Multiscale Geometry of the Olsen Model and Non-classical Relaxation Oscillations

Abstract

We study the Olsen model for the peroxidase–oxidase reaction. The dynamics is analyzed using a geometric decomposition based on multiple timescales. The Olsen model is four-dimensional, not in a standard form required by geometric singular perturbation theory and contains multiple small parameters. These three obstacles are the main challenges we resolve by our analysis. Scaling and the blow-up method are used to identify several subsystems. The results presented here provide a rigorous analysis for two oscillatory modes. In particular, we prove the existence of non-classical relaxation oscillations in two cases. The analysis is based on desingularization of lines of transcritical and submanifolds of fold singularities in combination with an integrable relaxation phase. In this context, our analysis also explains an assumption that has been utilized, based purely on numerical reasoning, in a previous bifurcation analysis by Desroches et al. (Discret Contin Dyn Syst S 2(4):807–827, 2009). Furthermore, the geometric decomposition we develop forms the basis to prove the existence of mixed-mode and chaotic oscillations in the Olsen model, which will be discussed in more detail in future work.

Appendices

Appendix 1: Normally Hyperbolicity and Fast–Slow Systems

We only recall the basic definitions and results about fast–slow systems. There are several standard references that detail many parts of the theory (Jones 1995; Kaper and Jones 2001; Mishchenko and Rozov 1980; Desroches et al. 2012; Arnold 1994). A fast–slow system of ordinary differential equations (ODEs) is given by:

$$\begin{aligned} \begin{array}{rcrcl} \epsilon \dot{x}&{}=&{}\epsilon \frac{\mathrm{d}x}{\mathrm{d}\tau }&{}=&{}f(x,y,\epsilon ),\\ \dot{y}&{}=&{}\frac{\mathrm{d}y}{\mathrm{d}\tau }&{}=&{}g(x,y,\epsilon ),\\ \end{array} \end{aligned}$$

(76)

where $x\in {\mathbb {R}}^m$ are fast variables, $y\in {\mathbb {R}}^n$ are slow variables, and $0<\epsilon \ll 1$ is a small parameter representing the ratio of timescales. The maps $f,g$ are assumed to be sufficiently smooth. Equation (76) can be rewritten by changing from the slow timescale $\tau$ to the fast timescale $t=\tau /\epsilon$

$$\begin{aligned} \begin{aligned} x'&=\textstyle \frac{\mathrm{d}x}{\mathrm{d}t}=f(x,y,\epsilon ),\\ y'&=\textstyle \frac{\mathrm{d}y}{\mathrm{d}t}=\epsilon ~g(x,y,\epsilon ).\\ \end{aligned} \end{aligned}$$

(77)

The singular limit $\epsilon \rightarrow 0$ of (77) yields the fast subsystem ODEs parametrized by the slow variables $y$. Setting $\epsilon \rightarrow 0$ in (76) gives a differential–algebraic equation (DAE), called the slow subsystem, on the critical manifold ${\mathcal {C}}_0:=\{(x,y)\in {\mathbb {R}}^{m+n}:f(x,y,\epsilon )=0\}$. Concatenations of fast and slow subsystem trajectories are called candidates (Benoît 1990; Haiduc 2009).

A subset ${\mathcal {S}}\subset {\mathcal {C}}$ is called normally hyperbolic if the $m\times m$ total derivative matrix $(D_xf)(p)$ is hyperbolic for $p\in {\mathcal {S}}$. A normally hyperbolic subset ${\mathcal {S}}$ is attracting if all eigenvalues of $(D_xf)(p)$ have negative real parts for $p\in {\mathcal {S}}, {\mathcal {S}}$ is called repelling if all eigenvalues have positive real parts, and ${\mathcal {S}}$ is of saddle type if there are positive and negative eigenvalues. On normally hyperbolic parts of ${\mathcal {C}}$, the implicit function theorem applies to $f(x,y,0)=0$ providing a map $h_0(y)=x$ so that ${\mathcal {C}}$ can be expressed (locally) as a graph.

Theorem 9.1

(Fenichel’s Theorem Fenichel (1979); Jones (1995); Tikhonov (1952) Suppose ${\mathcal {S}}={\mathcal {S}}_0$ is a compact normally hyperbolic submanifold (possibly with boundary) of the critical manifold ${\mathcal {C}}_0$. Then, for $\epsilon >0$ sufficiently small, there exists a locally invariant manifold ${\mathcal {S}}_\epsilon$ diffeomorphic to ${\mathcal {S}}_0$. ${\mathcal {S}}_\epsilon$ has a distance of ${\mathcal {O}}(\epsilon )$ from ${\mathcal {S}}_0$ and the flow on ${\mathcal {S}}_\epsilon$ converges to the slow flow as $\epsilon \rightarrow 0$.

The distance between ${\mathcal {S}}_\epsilon$ and ${\mathcal {S}}_0$ can be expressed in the Hausdorff metric or a suitable $C^r$-norm (using the map $h_0$ and its perturbation $h_\epsilon$). A manifold ${\mathcal {S}}_\epsilon$ provided by Fenichel’s Theorem is called a slow manifold. Slow manifolds are usually not unique, but different slow manifolds lie at a distance ${\mathcal {O}}(e^{-K/\epsilon })$ for some constant $K>0$. Often we shall make a choice of compact subset and choice of slow manifold without further notice, indicating that the choice does not matter for the asymptotic analysis performed.

A trajectory is called a maximal canard if it lies in the intersection of an attracting and a repelling slow manifold. Canards were first investigated by a group of French mathematicians (Benoît et al. 1981) using nonstandard analysis. Later also asymptotic (Eckhaus 1983; Baer and Erneux 1986) and geometric (Dumortier and Roussarie 1996; Krupa and Szmolyan 2001) methods have been developed to understand canard orbits.

Appendix 2: Geometric Desingularization

Here we shall briefly review the basic strategy for the blow-up approach for geometric desingularization of fast–slow systems. Details on the classical, single-scale, method can be found, e.g., in Dumortier (1993). The classical blow-up was first introduced into fast–slow systems in Dumortier and Roussarie (1996). Further developments can be found in Krupa and Szmolyan (2001), see also the introduction in Krupa and Szmolyan (2001).

The starting point is to write the system (77) as follows

$$\begin{aligned} \begin{aligned} x'&=f(x,y,\epsilon ),\\ y'&=\epsilon ~ g(x,y,\epsilon ),\\ \epsilon '&=0.\\ \end{aligned} \end{aligned}$$

(78)

Let us denote the vector field defined by (78) as $X$, i.e., $X$ is a mapping

$$\begin{aligned} X:{\mathbb {R}}^{m+n}\times [0,\epsilon _0)\rightarrow T\left( {\mathbb {R}}^{m+n}\times [0,\epsilon _0)\right) \end{aligned}$$

where $T(\cdot )$ indicates the tangent bundle. Further equations for parameters could be appended to (78) as well, if necessary. Suppose (78) has an equilibrium point for $\epsilon =0$, or more generally a submanifold ${\mathcal {M}}=\{f=0\}$ of equilibria in ${\mathbb {R}}^{m+n}\times \{\epsilon =0\}$. If $(D_xf)(p)$ is not a hyperbolic matrix for each $p\in {\mathcal {M}}$, the equilibrium (manifold) ${\mathcal {M}}$ is degenerate and classical linearization results do not apply directly to (78).

The blow-up technique is based on replacing ${\mathcal {M}}$ by a, usually more complicated, manifold $\bar{{\mathcal {M}}}$ and using a map

$$\begin{aligned} \Phi :\bar{{\mathcal {M}}}\rightarrow {\mathcal {M}}\end{aligned}$$

which induces a vector field $\bar{X}$ on $\bar{{\mathcal {M}}}$ via the pushforward $\Phi _*$ and the condition $\Phi _*(\bar{X})=X$. Using a good choice for $\bar{{\mathcal {M}}}$, one may often analyze the blown-up vector field $\bar{X}$ as it is possible that invariant manifolds of $\bar{X}$ are now (partially) hyperbolic.

As an example, consider the classical case when $(x,y)\in {\mathbb {R}}^2$ and $f(x,y)=y-x^2$ is the (truncated) normal form of a fold bifurcation. The origin $(x,y,\epsilon )=(0,0,0)$ is the important non-hyperbolic point, and the standard choice is to use a sphere for geometric desingularization $\bar{{\mathcal {M}}}:=S^2\times [0,r_0)$ for some constant $r_0>0$ or $r_0=+\infty$. Therefore, one has essentially inserted a sphere at the origin, see also Dumortier and Roussarie (1996), Krupa and Szmolyan (2001).

Although one could try to find a suitable global parametrization of $\bar{{\mathcal {M}}}$, this is usually not very convenient for calculations. Instead, one uses charts $\kappa _j:\bar{{\mathcal {M}}}\rightarrow {\mathbb {R}}^{m+n+1}$ of $\bar{{\mathcal {M}}}$ for the calculations, which is illustrated by the following important diagram

which commutes. Hence, one may just try to calculate the map $\varphi _j$ and obtain a vector field on ${\mathbb {R}}^{m+n+1}$ by applying the coordinate change

$$\begin{aligned} (x_j,y_j,\epsilon _j)=\varphi ^{-1}(x,y,\epsilon ). \end{aligned}$$

One may often, via a good choice of $\bar{{\mathcal {M}}}$ and chart maps $\kappa _j$, compute the vector fields in $(x_j,y_j,\epsilon _j)$-coordinates. The same remark applies to the transition maps between different charts $\kappa _{jk}$. Section 3 carries out these calculations for a submanifold of fold points in the Olsen model.

Appendix 3: An Auxiliary Center Manifold Reduction

Here we present the details for the center manifold calculation for (27). We drop the sub- and superscripts of $(r_1,y_1,\epsilon _1)$ and $(\alpha _1^*,b_1^*)$ for notational convenience; all variables and constants used in this section are temporary and should not be confused with notation within the main manuscript. Reordering the variables and translating (27) via $Y=y-1/(3ab)$ yields

$$\begin{aligned} \begin{aligned} r'&= r\left[ \epsilon (b-\xi )+3abY\right] =:f_1(r,\epsilon ,Y),\\ \epsilon '&=-\epsilon \left[ \epsilon (b-\xi )+3abY\right] =:f_2(r,\epsilon ,Y)\\ Y'&= f_3(r,\epsilon ,Y),\\ \end{aligned} \end{aligned}$$

(79)

where the function $f_3$ is given by

$$\begin{aligned} f_3(r,\epsilon ,Y):=\kappa \epsilon (1-[Y+1/(3ab)][1+a^*_1b^*_1]) -2[Y+1/(3ab)]\left( \epsilon (b-\xi )+3aby\right) . \end{aligned}$$

Let $z:=(r,\epsilon ,Y)^T$ and consider

$$\begin{aligned} A:=\left. D_z(z')\right| _{(0,0,0)}= \left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} K &{} -2 \\ \end{array} \right) \quad \text {and}\quad M:=\left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} -2/K &{} 0 \\ 0 &{} 1 &{} 1 \\ \end{array} \right) , \end{aligned}$$

where $K:=-(2 b + \kappa - 2 a b \kappa - 2 \xi )/(3 a b)$. Let $(x_1,x_2,\tilde{y})^T=\tilde{z}=M^{-1}z$ and observe that $M^{-1}AM=J\in {\mathbb {R}}^{3\times 3}$ with $J_{33}=-2$ and $J_{ij}=0$ otherwise. Set $\tilde{z}=(x_1,x_2,\tilde{y})=M^{-1}z$ so that

$$\begin{aligned} \begin{array}{ccccc} \left( \begin{array}{c}x_1'\\ x_2'\\ \end{array}\right) &{}=&{} \left( \begin{array}{cc}0 &{} 0\\ 0&{} 0\\ \end{array}\right) \left( \begin{array}{c}x_1\\ x_2\\ \end{array}\right) &{}+&{} \left( \begin{array}{c}F_1(x_1,x_2,\tilde{y})\\ F_2(x_1,x_2,\tilde{y})\\ \end{array}\right) \\ \tilde{y}'&{}=&{} -2\tilde{y}&{}+&{} G(x_1,x_2,\tilde{y}),\\ \end{array} \end{aligned}$$

(80)

where $(F_{1},F_2,G)^T=(0,0,2\tilde{y})^T+M^{-1}(f_1(M\tilde{z}), f_2(M\tilde{z}),f_3(M\tilde{z}))^T$. The system (80) is in the standard form for center manifold theory (Guckenheimer and Holmes 1983). The usual perturbation ansatz is $\tilde{y}=h(x_1,x_2)=k_{11}x_1^2+k_{12}x_1x_2+k_{22}x_2^2+{\mathcal {O}}(3),$ where ${\mathcal {O}}(3):={\mathcal {O}}(x_1^3,x_1^2x_2,x_1x_2^2,x_2^3)$. The defining invariance equation for the center manifold with $x=(x_1,x_2)^T$ and $F=(F_1,F_2)^T$ is

$$\begin{aligned} Dh(x)F(x,h(x))=-2h(x)+G(x,h(x)) \end{aligned}$$

(81)

since the $x'$-equations in (80) have no linear term. Collecting terms of order ${\mathcal {O}}(x_1^2)$ in (81) gives $k_{11}=0$ and the ${\mathcal {O}}(x_1x_2)$-terms give $k_{12}=0$. For ${\mathcal {O}}(x_2^2)$ equation (81) and $k_{11}=0=k_{12}$ imply

$$\begin{aligned} k_{22}=\frac{3ab(1+4ab)\kappa }{4(b-\xi )+2\kappa (1-2ab)}. \end{aligned}$$

Transforming back to the variables $(r,\epsilon ,y)$ via the matrix $M$ and translation yields the center manifold

$$\begin{aligned} y=\frac{1}{3ab}+\epsilon \frac{2(\xi -b)+\kappa (2ab-1)}{6ab}+k_{22}\frac{K^2}{4}\epsilon ^2+{\mathcal {O}}(3), \end{aligned}$$

where ${\mathcal {O}}(3)={\mathcal {O}}(r^3,r^2\epsilon ,r\epsilon ^2,\epsilon ^3)$. Computing

$$\begin{aligned} k_{22}\frac{K^2}{4}=\frac{\kappa (1+4ab)}{24ab}(2(b-\xi )+\kappa (1-2ab)) \end{aligned}$$

yields Proposition 3.9.

Appendix 4: Another Auxiliary Center Manifold Reduction

As for the center manifold reduction in Appendix 3, we present some of the important details for the center manifold calculation. As before for the previous “Appendix,” the notation here only pertains to this calculation and should not be confused with variables within the main text. It is convenient to translate (33) via $B_2:=b_2-\xi$, to re-label $x_2=X_2, y_2=Y_2$ and change to the timescale $\tau =s/\epsilon ^2$ which yields

$$\begin{aligned} \begin{aligned} \dot{X}_2&= 3a_2(B_2+\xi )Y_2-X_2^2+B_2X_2+\delta ,\\ \dot{a}_2&= \epsilon ^2(\mu -\alpha a_2 -a_2(B_2+\xi )Y_2),\\ \dot{B}_2&= \epsilon ^2\epsilon _b(1-(B_2+\xi )X_2 -a_2(B_2+\xi )Y_2),\\ \dot{\epsilon }&= 0,\\ \dot{\delta }&= 0,\\ \dot{Y}_2&= \kappa (X_2^2-Y_2-a_2(B_2+\xi )Y_2).\\ \end{aligned} \end{aligned}$$

(82)

The system (82) has a line of equilibrium points

$$\begin{aligned} {\mathcal {E}}_2:=\{(X_2,a_2,B_2,\epsilon ,\delta ,Y_2)=(0,a_2,0,0,0,0)\}, \end{aligned}$$

which is degenerate since the linearization of (82) at ${\mathcal {E}}_2$ for fixed $a_2=a_0$ is

$$\begin{aligned} A=\left. D_{(X_2,a_2,b_2,\epsilon ,\delta ,Y_2)} \left( \begin{array}{c} \dot{X}_2 \\ \dot{a}_2 \\ \dot{B}_2 \\ \dot{\epsilon } \\ \dot{\delta } \\ \dot{Y}_2\\ \end{array}\right) \right| _{{\mathcal {E}}_2}= \left( \begin{array}{cccccc} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 3a_0\xi \\ 0 &{} 0 &{} 0 &{} \mu -\alpha a_0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \epsilon _b &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -\kappa (1+a_0\xi )\\ \end{array} \right) . \end{aligned}$$

(83)

This matrix has one negative eigenvalue $-\kappa (1+a_0\xi )$ and a quintuple zero eigenvalue. Hence a center manifold reduction to a five-dimensional center flow is required to resolve the dynamics near ${\mathcal {E}}_2$. However, we use a preliminary transformation to get the system into standard form. Let $Z=(X_2,Y_2)^T$ and set

$$\begin{aligned} A_{XY}:=\left( \begin{array}{cc} 0 &{} 3a_0\xi \\ 0 &{} -\kappa (1+a_0\xi ) \\ \end{array} \right) \quad \text {and} \quad M:=\left( \begin{array}{cc} 1 &{} -\frac{3 a_0 \xi }{\kappa (1 + a_0 \xi )} \\ 0 &{} 1 \\ \end{array} \right) . \end{aligned}$$

Then consider new coordinates via $M\tilde{Z}=Z$ and observe that in the coordinates $(\tilde{X}_2,\tilde{Y}_2)^T$ we have

$$\begin{aligned} \tilde{Z}'=M^{-1}A_{XY}M\tilde{Z}+\text {h.o.t.}=\left( \begin{array}{cc} 0 &{} 0 \\ 0 &{} -\kappa (1+a_0\xi ) \\ \end{array} \right) \left( \begin{array}{c} \tilde{X}_2\\ \tilde{Y}_2 \\ \end{array} \right) +\text {h.o.t.}, \end{aligned}$$

where $\text {h.o.t.}$ denotes higher-order terms. Let $x:=(x_1,x_2,x_3,x_4,x_5)=(\tilde{X}_2,a_0-a_2,b_2,\epsilon ,\delta )$ so that $y=\tilde{Y}_2$ is the transformation of (82) into new coordinates

$$\begin{aligned} \begin{aligned} x_1'&= \textstyle x_5 + 3 x_3 y (x_3 + \xi ) + x_3 \left( x_1 - \frac{3 a_0 y \xi }{\kappa + a_0 \kappa \xi } \right) - \left( x_1 - \frac{3 a_0 y \xi }{\kappa + a_0 \kappa \xi }\right) ^2\\&\quad \,\,- \textstyle 3 a_0 \xi \frac{y + (a_0-x_2) y (x_3 + \xi ) - \left( x_1 - \frac{3 a_0 y \xi }{\kappa + a_0 \kappa \xi }\right) ^2}{1 + a_0 \xi },\\ x_2'&= x_4 (\mu - (a_0-x_2) (\alpha + y (x_3 + \xi ))),\\ x_3'&= \textstyle x_4 \epsilon _b (1 - (a_0-x_2) y (x_3 + \xi ) - (x_3 + \xi ) \left( x_1 - \frac{3 a_0 y \xi }{\kappa + a_0 \kappa \xi }\right) ,\\ x_4'&=0,\\ x_5'&=0,\\ y'&= \textstyle \kappa \left( -y - (a_0-x_2) y (x_3 + \xi ) + \left( x_1 - \frac{3 a_0 y \xi }{\kappa + a_0 \kappa \xi }\right) ^2\right) , \end{aligned} \end{aligned}$$

which is a vector field we denote by $(Cx+F(x,y),Py+G(x,y))^T$ for $F(x,y)\in {\mathbb {R}}^5, G(x,y)\in {\mathbb {R}}$. Observe that

$$\begin{aligned} C=\{A_{ij}\}_{i,j=1}^5,\quad \text {and}\quad P= -\kappa (1+a_0\xi ). \end{aligned}$$

The vector field is now in the correct form for center manifold theory, applied along the entire line of points parametrized by $a_0$. The ansatz is

$$\begin{aligned} y=h(x)=\sum _{i+j=2,i\le j}c_{ij}x_ix_j. \end{aligned}$$

The usual invariance equation is given by

$$\begin{aligned} Dh(x)[Cx+F(x,h(x))]=Ph(x)+G(x,h(x)), \end{aligned}$$

where different powers $x_ix_j$ have to have equal coefficients on both sides. This procedure yields

$$\begin{aligned} c_{11}=\frac{1}{1+a_0\xi },\quad c_{15}=-\frac{1}{\kappa (1+a_0\xi )^2}, \quad c_{55}=-\frac{1}{\kappa ^2(1+a_0\xi )^3}. \end{aligned}$$

All other coefficients $c_{ij}$ have vanish. Hence, the center manifold is given to lowest order by

$$\begin{aligned} \tilde{Y}_2=\frac{\tilde{X}_2^2}{1+a_0\xi } -\frac{\tilde{X}_2\delta }{\kappa (1+a_0\xi )^2}-\frac{\delta ^2}{\kappa ^2(1+a_0\xi )^3}. \end{aligned}$$

(84)

Transforming back to original coordinates and keeping lowest order terms yields

$$\begin{aligned} Y_2=\frac{X_2^2}{1+a_0\xi }-\frac{\delta X_2}{\kappa (1+a_0\xi )^2} +\frac{\delta ^2}{\kappa ^2(1+a_0\xi )^3}+{\mathcal {O}}(Y_2^2,X_2^3,X_2Y_2,\delta Y_2,\delta ^3). \end{aligned}$$

(85)

Substituting the result into (82) gives, up to leading order, the center flow

$$\begin{aligned} \epsilon ^2\frac{dX_2}{ds}\!&= \! X_2^2\left( \frac{2a_0\xi -1}{1+a_0\xi }\right) \!+\!X_2\left( \frac{-\delta }{\kappa (1+a_0\xi )^2}+B_2\right) \!+\!\delta +\frac{\delta ^2}{\kappa ^2(1+a_0\xi )^3}+{\mathcal {O}}(3),\nonumber \\ \frac{da_2}{ds}&= \mu -\alpha a_2+{\mathcal {O}}(2),\\ \frac{dB_2}{ds}&= \epsilon _b+{\mathcal {O}}(2),\nonumber \end{aligned}$$

(86)

which is precisely the result we wanted to prove.