## Abstract

We show that a planar slow–fast piecewise-linear (PWL) system with three zones admits limit cycles that share a lot of similarity with van der Pol canards, in particular an explosive growth. Using phase-space compactification, we show that these *quasi-canard* cycles are strongly related to a bifurcation at infinity. Furthermore, we investigate a limiting case in which we show the existence of a continuum of canard homoclinic connections that coexist for a single-parameter value and with amplitude ranging from an order of *ε* to an order of 1, a phenomenon truly associated with the non-smooth character of this system and which we call *super-explosion*.

## 1. Introduction

In smooth systems, canards are periodic solutions of singularly perturbed systems of differential equations that stay *ε*-close to an attracting locally invariant manifold during an *O*(1) time interval and then to a repelling locally invariant manifold for an *O*(1) time interval as well, where *ε* is the small parameter measuring the time-scale separation in the system. They were first discovered in the van der Pol oscillator with constant forcing [1].

In the original context of planar slow–fast systems of the form
1.1and
1.2(where *f* and *g* are sufficiently smooth functions), when considering a one-parameter unfolding of a Hopf bifurcation, canard limit cycles (or *canard cycles*) exist for a range of *α* which is exponentially small in *ε*. Hence, canard cycles are said to be *short-lived,* and the segment of the branch of periodic solutions corresponding to the canard regime is quasi-vertical (see fig. 1 on p. 115 in Benoît *et al.* [1] for an illustration of this aspect), leading to the term *canard explosion* [2]. In terms of dynamics, canard cycles correspond to transition orbits that bridge the gap between small-amplitude cycles and large-amplitude *relaxation oscillations*; this transition occurs at an *O*(*ε*)-distance from the Hopf bifurcation point and occurs within an exponentially thin band of parameter values. The existence of canards in planar slow–fast systems relies on the conjunction of several key elements, one of which is the presence of non-hyperbolic points—with respect to the fast direction—on the *critical manifold* of the system, that is, the curve
1.3where *α** is the bifurcation value. Generically, these points are isolated in phase space, and the simplest case is that of quadratic fold points (*x*^{⋆},*y*^{⋆}), which correspond to the condition
1.4The great attraction of canard problems is not only their mathematical relevance as part of the singular perturbation theory but also that systems of the type (1.1)–(1.2) with a *cubic-shaped* critical manifold (which then possesses two quadratic fold points) are ubiquitous in applications. Examples include the well-known van der Pol oscillator, which describes the electrical behaviour of an RLC circuit, that is, with a resistor R, inductance—usually denoted by L—and a capacitor C, with a nonlinear resistor, nerve axon dynamics in neuroscience [3,4] and the Belousov–Zhabotinsky chemical reaction [5].

The existence of canards in a system such as (1.1)–(1.2) has important implications in applications. For instance, in neuronal models, such as the FitzHugh–Nagumo equation, or biophysical models, such as planar reductions of the Hodgkin–Huxley equations [6], canard cycles organize the transition between rest states of the membrane potential of the neuron and its spiking states, which correspond to *action potentials*. Hence, canards are deeply related to the excitability threshold of neurons [7].

Soon after the first existence results on canards were obtained by means of non-standard analytical techniques in Benoît *et al*. [1], similar results were proved using standard methods, first-matched asymptotics [8] and, later on, geometric desingularization or blow-up methods [9–11].

A decade after the seminal work of Benoît *et al.*, the question of the persistence of canard cycles in a *piecewise-linear* (PWL) context was investigated by Komuro & Saito [12]. Here, the cubic critical manifold is replaced by a PWL caricature consisting of three straight line segments. The corners now play the role of the fold points of system (1.1)–(1.2), and cycles resembling canards and evolving around these corners were easily identified by direct simulation. However, for reasons explained by Arima *et al*. [13], the equivalent of *canards with head* can arise only in systems with one more piece in between the two corners. Later on, another attempt to find canards in a PWL system with three segments appeared in Llibre *et al*. [14], but these authors obtained a negative answer and did not emphasize any explosive behaviour. Finally, we note the recent revival of interest in studying PWL systems with multiple time scales, linked in with canard solutions [15,16].

In the papers about PWL van der Pol systems, the notion of an equivalent of classical canards, as well as the mechanisms attached to their creation and destruction, remains unclear. In this paper, we show that the PWL system of Komuro & Saito [12] admits limit cycles which are very much reminiscent of van der Pol canards, in the sense that they share a number of similarities with their smooth ‘cousins’: explosive behaviour with respect to a distinguished bifurcation parameter and peculiar shape, with some portion of the cycle near the unstable section of the fast nullcline. We will therefore call these cycles *quasi-canards*.

We study the region of existence of quasi-canards as well as the mechanisms by which they are created and destroyed. In particular, we show that the region of existence of quasi-canards is bounded by a parabola in a suitable two-parameter plane. Furthermore, we study (in §4) the destruction of quasi-canard cycles for parameter values approaching this parabola and demonstrate that it corresponds to a bifurcation at infinity, which we study by means of phase-space compactification. Because only the lower corner of the critical manifold is needed to generate PWL quasi-canard cycles, we perform the analysis at infinity in the context of a bi-zonal PWL system. In the limiting case corresponding to the region in two-parameter space below this parabola, we show that there is a continuum of homoclinic canard connections that coexist at a single-parameter value and that follow trajectories similar to the quasi-canards; their amplitudes range from order *ε* to order 1 and, by moving ever so slightly the parameter controlling the position of the slow nullcline, this continuum is destroyed, and the unique periodic attractor of the system becomes a large relaxation cycle. We call this phenomenon a *super-explosion*. Note that we give the name ‘canard’ to these homoclinic connections because, for the associated parameter values, it is possible to make sense of repelling slow manifolds, unlike the case of quasi-canards. We finally give some elements of comparison with the simplest smooth planar slow–fast system displaying canard cycles (the *singular fold* system of [10,11]), where canard cycles also disappear in a bifurcation at infinity.

The paper is organized as follows. In §2, we introduce the van der Pol-type system that we study in the rest of the paper after recalling known facts from the smooth case, and we investigate the region of existence of quasi-canards and give the main result of the paper (theorem 2.1). In §3, we prove this theorem. In §4, we introduce a phase-space compactification to study the bifurcation at infinity responsible for the disappearance of the quasi-canards. Finally, we present a comparison with classical canard explosions for smooth systems in §5 and draw some conclusions on this work with possible extensions to three-dimensional PWL systems in §6.

## 2. Canards in the van der Pol system

### (a) The smooth case

The first results on canards were derived in the seminal paper published by a group of French mathematicians from Strasbourg in the early 1980s [1]. They considered the van der Pol equation
2.1with bifurcation parameter *a* and 0<*ε*≪1. The system displays a Hopf bifurcation at *a*=1. The branch of limit cycles that emanates from this Hopf bifurcation initially behaves ‘normally’ in the sense that the envelope of the family of limit cycles follows a square-root progression. But for ‘small enough’ *ε* after the bifurcation point, the branch increases quasi-vertically. This corresponds to an *O*(1) change in amplitude and period of the associated periodic solution within an interval of size , for some constant *c*>0. This dramatic event, which results in an almost vertical bifurcation branch, has been termed a ‘canard explosion’ [2]. It occurs in such a narrow band of *ε* that it is very difficult to observe numerically and, hence, is easy to confuse with a discontinuous event [17].

### (b) The piecewise-linear case

Following Komuro & Saito [12], we study a PWL equivalent of the van der Pol equation (2.1). The system takes the form of a classical slow–fast Liénard system
2.2and
2.3with a continuous PWL function *f*, given by
where *k*>0 is a constant. The function *f* has three straight line segments separated by two corners at *x*=±1 and, hence, (2.2)–(2.3) is a PWL dynamical system with three zones. The critical manifold is C:=C_{L}∪C_{M}∪C_{R} given by *y*=*f*(*x*) (figure 1). The outer segments C_{L},C_{R} of C are attractive, whereas the middle segment, C_{M}, is repelling. In what follows, to simplify the analysis, we rescale time by *ε*, so that our system is now
2.4and
2.5where the overdot denotes differentiation with respect to the *fast time* *τ*=*t*/*ε*. It is fairly easy to compute periodic orbits that follow both attracting and repelling parts of the critical manifold (at least for those containing a small repelling segment; longer repelling segments are more challenging to compute), reminiscent of the *canard cycles* of the smooth van der Pol system [1]. These cycles are shown in figure 1*a*. The fixed parameters are *ε*=0.2, *k*=0.885 (values taken from Komuro & Saito [12]), and the parameter *a* varies from 0.5 (large outer cycle) to 0.9999999 (small inner cycle).

A comparison with the original van der Pol system, also for *ε*=0.2, is presented in figure 1*b*. Note that this value of *ε* might not seem ‘small enough’ at first sight; however, there is no definite way to quantify how small *ε* should be for the system to deserve the term *singularly perturbed*. One measure of this smallness of *ε* was proposed in Desroches & Jeffrey [18] by looking at the transition from small convex cycles to large non-convex cycles, characteristic to canard explosions and related to the existence of inflection points of the flow near the middle branch of the critical manifold. For the smooth van der Pol system, it has been shown that the maximal value of *ε* for which such a transition exists is 0.25; in this respect, the value chosen here falls into this range, and the important convex/non-convex characteristic of canard explosions is preserved.

In matrix form, system (2.4)–(2.5) becomes
2.6where is a 2×2 matrix and Z={L,M,R}. We have
in the left zone (L) ({*x*<−1}), in the right zone (R) ({*x*>1}) and in the middle zone (M) ({|*x*|≤1}), respectively. In addition, *α*_{R}=−*α*_{L}=1+*k*, *α*_{M}=0 and *β*_{Z}=*aε* for Z={L,M,R}.

Therefore, in the middle zone (M), for |*a*|<1, the stability of the equilibrium of system (2.4)–(2.5) located at (*a*,*f*(*a*)), with *f*(*a*)=−*ka*, is determined by the eigenvalues of *A*_{M}, given by Because *k*>0, the equilibrium point for |*a*|<1 is always unstable, being a focus for 4*ε*−*k*^{2}>0 and a node otherwise.

When |*a*|>1, the equilibrium has eigenvalues So for |*a*|>1 and , the equilibrium is a stable node.

By moving down *a* through its critical value *a*=1, the equilibrium loses stability as the system undergoes a bifurcation, always leading to a periodic orbit. However, this bifurcation has different features depending on the relative position of the point (*k*,*ε*) with respect to the parabola 4*ε*−*k*^{2}=0, as shown in our main result below. In the *focus case* when 4*ε*−*k*^{2}>0, the dynamics in the central region is of focus type; otherwise, we speak of the *node case*. In Komuro & Saito [12], *ε*=0.2 and *k*=0.885, corresponding to a focus case (of low frequency).

In order to consider behaviour close to the critical value *a*=1, we introduce a new parameter *δ*:=1−*a*>0, to state the main result of this paper. We emphasize that the required upper bound is needed to warranty that the dynamics in both external regions is of node type.

### Theorem 2.1

*Consider system* (2.2)–(2.3), *or equivalently system* (2.4)–(2.5), *with* *and k*>0 *fixed. The following statements hold*:

*For a>1, the equilibrium point is globally asymptotically stable and, therefore, it is the global attractor of the system.**For a=1, the equilibrium point is always the global attractor of the system; it is globally asymptotically stable when 4ε−k*^{2}*>0 (the focus case), but it is unstable for 4ε−k*^{2}*≤0 (the node case). The instability of this latter case comes from the existence of a bounded continuum of homoclinic orbits to the equilibrium point, the most external homoclinic orbit being defined by the unstable invariant manifold that coincides for −1≤x≤1 with the straight line given by**and eventually coming back to the equilibrium using, successively, zones L, M and R, approaching it tangentially to the straight line**For*−1<*a*<1,*the equilibrium point is unstable and surrounded by a unique stable limit cycle*.(a)

*When 4ε−k*^{2}*≤0 (the node case), the limit cycle always has points in the three linearity zones.*(b)

*When 4ε−k*^{2}*>0 (the focus case), the limit cycle is born in a Hopf-like bifurcation at δ=0, using only the zones M and R when δ>0 and small.*

*Regarding the focus case, if we measure the peak-to-peak amplitude A of the limit cycle by taking its two intersections with the line x*=1,*namely the points*(1,*y*_{1}(*δ*))*and*(1,*y*_{2}(*δ*)),*where y*_{1}(*δ*)<−*k*<*y*_{2}(*δ*),*then A*(*δ*):=*y*_{2}(*δ*)−*y*_{1}(*δ*)*is a linear function of δ as long as the limit cycle has no points in the region x*<−1.*More precisely, there exist two positive constants δ*_{G}(*k,ε*)*and S*_{F}(*k,ε*)*such that**for*0<δ≤δ_{G}.*Furthermore, the length of the linear range δ*_{G}*and the linear growth ratio S*_{F}*satisfy*

We remark that some of the earlier mentioned statements agree with the results obtained, following a different approach, for a similar PWL system in §5 of Llibre *et al*. [14], so that the bifurcations involved in theorem 2.1 are not entirely new. We should mention, however, that the quoted authors did not pay attention to the explosive behaviour that can be found in the focus case, which is an essential contribution of this work. Furthermore, certain key results of the quoted paper about limit cycles of their PWL system (namely proposition 10 and theorem 11) are unfortunately compromised by a serious gap in the proof of corollary 9.

From theorem 2.1, proved in §3, we clearly see a different behaviour in the transition to oscillation that happens when passing from *a*=1 to *a*<1, depending on the sign of 4*ε*−*k*^{2}. In the *node* case, that is, for (*ε*,*k*) under the parabola, this transition is instantaneous, because the limit cycle appears suddenly in a global bifurcation of the invariant object formed by the continuum of homoclinic canard orbits that exists for *a*=1. In this respect, we propose the term *super-explosion* to describe the continuum of canard homoclinic orbits that coexist at *a*=1 and are responsible for the instantaneous jump to a large relaxation oscillation for any *a*<1. This feature of the PWL case is clearly different from the smooth case, where the transition from stationary to periodic attractors via canard solutions is abrupt but never instantaneous. We will return to this point in §5.

In the *focus* case, the evolution of the amplitude is not abrupt but can be, indeed, very sharp before the limit cycle enters zone L (*x*<−1); the limiting situation (*δ*=*δ*_{G}), where the limit cycle touches tangentially the straight line *x*=−1 through the point (−1,*k*), will be called the *grazing bifurcation*. In figure 2, we show the amplitude evolution with *δ* for our case of reference (*ε*=0.2 and *k*=0.885), where such an explosive behaviour is observed.

It is well known that a key parameter to control the focus dynamics is the ratio *γ* of real and imaginary parts of its eigenvalues. Writing , we obtain
2.7and, to measure the proximity to the parabola *ε*=*k*^{2}/4, we introduce a new auxiliary parameter *η*>0 defined by *ε*=(1+*η*^{2})*k*^{2}/4, so that *η*=1/*γ* (figure 3). As shown later, an upper bound for *δ*_{G} (see §3) and a rough estimate for *S*_{F} (see appendix A) are
2.8Although for our case of reference the corresponding values of *γ*≈6.8330 and *η*≈0.14635 are moderate, the explosive behaviour is obvious, as we obtain from numerical simulation *δ*_{G}≈8.6332×10^{−10} and *S*_{F}≈2.565×10^{9}. For larger values of *δ*, that is, after the grazing bifurcation (*δ*>*δ*_{G}), the amplitude growth becomes much smaller as the limit cycle uses the three zones and is then confined by the invariant manifolds of the two external node dynamics.^{1}

In short, statement (3) of our theorem (2.1) says that an explosion of quasi-canard cycles occurs when we are in the parameter plane (*k*,*ε*) near the parabola 4*ε*−*k*^{2}=0 and above it (for *η* small). We want to emphasize that, different from the smooth case, this explosion is linear and starts from the very first moment when *δ*>0 (figure 2), which is related to the existence of a PWL Hopf-like bifurcation [19,20].

## 3. Proof of theorem (2.1)

We start by recalling that our PWL system is governed by Lipschitz vector fields and hence satisfies the standard results about existence and uniqueness of solutions, as well as continuous dependence on initial conditions and parameters. Note that system (2.4)–(2.5) can be written in the form
3.1and
3.2where *g*(*x*):=*x*−*f*(*x*) is a bounded function. It is then easy to show that the system is *dissipative*, which guarantees that its solutions are ultimately bounded.

### (a) Part 1: *a*>1

It is straightforward to prove part 1 because the only equilibrium is a stable node, located in zone R (*x*>1). Because the vector field is continuous, periodic orbits, if they exist, should surround the equilibrium point (theorem 3.1 of Hartman [21]). If we assume the existence of a periodic orbit, then the invariant manifolds of the node, which extend to infinity as straight lines in zone R, would intersect the periodic orbit, hence yielding a contradiction with the uniqueness of solutions.

### (b) Part 2: *a*=1

To prove part 2, we first notice that the equilibrium point, being at the corner (1,−*k*), has a different stability character depending on whether we view it from zone R or from zone M. From R, it behaves as an attractive node with two invariant manifolds defined by the straight lines We can therefore ensure that, for all orbits (*x*(*t*),*y*(*t*)) starting from *x*(0)≥1 and we have *x*(*t*)>1 for all *t*>0 and that .

However, orbits with *x*(0)≥1 and will reach a point (1,*y*) with *y*<−*k*, where on the switching line {*x*=1}. Hence, the trajectory enters zone M and its future evolution in the phase plane is then controlled by the associated vector field.

Because the equilibrium is unstable when seen from zone M, orbits starting with *x*(0)<1 can go far from the equilibrium point using zones M and L. In any case, however, because orbits are ultimately bounded, and there are no other equilibrium points, after crossing the vertical isocline if needed, they will eventually arrive at the vertical line *x*=1 in a point (1,*y*) with *y*>−*k*, where the stable node then takes control. Thus, the equilibrium is always globally attractive.

When 4*ε*−*k*^{2}≤0 (the node case), we have two unstable linear invariant manifolds that emanate from the equilibrium point to the left, so that in every neighbourhood of (1,−*k*) there are orbits escaping from it, and the equilibrium is unstable. The lower of these two invariant manifolds, that is, the one that coincides for −1≤*x*≤1 with becomes a maximal homoclinic orbit of a bounded set of orbits that have the equilibrium point both as *ω*-limit and *α*-limit (figure 4). Hence, part 2 of theorem 2.1 is proved.

### (c) Part 3*a*: −1<*a*<1: the node case

The existence and uniqueness of the limit cycle comes directly from theorem 1 of Llibre *et al.* [22]. In the node case, such a limit cycle must contain in its interior not only the equilibrium point, now located at (*a*,−*ka*), but also the portion of the invariant straight lines corresponding to −1≤*x*≤1 (figure 5). In this case, the limit cycle is forced to use all three zones.

### (d) Part 3*b*: −1<*a*<1: the focus case

For the focus case when *δ*=1−*a*>0, the dynamics in zone R is still governed by a *virtual* stable node at (*a*,−*k*−*δ*). Consider an orbit starting at a point with *x*>1 belonging to the upper linear invariant manifold of the right virtual node. After entering zone M at a point (1,*y*) with , this orbit must surround the unstable focus located at (1−*δ*,−*k*+*kδ*), spiralling up to intersect again the line *x*=1 and subsequently approaching again the upper invariant manifold of the right virtual node. If *δ*>0 is sufficiently small, then the focus is so near the vertical line *x*=1 that such an orbit does not enter zone L and allows us to build a positive invariant set containing the stable limit cycle. Thus, the limit cycle only uses zones M and R when *δ*>0 is sufficiently small, as stated in the theorem.

In order to understand the quasi-canard explosion in system (2.4)–(2.5), we will measure the growth rate of the amplitude for the limit cycle that is born at *a*=1. We take as the new origin the right corner at (1,−*k*) by introducing new variables *X*=*x*−1 and *Y* =*y*+*k*. We neglect zone L, because it is not involved in the birth of the limit cycle.

The system then becomes
3.3and
3.4with the unstable focus at (−*δ*,*kδ*).

Now for all values of *δ*>0, we can see that the corresponding limit cycles of this bi-zonal system are homothetic, all of them being a *δ*-scaled version of the limit cycle which exists for *δ*=1. Effectively, after the new rescaling *δx*=*X*, *δy*=*Y* (note that *x* and *y* are not equal to the original ones) and cancelling a common factor *δ*>0 appearing in the new equations, we obtain a *template system* (not depending on *δ*), namely
3.5and
3.6with the unstable focus now fixed at (−1,*k*). Note that the *template system* can be obtained from (3.3)–(3.4) by taking *δ*=1. After the change of variables (*x*+1,*y*−*k*)→(*x*,−*y*), a direct application of theorem 2 in Llibre *et al.* [22] with *t*_{L}=*k*, *t*_{R}=−1 and *d*_{L}=*d*_{R}=*ε* allows us to write the following result.

### Proposition 3.1

*Assuming* *and* *k*>0, *the template system* (3.5)–(3.6) *has periodic orbits if and only if* 4*ε*−*k*^{2}>0; *in such a case, the equilibrium point is surrounded by a unique stable limit cycle*.

As a consequence, the template system (3.5)–(3.6) undergoes a bifurcation leading to the appearance of one stable limit cycle (or its destruction) when 4*ε*−*k*^{2}=0. We will see in §4 that this phenomenon is associated with a heteroclinic bifurcation at infinity, so that the limit cycle is very large for . We emphasize in the next result the usefulness of the template system, because the size of its limit cycle gives a measure of the linear growth ratio with respect to *δ*>0 of the limit cycle both for system (3.3)–(3.4) and for the original system (2.4)–(2.5) (in this case, only when *δ*≤*δ*_{G}).

### Proposition 3.2

*Assume for the template system* (3.5)–(3.6) *that* *and* 4*ε*−*k*^{2}>0, *and that its limit cycle (predicted by proposition 3.1) passes through the points* (0,*y*_{1}) *and* (0,*y*_{2}). *Then, the linear growth ratio with respect to* *δ*>0 *of the limit cycle both for system* (3.3)–(3.4) *and for system* (2.4)–(2.5) *is* *S*_{F}(*k*,*ε*):=*y*_{2}−*y*_{1}.

Effectively, by undoing the rescaling *δx*=*X*, *δy*=*Y* , needed to obtain the template system, we see that the size of the limit cycle for the bi-zonal system (3.3)–(3.4) will be *A*(*δ*)=*δy*_{2}−*δy*_{1}=(*y*_{2}−*y*_{1})*δ*. Moreover, for sufficiently small *δ*>0, the bi-zonal system (3.3)–(3.4) reproduces the behaviour of our original tri-zonal system (2.4)–(2.5) locally near the lower corner, so that the assertion of the proposition is obvious.

We, therefore, deduce from proposition 3.2 that the quasi-canard explosion comes from the fact that, when (*ε*,*k*) leads to a focus case (but near the parabola that corresponds to the transition from focus to node), the size *S*_{F} of the template limit cycle is very large (figure 6). Of course, such an explosion does not give rise to a large limit cycle in system (2.4)–(2.5) because we have the *grazing* bifurcation, when the limit cycle grazes the line *x*=−1 at the point (−1,*k*), and afterwards uses the three zones.

To get an upper estimate for the grazing value *δ*_{G}, we assume in system (2.4)–(2.5) fixed values of *k* and *ε* and consider the value *δ*^{⋆} such that the orbit starting at (1,−*k*) passes, after a half-turn around the unstable focus, through the point (−1,*k*). Obviously, for such a value of *δ* the limit cycle is larger than the grazing cycle and so *δ*_{G}<*δ*^{⋆}. We equivalently work with system (3.3)–(3.4) by looking for the orbit starting at the origin that passes, after a half-turn around the unstable focus, now located at (−*δ*,−*kδ*), through the point (−2,2*k*), that is, the translation of (−1,*k*). The computations involve only points with *X*≤0, and it is more convenient to rewrite the system using a scale factor of *ω*, where , both for time and for *X*. Putting *x*=*ωX*,*y*=*Y* and recalling the definition of *γ* in (2.7), from (3.3)–(3.4), we find
3.7and
3.8where the dot now denotes the derivative with respect to a new time *θ*. Note that 1+*γ*^{2}=*ε*/*ω*^{2} and that the equilibrium point is now at (−*ωδ*,2*γωδ*)=(−*ωδ*,*kδ*). Thus, any solution corresponding to an orbit totally contained in *x*≤0 and joining the point (*x*_{1},*y*_{1}) with (*x*_{2},*y*_{2}) after a time interval *θ* satisfies
3.9The required orbit for computing *δ*^{⋆} must join here the point (*x*_{1},*y*_{1})=(0,0) and (*x*_{2},*y*_{2})=(−2*ω*,2*k*) and corresponds to *θ*=*π*. Thus, we have the condition , obtaining the upper estimate in (2.8). Now it can be seen immediately that *δ*_{G}→0 when since .

In appendix A, we give a procedure for computing the numerical value of *S*_{F}, which allows us to show quantitatively the remaining part of statement (3) of theorem 2.1. Thus, the proof is complete.

We can add, however, qualitative arguments to show that the condition 4*ε*−*k*^{2}=0 is associated with the existence of a saddle–node (SN) bifurcation at infinity in the template system (see §4). Such a bifurcation explains why the limit cycle of the template system disappears with infinite amplitude in a heteroclinic connection involving points at infinity, that is, in a closed loop joining different singular points of the equator of a Poincaré sphere, and consequently why the growth ratio with *δ*>0 of the periodic orbits is so large.

## 4. The template system: phase portrait and analysis at infinity

In order to understand how the large limit cycle is created or destroyed through the bifurcation at 4*ε*−*k*^{2}=0, we compactify the phase space and search for equilibria at infinity. Following M. Román & E. Ponce (2002, unpublished data), we introduce a new set of coordinates on the unit disc , and a map that transforms the points of the phase plane into the points . This transformation from the plane to the disc and its inverse are given by
4.1

We want to consider the image, under this transformation, of the template PWL vector field (3.5)–(3.6), which allows us to reproduce after rescaling the behaviour of the bi-zonal system (3.3)–(3.4), and also (locally near the lower corner) that of original system (2.4)–(2.5). The template vector field (3.5)–(3.6) can be written as 4.2and, following Román & Ponce (unpublished data), we find that the vector field on is given by 4.3where 4.4

This new vector field has the unit circle as an invariant curve (figure 7), which represents the points at infinity of the original phase plane, and it can be simplified by an implicitly defined time reparametrization that amounts to multiplying by the positive factor . Putting the expression for *f*(*x*,*y*) from (4.2) into (4.3), and carrying out such a time reparametrization, we obtain the topologically equivalent vector field
4.5

Apart from the transformed points corresponding to equilibrium points of the template system, we can now look for their singular points at infinity, that is, for equilibria of the transformed vector field on the boundary of , where . Clearly, we can put in , and then equilibria in the boundary of are given by the solutions of the equation Thus, the equilibria at infinity of the original system correspond to the angles with that is, with the slopes of the invariant manifolds of the possible real or virtual nodes in each zone.

For we have *m*=−1 and we obtain the two points denoted by S (a saddle) and UN (an unstable node) in figure 7, where we also show the invariant manifolds associated in this case with a virtual node. For we have *m*=*k*, however, and we pass from two equilibrium points when 4*ε*−*k*^{2}<0 to none for 4*ε*−*k*^{2}>0. At 4*ε*−*k*^{2}=0, we have the critical situation depicted in figure 7, where the two equilibria on the left-hand part of merge to disappear in a SN, also responsible for the existence of a heteroclinic cycle joining the saddle S and the point SN both through an orbit in the finite plane and an orbit in . This heteroclinic connection at infinity exists for *ε*=*k*^{2}/4 and then bifurcates when *ε* increases into a family of very large limit cycles decreasing in amplitude (figures 1 and 7). Simultaneously, the left node that at *ε*=*k*^{2}/4 is an improper node (IN) becomes an unstable focus to be surrounded by the limit cycles.

## 5. Comparison with canards in smooth systems

### (a) Similarities with the piecewise-linear case

We can find (at least) four similarities between the behaviour of canard cycles in smooth and in PWL planar slow–fast systems.

Canards in smooth slow–fast systems have been extensively studied over the past four decades, in particular in planar systems where the main mechanism for their appearance is the canard explosion, following a Hopf bifurcation that occurs very close (typically

*ε*-close) to the fold of the critical manifold. As we have shown in this paper, PWL quasi-canards also occur by an explosive transition close to a Hopf-like bifurcation (the crossing of the switching line by the equilibrium, which makes it lose its stability) that creates a stable limit cycle in the system.Canard solutions, in particular canard cycles, are characterized by the counterintuitive property that they follow, for an

*O*(1) interval of time, a repelling part of the fast nullcline, and this also happens in the PWL case. Furthermore, they are ‘created’ by the passage, upon parameter variation, of the slow nullcline through a point of the fast nullcline that separates an attracting from a repelling part of it. This is true in the smooth case, where the point is a*fold point*of the fast nullcline. This particular movement around the non-hyperbolic point is preserved in the PWL case, where the point is the intersection between a switching line and the fast nullcline of two consecutive zones, that is, a*corner point*. However, the absence of a repelling slow manifold in this PWL case constitutes an important difference from the smooth case; this is why we call only these cycles quasi-canards.In order to observe a quasi-canard cycle near the singular limit, one must move two parameters simultaneously, one parameter being the time-scale separation

*ε*. This was already noticed by Komuro & Saito [12]. This is also the situation in the smooth case because one must adjust the parameter*a*when decreasing*ε*because the location—in terms of*a*—of the canard explosion depends on*ε*. Furthermore, this two-parameter variation is deeply related to the presence of a focus equilibrium near the point that separates attracting and repelling parts of the fast nullcline, whether this point is a fold (smooth case) or a corner (PWL case).We further note that when decreasing

*ε*(keeping other parameters fixed) in systems for which the critical manifold has only one attracting part—a ‘corner system’ in the PWL context, a ‘singular fold’ in the smooth context—then canard cycles in the latter case, quasi-canard cycles in the former, disappear together with the focus equilibrium in a heteroclinic bifurcation at infinity.

### (b) Differences from the piecewise-linear case

We can find (at least) three main differences between canard cycles in both contexts.

*The behaviour of the bifurcation branch.*It is a nonlinear function of parameter*a*in the smooth case with a steep increase in an exponentially small interval during the canard explosion. On the other hand, it is linear in the PWL case, but also very steep during the explosion.*The quasi-canards with heads.*Unlike the smooth van der Pol-like case, that is, with two attracting parts on the critical manifold, which creates canards with heads as well as headless canards, PWL systems with three zones can only sustain headless canards. This is because of the nature of the dynamics, which is linear in each zone and, hence, generates linear invariant manifolds that prevent orbits following the fast nullcline of the middle zone for some time and then curve towards the fast nullcline of the left zone. This is very much related to the fact that one cannot make sense of repelling slow manifolds in the three-pieces PWL case. As indicated in Arima*et al.*[13], a fourth piece in the critical manifold creates repelling slow manifolds and, hence, true PWL canards, both without and with heads (see also Rotstein*et al.*[15]).*The phenomenon of super-explosion.*We use this term to describe*the instant transition to relaxation*that occurs in PWL systems (of the type we considered in this paper) when the equilibrium of the full system lies at the corner, on the switching surface, and is not of the focus type. We have shown that there exists a continuum of canard homoclinic connections with this equilibrium, owing to the fact that its topological type is different whether it is seen from one zone or the other. We have also shown that, for any value of the parameter lower than this critical value, there exists a unique large cycle that visits all three zones, that is, a relaxation oscillation. This jump in parameter space, from stationary dynamics to relaxation oscillation, can be seen as the limit of the smooth canard explosion when the parameter variation, of order , tends to 0 as*ε*tends to 0. It is then purely owing to the piecewise nature of the system, similar to the phenomena of*instant chaos*[23] or*big bang bifurcations*[24].

## 6. Discussion

In this paper, we investigated the canard phenomenon in prototype PWL continuous planar slow–fast systems with two and three zones. We complemented earlier work, in particular that by Komuro & Saito [12], in which canard-like behaviour had been reported but in which some questions remained unanswered. In particular, we showed that quasi-canard cycles exist in such two- and three-zone systems even if one has to be careful in the way one takes the singular limit *ε*=0. Indeed, decreasing the small parameter *ε* requires an adjustment in the value of the slope *k* of the fast repelling nullcline if one wants to remain in the quasi-canard regime. We call them quasi-canards because they are short-lived and follow the ‘ghosts’ of canard homoclinic connections, although they do not possess all the characteristics of their smooth cousins. The main theoretical result (theorem 2.1) aimed to understand what happens close to the boundary of the quasi-canard regime in the (*k*,*ε*)-plane. We found that quasi-canards disappear through a phenomenon related to a heteroclinic bifurcation at infinity in an associated bi-zonal template system, which we could understand by compactifying the phase plane. Furthermore, we analysed the extreme transition that takes place on the other side of this boundary in the (*k*,*ε*)-plane, where a large-amplitude cycle is born instantaneously as the main bifurcation parameter of the model (controlling the position of the slow nullcline) crosses the corner. We named this extreme transition ‘super-explosion’ as it involves a continuum of homoclinic connections, coexisting at one parameter value (for which the slow nullcline is exactly at the corner) and containing canard segments. This super-explosion is an effect of the non-smoothness of the system and does not persist in the smooth case. Similarly, the end of the quasi-canard explosion corresponds to a grazing bifurcation, which only has meaning in the non-smooth context. Numerically, we computed branches of the limit cycle in the PWL system, undergoing quasi-canard explosion, using a path-following technique as implemented in the package TC-HAT [25]. Such numerical continuation allowed us to follow the unbounded growth of quasi-canard cycles in the associated bi-zonal template system as they approach the boundary in the (*k*,*ε*)-plane with good precision, as well as to detect the grazing event that terminates the quasi-canard explosion in the normal regime, that is, for the tri-zonal original system.

Our main objective in future work is to study PWL slow–fast systems in three dimensions, especially those with two slow variables. This research direction has been recently initiated [26,27] but many questions remain unanswered. Canard solutions in three-dimensional slow–fast (smooth) systems with two slow variables are robust in the sense that they can be proved to exist for order-one intervals of parameters [28]. They are also crucial in the understanding of so-called *mixed-mode oscillations*, which are complex periodic solutions of these kinds of systems and have been encountered in many biological applications (see [29] for a recent review on this topic). It is then of direct interest to look at possible canards and mixed-mode dynamics in PWL systems in three dimensions. Adding one fast variable instead of one slow could also be interesting and could lead to the study of *bursting oscillations* in three-dimensional PWL systems (with two fast variables).

## Acknowledgements

The authors thank Martin Krupa for fruitful discussions as well as the excellent work of the anonymous referees, which greatly improved the preliminary version of the manuscript. E.F. and E.P. were partially supported by the Spanish Ministerio de Ciencia y Tecnología, in the frame of projects MTM2009-07849, MTM2010-20907, MTM2012-31821, and by the Consejería de Educación y Ciencia de la Junta de Andalucía grants TIC-0130 and P08-FQM-03770. M.D. acknowledges the support of EPSRC through grant no. EP/E032249/1 and the Department of Engineering Mathematics at the University of Bristol, where part of this work was carried out.

## Appendix A

A.1. Computation of the growth ratio *S*_{F}

From proposition 3.2, to compute the growth ratio *S*_{F}(*k*,*ε*), we need to know the intersection points (0,*y*_{1}) and (0,*y*_{2}) of the limit cycle of the template system (3.5)–(3.6) with the *y*-axis. Thus, we can rescale differently the time and the *x*-variable in each half plane. For *x*<0, we use the same factor *ω* in obtaining (3.7)–(3.8), so that, by putting *δ*=1 and *x*_{1}=*x*_{2}=0 in (3.9), we have
and *y*_{2}(*θ*)=*y*_{1}(−*θ*) for a certain *π*<*θ*<2*π*. We note that, because *y*_{1} must be negative, the range of admissible values for *θ* is , where is the first positive zero of the auxiliary function (see [30] for more details). Thus, provided the value of *θ* corresponding to the limit cycle is known, we can write

Similarly, for *x*>0, we rescale *x* and the time by a factor *Ω*, where . From (3.5) and (3.6), we find
A1and
A2where the dot now denotes derivatives with respect to the new time *ϕ* and
A3Any solution corresponding to an orbit totally contained in *x*≥0 and joining the point (0,*y*_{2}) with (0,*y*_{1}) satisfies
Here, the equilibrium point is now at (−*Ω*,−2*βΩ*)=(−*Ω*,−1). Solving for *y*_{1}, *y*_{2}, we obtain
and *y*_{1}(*ϕ*)=*y*_{2}(−*ϕ*), so that
Therefore, to determine *S*_{F}, it remains to find the two flight times *θ* and *ϕ* that correspond to the limit cycle, that is, we must impose the two conditions *y*_{1}(*θ*)=*y*_{1}(*ϕ*) and *y*_{2}(*θ*)=*y*_{2}(*ϕ*), where and *ϕ*>0. Equivalently, by noticing that the two expressions for *y*_{1}+*y*_{2} are more compact, the problem of computing *S*_{F} amounts to solving for and *ϕ*>0 the two equations
A4and
A5

It is now possible to show that if then , and for the solution of (A4)–(A5) we have , so that , and . Therefore, we obtain , as stated in theorem 2.1.

By using the parameter *η*=1/*γ*>0, which measures the proximity to the parabola *ε*=*k*^{2}/4, we can obtain the estimate for *S*_{F} given in (2.8). In fact, it suffices to make in (A4) the approximation for large *γ* (small *η*).

## Footnotes

↵1 In fact, the limit cycle amplitude grows as long as 0<

*δ*≤1, i.e. up to*a*=0, which corresponds to a symmetrical vector field with the equilibrium point at the origin; for*δ*>1, the limit cycle decreases in a reversal process up to*δ*=2 (i.e.*a*=−1), where the equilibrium is located at the left corner of the graph of*f*.

- Received October 14, 2012.
- Accepted March 18, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.