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

  • * Corresponding author: Paul Carter

    * Corresponding author: Paul Carter 

The first author is supported by NSF grant DMS-2016216

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  • 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 acanard 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.

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


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  • 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

    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

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

    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}) $

    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

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

    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 $

    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}) $

    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

    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

    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

    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

    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

    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)

    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)

    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

    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 $

    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

    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

    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
     | Show Table
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    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
     | Show Table
    DownLoad: CSV
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