Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci

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doi: 10.3934/dcdss.2022036

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Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci

Paul Carter ^1,, and Alan R. Champneys ^2,

1.	Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA
2.	Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK

^* Corresponding author: Paul Carter

Received August 2021 Revised January 2022 Early access February 2022

Fund Project: The first author is supported by NSF grant DMS-2016216

A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a canard explosion. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.

Keywords: Canards, traveling waves, periodic orbits, FitzHugh–Nagumo system, geometric singular perturbation theory.

Mathematics Subject Classification: Primary: 34E17; Secondary: 34C23, 37G15, 35C07.

Citation: Paul Carter, Alan R. Champneys. Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022036

References:

[1]	E. Benoît, J.-L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math., 32 (1981), 37-119.
[2]	P. Carter, Spike-adding canard explosion in a class of square-wave bursters, J. Nonlinear Sci., 30 (2020), 2613-2669. doi: 10.1007/s00332-020-09631-y.
[3]	P. Carter, J. D. M. Rademacher and B. Sandstede, Pulse replication and accumulation of eigenvalues, SIAM J. Math. Anal., 53 (2021), 3520-3576. doi: 10.1137/20M1340113.
[4]	P. Carter and B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh–Nagumo system, SIAM J. Math. Anal., 47 (2015), 3393-3441. doi: 10.1137/140999177.
[5]	P. Carter and B. Sandstede, Unpeeling a homoclinic banana in the FitzHugh–Nagumo system, SIAM J. Appl. Dyn. Syst., 17 (2018), 236-349. doi: 10.1137/16M1080707.
[6]	P. Carter and A. Scheel, Wave train selection by invasion fronts in the FitzHugh–Nagumo equation, Nonlinearity, 31 (2018), 5536-5572. doi: 10.1088/1361-6544/aae1db.
[7]	A. R. Champneys, Homoclinic orbits in the dynamics of articulated pipes conveying fluid, Nonlinearity, 4 (1991), 747-774. doi: 10.1088/0951-7715/4/3/007.
[8]	A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When shil'nikov meets hopf in excitable systems, SIAM J. Appl. Dyn. Syst., 6 (2007), 663-693. doi: 10.1137/070682654.
[9]	M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Rev., 54 (2012), 211-288. doi: 10.1137/100791233.
[10]	M. Desroches, T. J. Kaper and M. Krupa, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 046106, 13 pp. doi: 10.1063/1.4827026.
[11]	E. Doedel, B. Oldeman et al., Auto-07p: Continuation and bifurcation software for ordinary differential equations, 2020, Latest version at https://github.com/auto-07p.
[12]	F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), no. 577,100 pp. doi: 10.1090/memo/0577.
[13]	C. Fall, E. Marland, J. Wagner and J. Tyson, Computational Cell Biology, Springer-Verlag, New York, 2002.
[14]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical journal, 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.
[15]	P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems; A two-parameter analysis, J. Stat. Phys., 35 (1984), 697-727. doi: 10.1007/BF01010829.
[16]	P. Glendinning and C. Sparrow, Local and global behaviour near homoclinic orbits, J. Stat. Phys., 35 (1984), 645-696. doi: 10.1007/BF01010828.
[17]	J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh–Nagumo equation: The singular-limit, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 851-872. doi: 10.3934/dcdss.2009.2.851.
[18]	J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst., 9 (2010), 138-153. doi: 10.1137/090758404.
[19]	S. P. Hastings, On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations, Quart. J. Math. Oxford Ser., 27 (1976), 123-134. doi: 10.1093/qmath/27.1.123.
[20]	S. P. Hastings, Single and multiple pulse waves for the FitzHugh–Nagumo equations, SIAM J. Appl. Math., 42 (1982), 247-260. doi: 10.1137/0142018.
[21]	C. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, in Patterns and Dynamics in Reactive Media, Springer, 1991,101–115. doi: 10.1007/978-1-4612-3206-3_7.
[22]	T. Kaper and C. Jones, A primer on the exchange lemma for fast-slow systems, vol. 122 of The IMA Volumes in Mathematics and its Applications, 65–87, Springer, New York, 2001. doi: 10.1007/978-1-4613-0117-2_3.
[23]	J. Keener and J. Sneyd, Mathematical Physiology, 2nd edition, Springer-Verlag, New York, 2009.
[24]	M. Krupa, B. Sandstede and P. Szmolyan, Fast and slow waves in the FitzHugh–Nagumo equation, J. Differential Equations, 133 (1997), 49-97. doi: 10.1006/jdeq.1996.3198.
[25]	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.
[26]	M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368. doi: 10.1006/jdeq.2000.3929.
[27]	C. Kuehn, Multiple Time Series Dynamical Systems, Springer-Verlag, Heidelberg, 2015, Applied Mathematical Sciences, vol. 191.
[28]	Y. Kuznetsov and A. Panfilov, Stochastic waves in the FitzHugh-Nagumo system, 1981, Research Computing Centre, USSR Academy of Sciences, Pushchino. In Russian.
[29]	D. Linaro, A. Champneys, M. Desroches and M. Storace, Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster, SIAM J. Appl. Dyn. Syst., 11 (2012), 939-962. doi: 10.1137/110848931.
[30]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.
[31]	H. Osinga and K. Tsaneva-Atanasova, Dynamics of plateau bursting depending on the location of its equilibrium, Journal of Neuroendocrinology, 22 (2010), 1301-1314. doi: 10.1111/j.1365-2826.2010.02083.x.
[32]	J. D. M. Rademacher, Homoclinic Bifurcation from Heteroclinic Cycles with Periodic Orbits and Tracefiring of Pulses, Ph.D. thesis, University of Minnesota, 2004, http://www.math.uni-bremen.de/~jdmr/pub/dissMay7Web.pdf.
[33]	L. Shilnikov, On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Math. USSR Sb., 6 (1968), 427-438.
[34]	L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Method of Qualitative Theory of Nonlinear Dynamics: Part II, World Scientific, Singapore, 2001. doi: 10.1142/9789812798558_0001.
[35]	C. Soto-Trevino, Geometric Methods for Periodic Orbits in Singularly Perturbed Systems, Ph.D. Thesis, Boston University, 1998.
[36]	P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001.
[37]	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.
[38]	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.

show all references

References:

[1]	E. Benoît, J.-L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math., 32 (1981), 37-119.
[2]	P. Carter, Spike-adding canard explosion in a class of square-wave bursters, J. Nonlinear Sci., 30 (2020), 2613-2669. doi: 10.1007/s00332-020-09631-y.
[3]	P. Carter, J. D. M. Rademacher and B. Sandstede, Pulse replication and accumulation of eigenvalues, SIAM J. Math. Anal., 53 (2021), 3520-3576. doi: 10.1137/20M1340113.
[4]	P. Carter and B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh–Nagumo system, SIAM J. Math. Anal., 47 (2015), 3393-3441. doi: 10.1137/140999177.
[5]	P. Carter and B. Sandstede, Unpeeling a homoclinic banana in the FitzHugh–Nagumo system, SIAM J. Appl. Dyn. Syst., 17 (2018), 236-349. doi: 10.1137/16M1080707.
[6]	P. Carter and A. Scheel, Wave train selection by invasion fronts in the FitzHugh–Nagumo equation, Nonlinearity, 31 (2018), 5536-5572. doi: 10.1088/1361-6544/aae1db.
[7]	A. R. Champneys, Homoclinic orbits in the dynamics of articulated pipes conveying fluid, Nonlinearity, 4 (1991), 747-774. doi: 10.1088/0951-7715/4/3/007.
[8]	A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When shil'nikov meets hopf in excitable systems, SIAM J. Appl. Dyn. Syst., 6 (2007), 663-693. doi: 10.1137/070682654.
[9]	M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Rev., 54 (2012), 211-288. doi: 10.1137/100791233.
[10]	M. Desroches, T. J. Kaper and M. Krupa, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 046106, 13 pp. doi: 10.1063/1.4827026.
[11]	E. Doedel, B. Oldeman et al., Auto-07p: Continuation and bifurcation software for ordinary differential equations, 2020, Latest version at https://github.com/auto-07p.
[12]	F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), no. 577,100 pp. doi: 10.1090/memo/0577.
[13]	C. Fall, E. Marland, J. Wagner and J. Tyson, Computational Cell Biology, Springer-Verlag, New York, 2002.
[14]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical journal, 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.
[15]	P. Gaspard, R. Kapral and G. Nicolis, Bifurcation phenomena near homoclinic systems; A two-parameter analysis, J. Stat. Phys., 35 (1984), 697-727. doi: 10.1007/BF01010829.
[16]	P. Glendinning and C. Sparrow, Local and global behaviour near homoclinic orbits, J. Stat. Phys., 35 (1984), 645-696. doi: 10.1007/BF01010828.
[17]	J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh–Nagumo equation: The singular-limit, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 851-872. doi: 10.3934/dcdss.2009.2.851.
[18]	J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst., 9 (2010), 138-153. doi: 10.1137/090758404.
[19]	S. P. Hastings, On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations, Quart. J. Math. Oxford Ser., 27 (1976), 123-134. doi: 10.1093/qmath/27.1.123.
[20]	S. P. Hastings, Single and multiple pulse waves for the FitzHugh–Nagumo equations, SIAM J. Appl. Math., 42 (1982), 247-260. doi: 10.1137/0142018.
[21]	C. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, in Patterns and Dynamics in Reactive Media, Springer, 1991,101–115. doi: 10.1007/978-1-4612-3206-3_7.
[22]	T. Kaper and C. Jones, A primer on the exchange lemma for fast-slow systems, vol. 122 of The IMA Volumes in Mathematics and its Applications, 65–87, Springer, New York, 2001. doi: 10.1007/978-1-4613-0117-2_3.
[23]	J. Keener and J. Sneyd, Mathematical Physiology, 2nd edition, Springer-Verlag, New York, 2009.
[24]	M. Krupa, B. Sandstede and P. Szmolyan, Fast and slow waves in the FitzHugh–Nagumo equation, J. Differential Equations, 133 (1997), 49-97. doi: 10.1006/jdeq.1996.3198.
[25]	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.
[26]	M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368. doi: 10.1006/jdeq.2000.3929.
[27]	C. Kuehn, Multiple Time Series Dynamical Systems, Springer-Verlag, Heidelberg, 2015, Applied Mathematical Sciences, vol. 191.
[28]	Y. Kuznetsov and A. Panfilov, Stochastic waves in the FitzHugh-Nagumo system, 1981, Research Computing Centre, USSR Academy of Sciences, Pushchino. In Russian.
[29]	D. Linaro, A. Champneys, M. Desroches and M. Storace, Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster, SIAM J. Appl. Dyn. Syst., 11 (2012), 939-962. doi: 10.1137/110848931.
[30]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.
[31]	H. Osinga and K. Tsaneva-Atanasova, Dynamics of plateau bursting depending on the location of its equilibrium, Journal of Neuroendocrinology, 22 (2010), 1301-1314. doi: 10.1111/j.1365-2826.2010.02083.x.
[32]	J. D. M. Rademacher, Homoclinic Bifurcation from Heteroclinic Cycles with Periodic Orbits and Tracefiring of Pulses, Ph.D. thesis, University of Minnesota, 2004, http://www.math.uni-bremen.de/~jdmr/pub/dissMay7Web.pdf.
[33]	L. Shilnikov, On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Math. USSR Sb., 6 (1968), 427-438.
[34]	L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Method of Qualitative Theory of Nonlinear Dynamics: Part II, World Scientific, Singapore, 2001. doi: 10.1142/9789812798558_0001.
[35]	C. Soto-Trevino, Geometric Methods for Periodic Orbits in Singularly Perturbed Systems, Ph.D. Thesis, Boston University, 1998.
[36]	P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001.
[37]	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.
[38]	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.

Figure 1. Traveling wave trains obtained in the FitzHugh–Nagumo equation (2) for $ \varepsilon = 0.005, \gamma = p = 0 $ with wave speed $ s = 0.65 $: small amplitude canard cycle (left), and a large amplitude relaxation oscillation (right), corresponding to the blue and green orbits, respectively, depicted in Figure 2

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Figure 2. Plotted is the traveling canard explosion which emerges from the Hopf bifurcation at the equilibrium $ (v,d,w) = (0,0,0) $ at the parameter values $ p = 0 $, $ s = 0.65 $, $ \gamma = 0 $, and $ \varepsilon = 0.005 $, when continuing in the parameter $ \lambda $. The left panel shows the bifurcation diagram obtained by plotting the $ L^2 $-norm of the periodic orbits along the explosion vs. the parameter $ \lambda $; a zoomed in portion of this bifurcation diagram showing the folds along the canard explosion is shown in the inset. The colored circles in the bifurcation diagram correspond to the periodic orbits in the right panel plotted in $ (v,d,w) $ phase space along with the cubic critical manifold (shown in red). The explosion encompasses the transition from small amplitude oscillations (blue) born locally at the Hopf bifurcation, to canards "without head" (yellow) and "with head" (purple), to large amplitude "relaxation oscillation"-type orbits (green). The folds in the bifurcation branch are observed primarily along the upper part of the branch, associated to the canards with head. We will discuss why these folds appear only on this part of the branch in §5.2

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Figure 3. Shown the singular slow-fast geometry for the FitzHugh–Nagumo system (3) for $ \lambda = \varepsilon = 0 $

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Figure 4. The global setup of the slow-fast system (16) according to Hypotheses 1-4. For convenience the origin is taken to coincide with the fold point $ \mathcal{F} = (v_\mathrm{f}, d_\mathrm{f}, w_\mathrm{f}) $

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Figure 5. The assumed heteroclinic orbits in the layer problem according to Hypothesis 2 in the case $ w<w_0 $ (left) and $ w>w_0 $ (right). In the left panel, the heteroclinic orbits $ \phi(w) $ are shown to converge to $ \mathcal{C}^\ell_0 $ in the generic (weak stable) direction, as is the case for the FitzHugh–Nagumo system (3), but this is not necessary to satisfy Hypothesis 2 for the construction of periodic orbits

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Figure 6. Shown is the setup for Hypothesis 4 in the local coordinates $ (x,y,z) $ for the normal form (25) for $ \varepsilon = 0 $

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Figure 7. (a) Geometric setup for the construction of global canard orbits as in Theorem 3.1. (b) Setup for matching conditions in the section $ \Sigma $

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Figure 8. Splitting of the manifolds $ \mathcal{C}^{\ell,\mathrm{base}}_ \varepsilon $ and $ \mathcal{C}^{\ell,\mathrm{base}}_ \varepsilon $ in the center manifold $ \mathcal{W}^\mathrm{c}(\mathcal{F}) $

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Figure 9. Shown is the forward/backward evolution of the curve $ \Lambda $ along the manifolds $ \mathcal{C}^{\ell,r}_ \varepsilon $. The right panel shows the forward and backwards intersections $ \Lambda_\mathrm{start},\Lambda_\mathrm{end} $ of $ \Lambda $ with the section $ \Sigma $. The rotation along the left branch is accumulated in the region $ w>w_0 $ where the fast dynamics along $ \mathcal{C}^\ell_ \varepsilon $ are oscillatory

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Figure 10. Schematic wiggly canard bifurcation diagram of fast jump height $ \bar{w} $ versus $ \lambda $. The fold envelope and distance between successive folds are given by (53) and (61), respectively

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Figure 11. (Left) Results of one-parameter continuation in $ \lambda $ of periodic orbits of the synthetic example (72) for $ \delta = 2 $ and $ \varepsilon = 0.1 $. (Right) orbits at each of the fold points in the left-hand panel

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Figure 12. Similar to Fig. 12 but for $ \delta = 0.5 $ and $ \varepsilon = 0.02 $. The orbits in the right-hand plot are depicted for even increments in period between 60 and 120

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Figure 13. (Top) 2-parameter continuation of folds of solutions in Fig. 11. (Bottom) two successive zooms of the data in the top right plot

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Figure 14. Computed differences in $ w $-values of successive folds as a function of $ \varepsilon $, plot on a log-log scale. Also plotted for reference are straight lines with slopes 1 (dashed line) and 2/3 (dotted lines)

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Figure 15. Results of continuation of folds in the FitzHugh–Nagumo system (3) for $ \gamma = 0, s = 0.65 $. Top left: Plot of $ L^2 $-norm versus the parameter $ \lambda $ for the canard explosion of traveling wavetrains in (3) for $ \varepsilon = 0.005 $. Shown in the inset is a zoom of the location of the first (highest) six folds. Top right: Plotted is the jump height $ \bar{w} $ of the first six folds for decreasing $ \epsilon $. Bottom panels: Plotted are the differences between the jump heights of successive folds for decreasing $ \varepsilon $ as a standard plot (left) and log-log plot (right)

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Figure 16. (Top) Shown is the way-in-way-out function given by the difference $ R_\ell(w_*)-R_r(\bar{w}) $ (dashed red) plotted versus $ \bar{w} $ for the case of canards with head, as well as the difference $ R_\ell(\bar{w})-R_r(\bar{w}) $ (solid blue) in the case of canards without head. In the latter case, the quantity $ R_\ell(\bar{w})-R_r(\bar{w}) $ is briefly positive between $ \bar{w} = 0 $ and $ \bar{w}\approx 0.039 $. The bottom left and right panels, respectively, depict plots of the $ L^2 $ norm versus $ \lambda $ and the maximum height $ w $ along the orbit versus $ \lambda $ for the canard explosion of wave trains in (3) for $ \gamma = 0, s = 0.65, \varepsilon = 0.005 $. The square denotes the location along the bifurcation branch where the way-in-way-out quantity $ R_\ell(\bar{w})-R_r(\bar{w}) $ switches from positive to negative; hence one does not expect to observe folds below this location. The triangle denotes the approximate transition point above which the orbits correspond to canards with head and below which correspond to canards without head

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Figure 17. Shown are orbits in $ (v,w) $-space along the continuation of the fold LP$ 1 $ for values of $ \varepsilon = \{0.005, 0.004, 0.003, 0.002, 0.001\} $ (blue, red, yellow, purple, green). The dashed black line denotes the location of the Airy transition at $ w\approx 0.0031 $

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Figure 18. Canard explosion in the FitzHugh–Nagumo equation (3) for $ \varepsilon = 0.001 $ in the case $ s = 0.72 $ (left) and $ s = 0.65 $ (right). In the latter case, the canard explosion does not result in a family of relaxation oscillations, but rather a continuous spike-adding sequence through which additional large amplitude oscillations are accumulated via repeated canard explosions. The inset shows a zoom of this family of canard explosions along the upper portion of the branch

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Figure 19. Shown is a plot of period versus $ \lambda $ for the canard explosion in the case $ \varepsilon = 0.001, s = 0.65 $ as in Figure 18. The $ v $-profiles for periodic wave trains with $ 1,3,5,7 $, and $ 9 $ spikes, respectively, are depicted in the insets at various points along the spike adding branch. Each such profile is plotted over one period, which increases when moving vertically along the branch as additional spikes are added

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Table 1. Computation versus asymptotic prediction for $ w_{N+1}-w_N $ and for the accummulation rate of $ \lambda $-values for the explicitly constructed model with $ \delta = 2 $ and $ \varepsilon = 0.02 $. Here the second column gives the computed values of the difference between successive $ w_N $-values and the third column gives the value that this ratio should have according to (61), where $ h_\ell $ is estimated using (74). The fourth column gives the absolute error between these two quantities. The fifth column gives the computed value of the left-hand-side of the expression (62) that determines the rate of accummulation of $ \lambda $-values for the case that $ N = \mathcal{O}(1/ \varepsilon) $. The final column of the table gives the value of the right-hand-side of (62) for the given value of $ w_N $

$ N $	$ w_{N+1} - w_N $	$ - \varepsilon \pi h_\ell/\omega $	abs. error	$ \log\left ( \frac{\| \lambda(w_{N+1})-\lambda^{\rm mc} \|}{\| \lambda (w_N)-\lambda^{\rm mc} \|} \right) $	$ {\pi/\alpha/\omega} $
2	0.007800	0.009120	0.297041	-3.134495	-3.312145
3	0.007810	0.008331	0.143856	-2.766146	-2.914168
4	0.007659	0.007844	0.088228	-0.752243	-2.656183

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$ N $	$ w_{N+1} - w_N $	$ - \varepsilon \pi h_\ell/\omega $	abs. error	$ \log\left ( \frac{\| \lambda(w_{N+1})-\lambda^{\rm mc} \|}{\| \lambda (w_N)-\lambda^{\rm mc} \|} \right) $	$ {\pi/\alpha/\omega} $
2	0.007800	0.009120	0.297041	-3.134495	-3.312145
3	0.007810	0.008331	0.143856	-2.766146	-2.914168
4	0.007659	0.007844	0.088228	-0.752243	-2.656183

Table 2. Similar to Table 1 but for $ \varepsilon = 0.03 $, in which case additional fold curves could be computed

$ N $	$ w_{N+1} - w_N $	$ - \varepsilon \pi h_\ell/\omega $	abs. error	$ \log\left ( \frac{\| \lambda(w_{N+1})-\lambda^{\rm mc} \|}{\| \lambda (w_N)-\lambda^{\rm mc} \|} \right) $	$ {\pi/\alpha/\omega} $
2	0.010474	0.013585	0.229015	-2.670591	-3.312145
3	0.010851	0.012412	0.125764	-2.500844	-2.914168
4	0.010742	0.011690	0.081075	-1.963004	-2.656183
5	0.010570	0.011208	0.056862	-2.607696	-2.474712
6	0.010398	0.010861	0.042625	0.821494	-2.338275

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$ N $	$ w_{N+1} - w_N $	$ - \varepsilon \pi h_\ell/\omega $	abs. error	$ \log\left ( \frac{\| \lambda(w_{N+1})-\lambda^{\rm mc} \|}{\| \lambda (w_N)-\lambda^{\rm mc} \|} \right) $	$ {\pi/\alpha/\omega} $
2	0.010474	0.013585	0.229015	-2.670591	-3.312145
3	0.010851	0.012412	0.125764	-2.500844	-2.914168
4	0.010742	0.011690	0.081075	-1.963004	-2.656183
5	0.010570	0.011208	0.056862	-2.607696	-2.474712
6	0.010398	0.010861	0.042625	0.821494	-2.338275

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