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

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.

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). $\square$