaos, J. Phys. Chem., 100 (1996), 13132-13147. doi: 10.1021/jp953547m.

[15]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226. doi: 10.1512/iumj.1972.21.21017.

[16]

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.

[17]

J. GinouxB. Rossetto and L. Chua, Slow invariant manifolds as curvature of the flow of dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 3409-3430. doi: 10.1142/S0218127408022457.

[18]

A. GiraldoB. Krauskopf and H. M. Osinga, Saddle invariant objects and their global manifolds in a neighborhood of a homoclinic flip bifurcation of case B, SIAM J. Appl. Dyn. Sys., 16 (2017), 640-686. doi: 10.1137/16M1097419.

[19]

A. GoryachevP. Strizhak and R. Kapral, Slow manifold structure and the emergence of mixed-mode oscillations, J. Chem. Phys., 107 (1997), 2881-2889.

[20]

J. Guckenheimer, Singular Hopf bifurcation in systems with two slow variables, SIAM J. Appl. Dyn. Sys., 7 (2008), 1355-1377. doi: 10.1137/080718528.

[21]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, Berlin, New York, 1983.

[22]

J. GuckenheimerK. Hoffman and W. Weckesser, The forced Van der Pol equation Ⅰ: The slow flow and its bifurcations, SIAM J. Appl. Dyn. Sys., 2 (2003), 1-35. doi: 10.1137/S1111111102404738.

[23]

J. Guckenheimer and R. Haiduc, Canards at folded nodes, Mosc. Math. J., 5 (2005), 91-103.

[24]

J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type, SIAM J. Appl. Dyn. Sys., 8 (2009), 854-879. doi: 10.1137/080741999.

[25]

A. Haro, M. Canadell, J. Figueras, A. Luque and J. Mondelo, The Parameterization Method for Invariant Manifolds, Applied Mathematical Sciences, vol. 195, Springer, Berlin, New York, 2016.

[26]

C. HasanB. Krauskopf and H. M. Osinga, Mixed-mode oscillations and twin canard orbits in an autocatalytic chemical reaction, SIAM J. Appl. Dyn. Sys., 16 (2017), 2165-2195. doi: 10.1137/16M1099248.

[27]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems III (Edited by H. Broer, F. Takens and B. Hasselblatt), Elsevier, 2010,379-524. doi: 10.1016/S1874-575X(10)00316-4.

[28]

J. L. HudsonM. Hart and J. Marinko, An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 71 (1979), 1601-1606. doi: 10.1063/1.438487.

[29]

E. M. Izhikevich, Neural excitability, spiking and bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.

[30]

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, Cambridge, 2007.

[31]

C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems (Montecatini Terme, 1994) (ed. J. Russell), Springer-Verlag, Berlin, New York, 1995, 44-118.

[32]

J. Knobloch, Lin's Method for Discrete and Continuous Dynamical Systems and Applications, Habilitationsschrift, TU Ilmenau, 2004.

[33]

J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.

[34]

M. T. M. Koper, Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram, Phys. D, 80 (1995), 72-94. doi: 10.1016/0167-2789(95)90061-6.

[35]

M. T. M. Koper and P. Gaspard, Mixed-mode oscillations in a simple model of an electrochemical oscillator, J. Phys. Chem., 95 (1991), 4945-4947. doi: 10.1021/j100166a009.

[36]

B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, in Numerical Continuation Methods for Dynamical Systems: Path following and boundary value problems (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), SpringerVerlag, Berlin, New York, 2007,117-154.

[37]

B. KrauskopfH. M. OsingaE. J. DoedelM. HendersonJ. GuckenheimerA. VladimirskyM. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791. doi: 10.1142/S0218127405012533.

[38]

B. Krauskopf and T. Rieß, A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690. doi: 10.1088/0951-7715/21/8/001.

[39]

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.

[40]

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

[41]

C. Kuehn, Multiple Time Scale Dynamics, Applied Mathematical Sciences, vol. 191, Springer, Berlin, New York, 2015.

[42]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, Berlin, New York, 2004.

[43]

X.-B. Lin, Using Melnikov's method to solve Shilnikov's problem, Proc. R. Soc. Edinb, 116 (1990), 295-325. doi: 10.1017/S0308210500031528.

[44]

A. Milik and P. Szmolyan, Multiple time scales and canards in a chemical oscillator, in Multiple-Time-Scale Dynamical Systems (eds. B. Krauskopf, C. Jones and A. Khibnik), IMA Vol. Math. Appl., vol. 122, Springer-Verlag, Berlin, New York, 2001,117-140.

[45]

J. Mujica, B. Krauskopf and H. M. Osinga, Tangencies between global invariant manifolds and slow manifolds near a singular Hopf bifurcation, SIAM J. Appl. Dyn. Syst. (in press).

[46]

B. OldemanA. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977-2999. doi: 10.1142/S0218127403008326.

[47]

V. PetrovS. K Scott and K. Showalter, Mixed-mode oscillations in chemical systems, J. Chem. Phys., 97 (1992), 6191-6198. doi: 10.1063/1.463727.

[48]

J. D. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Diff. Eq., 218 (2005), 390-443. doi: 10.1016/j.jde.2005.03.016.

[49]

H. G. RotsteinN. KopellA. M. Zhabotinsky and I. R. Epstein, Canard phenomenon and localization of oscillations in the Belousov-Zhabotinsky reaction with global feedback, J. Chem. Phys., 119 (2003), 8824-8832. doi: 10.1063/1.1614752.

[50]

R. A. SchmitzK. R. Graziani and J. L. Hudson, Experimental evidence of chaotic states in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 67 (1977), 3040-3044. doi: 10.1063/1.435267.

[51]

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Adison-Wesley, Reading, MA, 1994.

[52]

P. Szmolyan and M. Wechselberger, Canards in $\mathbb R^3$, J. Diff. Eq., 177 (2001), 419-453.

[53]

F. Tracqui, Organizing centres and symbolic dynamic in the study of mixed-mode oscillations generated by models of biological autocatalytic processes, Acta Biotheoretica, 42 (1994), 147-166. doi: 10.1007/BF00709487.

[54]

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. doi: 10.1137/030601995.

[55]

M. Wechselberger, J. Mitry and J. Rinzel, Canard theory and excitability, in Nonautonomous Dynamical Systems in the Life Sciences (eds. P. Kloeden and C. Poetzsche), Lecture Notes in Mathematics, vol. 2102, Springer, Berlin, New York, 2013, 89-132.

[56]

A. C. Yew, Multipulses of nonlinearly coupled Schrödinger equations, J. Diff. Eq., 173 (2001), 92-137. doi: 10.1006/jdeq.2000.3922.

show all references

References:
[1]

V. I. Arnol'd, ed., Encyclopedia of Mathematical Sciences: Dynamical Systems V, Springer-Verlag, Berlin, New York, 1994.

[2]

E. Benoît, Systèmes lents-rapides dans $\mathbb R^3$ et leurs canards, Astérisque, 2 (1983), 159-191.

[3]

B. Braaksma, Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlinear Sci., 8 (1998), 457-490. doi: 10.1007/s003329900058.

[4]

M. Brøns and K. Bar-Eli, Canard explosion and excitation in a model of the Belousov-Zhabotinskii reaction, J. Phys. Chem, 95 (1991), 8706-8713.

[5]

M. BrønsM. Krupa and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon, Fields Institute Communications, 49 (2006), 39-63.

[6]

P. De Maesschalck and M. Wechselberger, Neural excitability and singular bifurcations, Journal of Mathematical Neuroscience, 5 (2015), 16.

[7]

M. DesrochesB. 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.

[8]

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

[9]

M. DesrochesJ. GuckenheimerB. KrauskopfC. KuehnH. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Review, 54 (2012), 211-288. doi: 10.1137/100791233.

[10]

E. J. Doedel and B. E. Oldeman, AUTO-07p: Continuation and bifurcation software for ordinary differential equations, with major contribution from A. R. Champneys, F. Dercole, T. F. Fairgrieve, Yu. A. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. Zhang, 2012. Available from: http://cmvl.cs.concordia.ca/auto/.

[11]

J. DroverJ. RubinJ. Su and B. Ermentrout, Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies, SIAM J. Appl. Math., 65 (2004), 69-92. doi: 10.1137/S0036139903431233.

[12]

F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Memoirs of the AMS, 121, 1996.

[13]

J. P. EnglandB. Krauskopf and H. M. Osinga, Computing two-dimensional global invariant manifolds in slow-fast systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 805-822. doi: 10.1142/S0218127407017562.

[14]

I. R. Epstein and K. Showalter, Nonlinear chemical dynamics: Oscillations, patterns, and chaos, J. Phys. Chem., 100 (1996), 13132-13147. doi: 10.1021/jp953547m.

[15]

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226. doi: 10.1512/iumj.1972.21.21017.

[16]

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.

[17]

J. GinouxB. Rossetto and L. Chua, Slow invariant manifolds as curvature of the flow of dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 3409-3430. doi: 10.1142/S0218127408022457.

[18]

A. GiraldoB. Krauskopf and H. M. Osinga, Saddle invariant objects and their global manifolds in a neighborhood of a homoclinic flip bifurcation of case B, SIAM J. Appl. Dyn. Sys., 16 (2017), 640-686. doi: 10.1137/16M1097419.

[19]

A. GoryachevP. Strizhak and R. Kapral, Slow manifold structure and the emergence of mixed-mode oscillations, J. Chem. Phys., 107 (1997), 2881-2889.

[20]

J. Guckenheimer, Singular Hopf bifurcation in systems with two slow variables, SIAM J. Appl. Dyn. Sys., 7 (2008), 1355-1377. doi: 10.1137/080718528.

[21]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, Berlin, New York, 1983.

[22]

J. GuckenheimerK. Hoffman and W. Weckesser, The forced Van der Pol equation Ⅰ: The slow flow and its bifurcations, SIAM J. Appl. Dyn. Sys., 2 (2003), 1-35. doi: 10.1137/S1111111102404738.

[23]

J. Guckenheimer and R. Haiduc, Canards at folded nodes, Mosc. Math. J., 5 (2005), 91-103.

[24]

J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type, SIAM J. Appl. Dyn. Sys., 8 (2009), 854-879. doi: 10.1137/080741999.

[25]

A. Haro, M. Canadell, J. Figueras, A. Luque and J. Mondelo, The Parameterization Method for Invariant Manifolds, Applied Mathematical Sciences, vol. 195, Springer, Berlin, New York, 2016.

[26]

C. HasanB. Krauskopf and H. M. Osinga, Mixed-mode oscillations and twin canard orbits in an autocatalytic chemical reaction, SIAM J. Appl. Dyn. Sys., 16 (2017), 2165-2195. doi: 10.1137/16M1099248.

[27]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems III (Edited by H. Broer, F. Takens and B. Hasselblatt), Elsevier, 2010,379-524. doi: 10.1016/S1874-575X(10)00316-4.

[28]

J. L. HudsonM. Hart and J. Marinko, An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 71 (1979), 1601-1606. doi: 10.1063/1.438487.

[29]

E. M. Izhikevich, Neural excitability, spiking and bursting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.

[30]

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, Cambridge, 2007.

[31]

C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems (Montecatini Terme, 1994) (ed. J. Russell), Springer-Verlag, Berlin, New York, 1995, 44-118.

[32]

J. Knobloch, Lin's Method for Discrete and Continuous Dynamical Systems and Applications, Habilitationsschrift, TU Ilmenau, 2004.

[33]

J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.

[34]

M. T. M. Koper, Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram, Phys. D, 80 (1995), 72-94. doi: 10.1016/0167-2789(95)90061-6.

[35]

M. T. M. Koper and P. Gaspard, Mixed-mode oscillations in a simple model of an electrochemical oscillator, J. Phys. Chem., 95 (1991), 4945-4947. doi: 10.1021/j100166a009.

[36]

B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, in Numerical Continuation Methods for Dynamical Systems: Path following and boundary value problems (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), SpringerVerlag, Berlin, New York, 2007,117-154.

[37]

B. KrauskopfH. M. OsingaE. J. DoedelM. HendersonJ. GuckenheimerA. VladimirskyM. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791. doi: 10.1142/S0218127405012533.

[38]

B. Krauskopf and T. Rieß, A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690. doi: 10.1088/0951-7715/21/8/001.

[39]

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.

[40]

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

[41]

C. Kuehn, Multiple Time Scale Dynamics, Applied Mathematical Sciences, vol. 191, Springer, Berlin, New York, 2015.

[42]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, Berlin, New York, 2004.

[43]

X.-B. Lin, Using Melnikov's method to solve Shilnikov's problem, Proc. R. Soc. Edinb, 116 (1990), 295-325. doi: 10.1017/S0308210500031528.

[44]

A. Milik and P. Szmolyan, Multiple time scales and canards in a chemical oscillator, in Multiple-Time-Scale Dynamical Systems (eds. B. Krauskopf, C. Jones and A. Khibnik), IMA Vol. Math. Appl., vol. 122, Springer-Verlag, Berlin, New York, 2001,117-140.

[45]

J. Mujica, B. Krauskopf and H. M. Osinga, Tangencies between global invariant manifolds and slow manifolds near a singular Hopf bifurcation, SIAM J. Appl. Dyn. Syst. (in press).

[46]

B. OldemanA. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977-2999. doi: 10.1142/S0218127403008326.

[47]

V. PetrovS. K Scott and K. Showalter, Mixed-mode oscillations in chemical systems, J. Chem. Phys., 97 (1992), 6191-6198. doi: 10.1063/1.463727.

[48]

J. D. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Diff. Eq., 218 (2005), 390-443. doi: 10.1016/j.jde.2005.03.016.

[49]

H. G. RotsteinN. KopellA. M. Zhabotinsky and I. R. Epstein, Canard phenomenon and localization of oscillations in the Belousov-Zhabotinsky reaction with global feedback, J. Chem. Phys., 119 (2003), 8824-8832. doi: 10.1063/1.1614752.

[50]

R. A. SchmitzK. R. Graziani and J. L. Hudson, Experimental evidence of chaotic states in the Belousov-Zhabotinskii reaction, J. Chem. Phys., 67 (1977), 3040-3044. doi: 10.1063/1.435267.

[51]

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Adison-Wesley, Reading, MA, 1994.

[52]

P. Szmolyan and M. Wechselberger, Canards in $\mathbb R^3$, J. Diff. Eq., 177 (2001), 419-453.

[53]

F. Tracqui, Organizing centres and symbolic dynamic in the study of mixed-mode oscillations generated by models of biological autocatalytic processes, Acta Biotheoretica, 42 (1994), 147-166. doi: 10.1007/BF00709487.

[54]

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. doi: 10.1137/030601995.

[55]

M. Wechselberger, J. Mitry and J. Rinzel, Canard theory and excitability, in Nonautonomous Dynamical Systems in the Life Sciences (eds. P. Kloeden and C. Poetzsche), Lecture Notes in Mathematics, vol. 2102, Springer, Berlin, New York, 2013, 89-132.

[56]

A. C. Yew, Multipulses of nonlinearly coupled Schrödinger equations, J. Diff. Eq., 173 (2001), 92-137. doi: 10.1006/jdeq.2000.3922.

Figure 1.  Lin's method setup for finding canard orbits in system (16) with $\mu = 8.5$. Panel (a1) shows $S^a_{\varepsilon}$ (red surface) and $S^r_{\varepsilon}$ (blue surface) computed from $L^a$ and $L^r$, respectively, up to $\Sigma$. The initial orbit segments $u_a$ (red curve) and $u_r$ (blue curve) are each other's symmetric counterparts and define the Lin space $Z = span\{(0, 0, 1)\}$ (dark-gray line) that defines the Lin gap $\eta$. Panel (b1) shows the situation when the Lin gap is closed and the canard orbit $\xi_1$ (orange) is detected. The relevant objects in $\Sigma$ are shown in panels (a2) and (b2), respectively.
Figure 2.  Illustration of the Lin's method approach to detect canard orbits for (16) with $\mu = 8.5$ in the section $\Sigma$, which is the $(x, z)$-plane. Shown are the intersection sets $\widehat{S}^a_{\varepsilon}$ (red curve) and $\widehat{S}^r_{\varepsilon}$ (blue curve), together with the Lin space $Z$ (vertical dark-gray line). Panels (a1) and (a2) show the detection of the canard orbit $\xi_0$ (cyan), and panels (b1) and (b2) show the detection of $\xi_1$ (orange).
Figure 3.  Three-dimensional view of the slow manifolds $S^a_{\varepsilon}$ and $S^r_{\varepsilon}$, and all canard orbits $\xi_0$-$\xi_4$ of the normal form (16) for $\mu = 8.5$.
Figure 4.  Illustration of the Lin's method approach for (16) with $\mu = 8.5$ and a Lin space $Z$ in general position. Panel (a1) shows when $Z$ becomes tangent to $\widehat{S}^a_{\varepsilon}$ as the end points tracing $\widehat{S}^a_{\varepsilon}$ and $\widehat{S}^r_{\varepsilon}$ move to the right, and panel (a2) shows that it is not possible to detect a canard orbit by keeping $Z$ fixed. Panels (b1) and (b2) show a similar situation when the end points tracing $\widehat{S}^a_{\varepsilon}$ and $\widehat{S}^r_{\varepsilon}$ move to the left.
Figure 5.  Three-dimensional view of the slow manifolds computed up to section $\Sigma \subset \{z = -0.8\}$ of the Koper model (19) for the parameters values given by (20).
Figure 6.  Illustration of the Lin's method approach to detect canard orbits of (19) with parameter values as (20) in section $\Sigma$, represented by the $(x, y)$-plane. Shown are the intersection sets $\widehat{S}^a_{\varepsilon}$ (red curve) and $\widehat{S}^r_{\varepsilon}$ (blue curve), together with the corresponding Lin space (dark-gray line). Panel (a1) shows the detection of the canard orbit $\xi_0$ (cyan) and panel (a2) shows a tangency of the Lin space $Z_0$ with $\widehat{S}^r_{\varepsilon}$. Panels (b1) and (b2) show the detection of $\xi_1$ (orange) and $\xi_2$ (green), respectively.
Figure 7.  Intermediate step (Ⅲ) for the detection of a simultaneous tangency of the Lin space with $\widehat{S}^a_{\varepsilon}$ and $\widehat{S}^r_{\varepsilon}$, for the Koper model (19) with parameters as in (20). Panel (a1) shows the detection of the points defining the Lin space $Z_1$ and panel (a2) shows the corresponding fold of $\beta_a$. Panels (b1) and (b2) show step (Ⅲ) for the detection of the points defining the Lin space $Z_2$.
Figure 8.  Slow manifolds and the canard orbits $\xi_0$-$\xi_5$ of the Koper model (19) for the parameter values (20).
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