Canards

Post-publication activity

Curator: Martin Wechselberger

Figure 1: A canard (with hat) of the van der Pol oscillator
Canards were discovered and first analyzed by French mathematicians (Benoît et al. [1981]) who studied 2D relaxation oscillators, in particular the van der Pol oscillator. There the classical canard phenomenon explains the very fast transition upon variation of a parameter from a small amplitude limit cycle via canard cycles to a large amplitude relaxation cycle. This very fast transition called canard explosion happens within an exponentially small range of a control parameter. Thus this phenomenon is very hard to detect and it seems to be more like a canard in a newspaper. Furthermore, the shape of a canard cycle in the phase space resembles that of a duck (Figure 1). So the notion canard was born and the chase on these creatures began with either nonstandard (Benoît et al [1981]) or standard (Eckhaus [1983]) methods.

Methods

The main mathematical methods to analyze canards are nonstandard analysis (see, e.g., Benoît et al. [1981], Diener [1984]), matched asymptotic expansions (see, e.g., Eckhaus [1983], Mishchenko et al. [1994]) and the blow-up technique (see, e.g., Dumortier and Roussarie [1996], Krupa and Szmolyan [2001], Szmolyan and Wechselberger [2001]), which extends geometric singular perturbation theory known as Fenichel theory to non-hyperbolic points. Moreover, with complex analysis, the canard phenomenon can be understood with either a geometrical point of view (see, e.g. Callot [1993] or Benoît et al. [1998]) or the Gevrey theory which is convenient to study the divergent asymptotic expansions of canards (see, e.g. Canalis-Durand et al. [2000] or Benoît [2001]).

Definition

Canards are a phenomenon occurring in singularly perturbed systems (also known as slow-fast systems), i.e. they occur in systems of the form

$$\begin{array}{rcl} \varepsilon\dot{x}&=&f(x,z,\varepsilon)\\ \dot{z}&=&g(x,z,\varepsilon)\,, \end{array}$$

with fast variable $$x\in\mathbb{R}^n\ ,$$ slow variable $$z\in\mathbb{R}^m\ ,$$ sufficiently smooth functions $$g\ ,$$ $$f$$ and small parameter $$0\!<\!\varepsilon\!\ll\! 1\ .$$ This system evolves on a slow time scale $$t=\varepsilon \tau\ .$$ The limiting problem $$\varepsilon\to 0$$ on this slow time scale $$t$$ is called the reduced problem and describes the evolution of the slow variable $$z\in\mathbb{R}^m\ .$$ The phase space of the reduced problem is the critical manifold $$S$$ defined by $$S:=\{(x,z)\in\mathbb{R}^n\times\mathbb{R}^m\,:\,f(x,z,0)=0\}\ .$$ On the other hand, the limiting problem $$\varepsilon\to 0$$ on the fast time scale $$\tau$$ is called the layer problem and describes the evolution of the fast variable $$x\in\mathbb{R}^n$$ for fixed $$z\in\mathbb{R}^m\ .$$

By Fenichel theory (Fenichel [1979]), normally hyperbolic subsets of $$S$$ perturb to nearby slow invariant manifolds $$S_\varepsilon$$ of the singularly perturbed system with the flow given approximately by the flow of the reduced system. The most common case where normal hyperbolicity breaks down is given by a folded critical manifold $$S=S_a\cup L \cup S_r$$ where $$S_a$$ denotes the attracting part of the critical manifold $$S\ ,$$ $$S_r$$ the repelling part of $$S\ ,$$ and $$L$$ the fold of $$S$$ along which normal hyperbolicity is lost via a saddle-node bifurcation of the layer problem. Of course, any bifurcation of the layer problem leads to a loss of normal hyperbolicity of $$S\ .$$ Canards are a special class of solutions of singularly perturbed systems where normal hyperbolicity is lost.

Figure 2: Examples of canard solutions (green) near critical manifold S; normal hyperbolicity is lost at L, e.g., via saddle-node bifurcation of layer problem (Fig. A), or via transcritical bifurcation of layer problem (Fig. B)

Definition: A canard is a solution of a singularly perturbed system which follows an attracting slow manifold $$S_a\ ,$$ passes close to a bifurcation point $$p\in L$$ of the critical manifold, and then follows a repelling slow manifold $$S_r$$ for a considerable amount of time.

In geometric terms a canard solution corresponds to the intersection of an attracting and repelling slow manifold $$S_{a,\varepsilon}\cap S_{r,\varepsilon}$$ near a non-hyperbolic point $$p\in L\ .$$ This geometric object is called a maximal canard.

In the case of a Hopf bifurcation of the layer problem, the phenomenon is quite different: there is a delayed loss of stability but there exist no canards (see Wallet [1986], Neishtadt [1987]). This case will not be explained here.

Codimension of the Canard Phenomenon

Canards in singularly perturbed systems with just one slow variable ($$z\in\mathbb{R}$$) and one fast variable ($$x\in\mathbb{R}$$) are non generic, since maximal canards in such systems occur only for discrete values of a control parameter (Krupa and Szmolyan [2001]), i.e. a one parameter family of singularly perturbed systems is needed to unfold this canard problem. In general, if canards are considered in singular perturbation problems with $$z\in\mathbb{R}$$ and $$x\in\mathbb{R}^n$$ where normal hyperbolicity is lost via a saddle-node bifurcation of the layer problem, then the singularly perturbed system can be reduced to a one slow and one fast variable system by a center manifold reduction.

The Classical Canard Phenomenon

Canard Cycles and their Explosion

The classical canard phenomenon (discovered by Benoît et al. [1981]) occurs in singularly perturbed systems with $$x\in\mathbb{R}$$ and $$z\in\mathbb{R}\ .$$ Its prototypical example is the van der Pol oscillator given by

$$\begin{array}{rcl} \varepsilon\dot{x}&=& z-x^3/3+x\\ \dot{z}&=& a-x\,, \end{array}$$

where $$a\in\mathbb{R}$$ denotes a control parameter (external forcing). The critical manifold $$S$$ is cubic shaped consisting of two attracting outer branches and a middle repelling branch connected via two fold-points where normal hyperbolicity is lost.

Figure 3: Canard explosion of the van der Pol oscillator ($$\varepsilon=0.01$$) within an exponentially small neighbourhood of a=0.998740451245 where the transition from relaxation oscillations to small amplitude limit cycles happens via canard cycles. The small amplitude limit cycles then terminate at a=1 via a Hopf bifurcation.

Here, the nature of the classical canard phenomenon is the transition from a small amplitude oscillatory state to a (large amplitude) relaxation oscillatory state within an exponentially small range $$O(\exp \left(-1/\varepsilon \right))$$of the control parameter $$a$$ (see Figure 3). This transition, also called canard explosion, occurs through a sequence of canard cycles which can be asymptotically stable, but they are hard to observe in an experiment or simulation because of sensitivity to the control parameter and also because of sensitivity to noise. This is well known in chemical literature where a canard explosion is classified as a hard transition, because, for practical purposes, the transition from a small amplitude oscillation to a relaxation oscillation occurs immediately (see, e.g., Brøns and Bar-Eli [1991], Peng et al. [1991]).

Generic Canards

Canards in singularly perturbed systems with two or more slow variables ($$z\in\mathbb{R}^m,\, m\ge 2$$) and one fast variable ($$x\in\mathbb{R}$$) are robust, since maximal canards generically persist under small parameter changes (Benoît [1983], Mishchenko et al. [1994], Szmolyan and Wechselberger [2001], Benoît [2001], Wechselberger [2005]). In general, if canards are considered in singular perturbation problems with $$z\in\mathbb{R}^m,\,m\ge 2$$ and $$x\in\mathbb{R}^n$$ where normal hyperbolicity is lost via a saddle-node bifurcation of the layer problem, then the singularly perturbed system can be reduced to an $$m$$ slow and one fast variable system by a center manifold reduction. Therefore, a generic example with minimal dimension is given by a 3D singularly perturbed system with a 2D folded critical manifold.

Figure 4: A: The reduced flow near a saddle singularity, B-C: The corresponding reduced flow near a folded saddle singularity (B) and the same reduced flow shown on the 2D folded critical manifold (C)

Classification

There exist different types of canards in 3D systems with 2D folded critical manifolds (Benoît [1983], Szmolyan and Wechselberger [2001]). The classification of these canards is based on the analysis of the corresponding 2D reduced system. There exist (discrete) folded singularities on the fold $$F\ ,$$ called canard points, where the reduced flow crosses from the attracting to the repelling branch of the critical manifold. Generically, these folded singularities are either folded saddles (see Figure 4) or folded nodes (see Figure 5), resembling similar phase portraits as ordinary saddles or nodes in 2D phase space, besides that the reduced flow allow trajectories to cross the fold $$F$$ at these canard points. Note that in the folded saddle case (Figure 4B-4C) two solutions of the reduced flow cross via the folded saddle from the attracting branch to the repelling branch of the critical manifold or vice versa. More strikingly, a whole family of solutions (Figure 5B, shadowed sector) crosses via the folded node singularity from the attracting to the repelling branch. This is possible since existence and uniqueness results of ordinary differential equations are violated along the fold $$F\ .$$

Figure 5: The reduced flow near a node singularity, B-C: The corresponding reduced flow near a folded node singularity (B) and the same reduced flow shown on the 2D folded critical manifold (C)

The folded singularity corresponding to a saddle-node bifurcation of a folded saddle and a folded node is called a folded saddle-node type I. Another possible bifurcation scenario is a transcritical bifurcation of a folded singularity and an ordinary singularity. In this case, the ordinary singularity crosses via a transcritical bifurcation from the attracting branch $$S_a$$ to the repelling branch $$S_r$$ or vice versa. The corresponding folded singularity at this transcritical bifurcation is called a folded saddle-node type II.

The Generalized Canard Phenomenon

Mixed Mode Oscillations (MMOs)

MMOs correspond to switching between small amplitude oscillations and relaxation oscillations. These patterns were first discovered in the famous Belousov-Zhabotinsky reaction and, since then, have been frequently observed in experiments and models of chemical and biological rhythms. One way to explain these patterns is based on canards of folded node type. The reason is that canards of folded node type can be responsible for small amplitude oscillations (Wechselberger [2005]). A good intuition for MMOs is that a system moves dynamically from a small amplitude oscillatory state to a relaxation oscillatory state and the feature of the large relaxation oscillation is to bring the system back to the basin of attraction of the small amplitude oscillatory state. Other proposed mechanisms for MMOs are break-up/loss of stability of a Shilnikov homoclinic orbit (Koper [1995]), break-up of an invariant torus (Larter and Steinmetz [1991]) or slow passage through a delayed Hopf bifurcation (Larter et al. [1988]).

Figure 6: Explanation of MMOs via generalized canard phenomenon: the upper center figure shows a time trace of a $$1^4$$ MMO pattern ($$\epsilon=0.01$$); the lower right figure summarizes Assumptions 1-3 in the phase space such that a singularly perturbed system possesses MMOs; the lower left figure shows the 4 small amplitude oscillations in the phase space near the corresponding folded node singularity.

Figure 6 shows a $$1^4$$ MMO pattern consisting of 1 large amplitude oscillation and 4 small amplitude oscillations. In general, the symbol $$L^s$$ is assigned to a MMO pattern with $$L$$ large and $$s$$ small oscillations. The observed MMO pattern in Figure 6 can be explained as follows. Given a singularly perturbed system:

$$\begin{array}{rcl} \varepsilon\dot{x}&=&f(x,y,z,\varepsilon)\\ \dot{y}&=&g_1(x,y,z,\varepsilon)\\ \dot{z}&=&g_2(x,y,z,\varepsilon) \end{array}$$

• Assumption 1: The critical manifold $$S$$ of the singularly perturbed system is (locally) a folded surface.
• Assumption 2: The corresponding reduced problem possesses a folded node singularity.
• Assumption 3: There exist a singular periodic orbit (Figure 6, lower right) which consists of a segment on $$S_a$$ (blue) within the singular funnel (shadowed region) with the folded node singularity (black circle) as an endpoint, fast fibers (red) of the layer problem and a global return mechanism (green). The global return mechanism has to be specified.

Theorem (Brøns et al. [2006]): Given the above singularly perturbed system under Assumptions 1-3. Then, for sufficiently small $$\varepsilon\ ,$$ there exist MMOs of type $$1^s\ .$$ It is possible to calculate the number $$s$$ of small oscillations.

The number of small oscillations is given by $$s=[(\mu+1)/(2\mu)]$$ with $$\mu=\lambda_1/\lambda_2\le 1$$ where $$\lambda_1$$ and $$\lambda_2$$ are the corresponding eigenvalues of the folded node singularity whenever $$1/\mu \notin \mathbb{N}$$ (except resonance cases).

References

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