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Spike-Adding Canard Explosion in a Class of Square-Wave Bursters

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Abstract

This paper examines a spike-adding bifurcation phenomenon whereby small-amplitude canard cycles transition into large-amplitude bursting oscillations along a single continuous branch in parameter space. We consider a class of three-dimensional singularly perturbed ODEs with two fast variables and one slow variable and singular perturbation parameter \(\varepsilon \ll 1 \) under general assumptions which guarantee such a transition occurs. The primary ingredients include a cubic critical manifold and a saddle homoclinic bifurcation within the associated layer problem. The continuous transition from canard cycles to N-spike bursting oscillations up to \(N\sim {\mathcal {O}}(1/\varepsilon )\) spikes occurs upon varying a single bifurcation parameter on an exponentially thin interval. We construct this transition rigorously using geometric singular perturbation theory; critical to understanding this transition are the existence of canard orbits and slow passage through the saddle homoclinic bifurcation, which are analyzed in detail.

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Acknowledgements

The author gratefully acknowledges support through NSF Grant DMS-2016216 (formerly DMS-1815315).

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  1. School of Mathematics, University of Minnesota, Minneapolis, USA

    Paul Carter

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A Estimates Near the Saddle Homoclinic Point

A Estimates Near the Saddle Homoclinic Point

In this section, we present a proof of Lemma 3.11. We first quote the following result regarding the nature of solutions to the boundary value problem with entry/exit conditions in the sections \( \Sigma ^\mathrm {h}_A, \Sigma ^\mathrm {h}_B\).

Proposition A.1

(Schecter 2008a, Theorem 2.1) Fix \(\Delta >0\) small. There exists \(K_0,\eta >0\) such that the following holds. For any sufficiently small \(\varepsilon >0\), any \(T>0\) and any \(|Y^*|\le \delta _Y\), there exists a solution \((A,B,Y)(\xi ;Y^*,T)\) to (3.32) with \((A,B,Y)(0)\in \Sigma ^\mathrm {h}_A\) and \((A,B,Y)(T)\in \Sigma ^\mathrm {h}_B\) with \(Y(T;Y^*,T)=Y^*\). Furthermore,

$$\begin{aligned} \begin{aligned} |A(\xi ;Y^*,T)|&\le K_0e^{-\eta \xi }\\ |B(\xi ;Y^*,T)|&\le K_0e^{\eta (\xi -T)}\\ |Y(\xi ;Y^*,T)-\Phi (\xi ,Y^*,T)|&\le K_0\varepsilon e^{-\eta T}, \end{aligned} \end{aligned}$$
(A.1)

where \(\Phi (\xi ,Y^*,T)\) denotes the solution of \({\dot{Y}}=\varepsilon G_1(Y,k,\varepsilon )\) satisfying \(Y(T)=Y^*\). The partial derivatives of \((A,B,Y)(\xi ;Y^*,T)\) with respect to \(\xi ,Y^*,T\) up to order r satisfy the same estimates.

Remark A.2

We remark on the appearance of the factor of \(\varepsilon \) appearing in estimates (A.1) for the solution \(Y(\xi ;Y^*,T)\) which is not present in Schecter (2008a, Theorem 2.1). This is due to the fact that the Y-dynamics are of \({\mathcal {O}}(\varepsilon )\), in contrast to the more general case in Schecter (2008a), where there is no small parameter and hence the center dynamics are \({\mathcal {O}}(1)\).

Proof of Lemma 3.11

We use the formulation of Proposition A.1 to prove the estimates on the local map \(\Pi _\mathrm {loc}\). We fix \(\Delta >0\) and assume \(0<\delta _Y,\delta \ll \Delta \) are taken sufficiently small.

For a solution \((A,B,Y)(\xi ; Y^*,T)\) of Proposition A.1, we set \( {\tilde{A}}(Y^*,T):=A(T;Y^*,T)\) and \({\tilde{B}}(Y^*,T):=B(0;Y^*,T)={\mathcal {O}}(e^{-\eta T})\). The map \(\Pi _\mathrm {loc}\) is then determined by

$$\begin{aligned} \begin{aligned} B_\mathrm {loc}(R,Y^*)&= {\tilde{B}}(Y^*,T)\\ Y_\mathrm {loc}(R,Y^*)&= Y(0;Y^*,T). \end{aligned} \end{aligned}$$
(A.2)

where R is defined via the relation \(\Delta R^\rho = \tilde{A}(Y^*,T)\), and the exponent \(\rho \) is as yet to be determined.

Let \(\Phi (\xi ,Y^*,T)\) denote the solution of \({\dot{Y}}=\varepsilon G_1(Y,k,\varepsilon )\) satisfying \(Y(T)=Y^*\); in particular, \(\Phi (\xi ,Y^*,T)\) satisfies the integral equation

$$\begin{aligned} \Phi (\xi ,Y^*,T) =Y^*+\int _T^\xi \varepsilon G_1(\Phi (\xi ,Y^*,T),k,\varepsilon )\mathrm {d}\xi , \end{aligned}$$
(A.3)

and we have the estimates

$$\begin{aligned} \begin{aligned} \Phi (0,Y^*,T)&=Y^*+\varepsilon \gamma T(1+{\mathcal {O}}(\Delta ))\\ \partial _{Y^*} \Phi (\xi ,Y^*,T)&=1+{\mathcal {O}}( \Delta )\\ \partial _T \Phi (\xi ,Y^*,T)&={\mathcal {O}}( \varepsilon ). \end{aligned} \end{aligned}$$
(A.4)

We now define the functions

$$\begin{aligned} \begin{aligned} {\tilde{\alpha }}_0(Y^*,T)&:= \int _0^T F_1(0,0,\Phi (\xi ,Y^*,T),k,\varepsilon )\mathrm {d}\xi \\&=\int _0^T \alpha + {\mathcal {O}}(\Phi ,\varepsilon ) \mathrm {d}\xi \\ {\tilde{\beta }}_0(Y^*,T)&:= \int _0^T F_2(0,0,\Phi (\xi ,Y^*,T),k,\varepsilon )\mathrm {d}\xi \\&=\int _0^T \beta + {\mathcal {O}}(\Phi ,\varepsilon ) \mathrm {d}\xi , \end{aligned} \end{aligned}$$
(A.5)

where

$$\begin{aligned} \begin{aligned} \partial _{Y^*}{\tilde{\alpha }}_0(Y^*,T)&={\mathcal {O}}(T)\\ \partial _T{\tilde{\alpha }}_0(Y^*,T)&= \alpha + {\mathcal {O}}(\Delta )\\ \partial _{Y^*}{\tilde{\beta }}_0(Y^*,T)&={\mathcal {O}}(T)\\ \partial _T{\tilde{\beta }}_0(Y^*,T)&= \beta + {\mathcal {O}}(\Delta ). \end{aligned} \end{aligned}$$
(A.6)

We further define the functions

$$\begin{aligned} \begin{aligned} {\tilde{\alpha }}(Y^*,T)&:= \int _0^T F_1\left( A(\xi ;Y^*,T),B(\xi ;Y^*,T),Y(\xi ;Y^*,T),k,\varepsilon \right) \mathrm {d}\xi \\ {\tilde{\beta }}(Y^*,T)&:= \int _0^T F_2\left( A(\xi ;Y^*,T),B(\xi ;Y^*,T),Y(\xi ;Y^*,T),k,\varepsilon \right) \mathrm {d}\xi . \end{aligned} \end{aligned}$$
(A.7)

We use the estimates in Proposition A.1 combined with directly integrating Eq. (3.32) in reverse time and obtain

$$\begin{aligned} \begin{aligned} {\tilde{A}}&= \Delta \exp \left( -{\tilde{\alpha }}(Y^*,T) \right) \\ {\tilde{B}}&= \Delta \exp \left( -{\tilde{\beta }}(Y^*,T)\right) . \end{aligned} \end{aligned}$$
(A.8)

Using these expressions along with estimates (A.1), we have that

$$\begin{aligned} \begin{aligned} |{\tilde{\alpha }}(Y^*,T)-{\tilde{\alpha }}_0(Y^*,T)|&= {\mathcal {O}}(\Delta )\\ |{\tilde{\beta }}(Y^*,T)-{\tilde{\beta }}_0(Y^*,T)|&= {\mathcal {O}}(\Delta ) \end{aligned} \end{aligned}$$
(A.9)

and the partial derivatives of these expressions with respect to \(Y^*,T\) are also \({\mathcal {O}}(\Delta )\).

The ultimate goal is to express the quantities \({\tilde{B}}\) and \(Y(0;Y^*,T)\) in terms of the quantities \(R, Y^*\), where we define

$$\begin{aligned} \begin{aligned} R&= \left( \frac{{\tilde{A}}}{\Delta }\right) ^{{\tilde{\beta }}_0/{\tilde{\alpha }}_0}. \end{aligned} \end{aligned}$$
(A.10)

To achieve this, we recall (A.8) combined with (A.9)

$$\begin{aligned} \begin{aligned} {\tilde{A}}&=\Delta \exp \left( -\tilde{\alpha }_0(Y^*,T)+ {\mathcal {O}}(\Delta ) \right) \\&=\Delta \exp \left( -\tilde{\alpha }_0(Y^*,T) \right) (1+ {\mathcal {O}}(\Delta )), \end{aligned} \end{aligned}$$
(A.11)

where the derivatives of the \({\mathcal {O}}(\Delta )\) remainder terms with respect to \(Y^*,T\) are also \({\mathcal {O}}(\Delta )\). Hence,

$$\begin{aligned} \begin{aligned} R&= \left( \frac{{\tilde{A}}}{\Delta }\right) ^{{\tilde{\beta }}_0/{\tilde{\alpha }}_0}\\&=\exp \left( -\tilde{\beta }_0(Y^*,T) \right) (1+ {\mathcal {O}}(\Delta ))^{{\tilde{\beta }}_0/{\tilde{\alpha }}_0}. \end{aligned} \end{aligned}$$
(A.12)

This relation can be used to solve for \(T=T(R,Y^*)\), obtaining

$$\begin{aligned} \begin{aligned} T(R,Y^*)&=-\frac{\log R}{\beta }(1+ {\mathcal {O}}(\Delta )). \end{aligned} \end{aligned}$$
(A.13)

Note, due to the exponent \({{\tilde{\beta }}_0/{\tilde{\alpha }}_0}\) appearing in the remainder term of (A.12), the derivatives of the remainder terms in (A.13) with respect to \(R,Y^*\) no longer satisfy the same estimates. However, we are still able to estimate the first order partial derivatives

$$\begin{aligned} \begin{aligned} \partial _RT(R,Y^*)&=-\frac{1}{\beta R}(1+ {\mathcal {O}}(\Delta ))\\ \partial _{Y^*}T(R,Y^*)&={\mathcal {O}}(\log R), \end{aligned} \end{aligned}$$
(A.14)

by implicitly differentiating (A.12).

We set \(B_\mathrm {loc}(R,Y^*):={\tilde{B}}\) and determine

$$\begin{aligned} \begin{aligned} {\tilde{B}}&=\Delta \exp \left( -\tilde{\beta }_0(Y^*,T) \right) (1+ {\mathcal {O}}(\Delta ))\\&= \Delta R(1+ {\mathcal {O}}(\Delta )), \end{aligned} \end{aligned}$$
(A.15)

where the derivatives of the \({\mathcal {O}}(\Delta )\) remainder terms with respect to \(Y^*,T\) are also \({\mathcal {O}}(\Delta )\), and using the expressions (A.14), we obtain

$$\begin{aligned} \begin{aligned} \partial _RB_\mathrm {loc}(R,Y^*)&= \Delta (1+ {\mathcal {O}}(\Delta ))\\ \partial _{Y^*}B_\mathrm {loc}(R,Y^*)&={\mathcal {O}}(R\log R). \end{aligned} \end{aligned}$$
(A.16)

Next, using (A.1), (A.4), and (A.14), and setting \(Y_\mathrm {loc}(R,Y^*):=Y(0)\), we have that

$$\begin{aligned} \begin{aligned} Y_\mathrm {loc}(R,Y^*)&= Y^*-\frac{\varepsilon \gamma \log R}{\beta }\left( 1+{\mathcal {O}}(\Delta )\right) \\ \partial _RY_\mathrm {loc}(R,Y^*)&= -\frac{\varepsilon \gamma }{\beta R}\left( 1+{\mathcal {O}}(\Delta )\right) \\ \partial _{Y^*}Y_\mathrm {loc}(R,Y^*)&= 1+{\mathcal {O}}(\Delta ). \end{aligned}\end{aligned}$$
(A.17)

Finally, we define \(\rho (R,Y^*):={\tilde{\alpha }}_0/{\tilde{\beta }}_0\), and using (A.5) and (A.14), we have

$$\begin{aligned} \begin{aligned} \rho (R,Y^*)&= \alpha /\beta +{\mathcal {O}}(\Delta )\\ \partial _R\rho (R,Y^*)&= {\mathcal {O}}\left( \frac{\Delta }{R\log R}\right) \\ \partial _{Y^*}\rho (R,Y^*)&= {\mathcal {O}}(1), \end{aligned} \end{aligned}$$
(A.18)

which completes the proof of estimates (3.41). \(\square \)

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Carter, P. Spike-Adding Canard Explosion in a Class of Square-Wave Bursters. J Nonlinear Sci 30, 2613–2669 (2020). https://doi.org/10.1007/s00332-020-09631-y

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