Abstract
Canard cycles are periodic orbits that appear as special solutions of fastslow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blowup method, and control theory, to design controllers that stabilize canard cycles of planar fastslow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixedmode oscillations in the van der Pol oscillator.
Introduction
Fastslow systems (also known as singularly perturbed ordinary differential equations, see more details in Section 2) are often used to model phenomena occurring in two or more time scales. Examples of these are vast and range from oscillatory patters in biochemistry and neuroscience [6, 18, 25, 26], all the way to stability analysis and control of power networks [10, 14], among many others [41, Chapter 20]. The overall idea behind the analysis of fastslow systems is to separate the behavior that occurs at each time scale, understand such behavior, and then try to elucidate the corresponding dynamics of the full system. Many approaches have been developed, such as asymptotic methods [17, 34, 51, 52], numeric and computational tools [24, 32], and geometric techniques [20, 31, 33], see also [41, 46, 56]. In this article, we take a geometric approach.
Although the time scale separation approach has been very fruitful, there are some cases in which it does not suffice to completely describe the dynamics of a fastslow system, see the details in Section 2. The reason is that, for some systems, the fast and the slow dynamics are interrelated in such a way that some complex behavior is only discovered when they are not fully separated. An example of the aforementioned situation are the socalled canards [7, 8, 15], see Section 2.1 for the appropriate definition. Canards are orbits that, counterintuitively, stay close for a considerable amount of time to a repelling set of equilibrium points of the fast dynamics. Canards are extremely important not only in the theory of fastslow systems, but also in applied sciences, and especially in neuroscience, as they have allowed, for example, the detailed description of the very fast onset of large amplitude oscillations due to small changes of a parameter in neuronal models [18, 26] and of other complex oscillatory patterns [9, 12, 47]. Due to their very nature, canard orbits are not robust, meaning that small perturbations may drastically change the shape of the orbit.
On the other hand, the application of singular perturbation techniques in control theory is farreaching. Perhaps, as already introduced above, one of the biggest appeals of the theory of fastslow systems is the time scale separation, which allows the reduction of large systems into lower dimensional ones for which the control design is simpler [29, 37, 38]. Applications range from the control of robots [28, 53, 54], all the way to industrial biochemical processes, and large power networks [13, 35, 36, 43, 49, 50]. However, as already mentioned, not all fastslow systems can be analyzed by the convenient time scale separation strategy, and although some efforts from very diverse perspectives have been made [2,3,4,5, 22, 23, 29, 30], a general theory that includes not only the regulation problem but also the path following and trajectory planning problems is, to date, lacking.
The main goal of this article is to merge techniques of fastslow dynamical systems with control theory methods to develop controllers that stabilize canard orbits. The idea of controlling canards has already been explored in [16], where an integral feedback controller is designed for the FitzHughNagumo model to steer it towards the socalled canard regime. In contrast, here we take a more general and geometric approach by considering the folded canard normal form, see Section 2.1. Moreover, we integrate control techniques with Geometric Singular Perturbation Theory (GSPT) and propose a controller design methodology in the blowup space. Later, we apply such geometric insight to the van der Pol oscillator where we provide a controller that produces any oscillatory pattern allowed by the geometric properties of the model, see Section 4.
The rest of this document is arranged as follows: in Section 2, we present definitions and preliminaries of the geometric theory of fastslow systems and of folded canards, which are necessary for the main analysis. In Section 3, we develop a controller that stabilizes folded canard orbits, where the main strategy is to combine the blowup method with statefeedback control techniques to achieve the goal. Afterwards in Section 4, as an extension to our previously developed controller, we develop a controller that stabilizes several canard cycles and is able to produce robust complex oscillatory patters in the van der Pol oscillator. We finish in Section 5 with some concluding remarks and an outlook.
Preliminaries
A fastslow system is a singularly perturbed ordinary differential equation (ODE) of the form
where \(x\in \mathbb {R}^{m}\) is the fast variable, \(y\in \mathbb {R}^{n}\) the slow variable, 0 < ε ≪ 1 is a small parameter accounting for the time scale separation between the aforementioned variables, \(\lambda \in {\mathbb {R}}^{p}\) denotes other parameters, and f and g are assumed sufficiently smooth. In this document, the overdot is used to denote the derivative with respect to the slow time τ. It is wellknown that, for ε > 0, an equivalent way of writing Eq. 1 is
where now the prime denotes the derivative with respect to the fast time t := τ/ε.
One of the mathematical theories that is concerned with the analysis of Eqs. 1– 2 is Geometric Singular Perturbation Theory (GSPT) [41]. The overall idea of GSPT is to study the limit equations that result from setting ε = 0 in Eqs. 1– 2. Then, one looks for invariant objects that can be shown to persist up to small perturbations. Such invariant objects give a qualitative description of the behavior of Eqs. 1– 2. Accordingly, setting ε = 0 in Eqs. 1– 2 one gets
known, respectively, as the reduced slow subsystem (which is a constrained differential equation [55] or a differential algebraic equation [42]) and the layer equation. The aforementioned limit systems are not equivalent any more, but they are related by the following important geometric object.
Definition 1 (The critical manifold)
The critical manifold is defined as
We note that the critical manifold is the phasespace of the reduced slow subsystem and the set of equilibrium points of the layer equation. The properties of the critical manifold are essential to GSPT, in particular the following.
Definition 2 (Normal hyperbolicity)
Let \(p\in \mathcal {C}_{0}\). We say that p is hyperbolic if the matrix \(\text {D}_{x}f(p,0,\lambda )_{\mathcal {C}_{0}}\) has all its eigenvalues away from the imaginary axis. If every point \(p\in {\mathcal {C}}_{0}\) is hyperbolic, we say that \({\mathcal {C}}_{0}\) is normally hyperbolic. On the contrary, if for some \(p\in {\mathcal {C}}_{0}\) the matrix \({\text {D}}_{x}f(p,0,\lambda )_{{\mathcal {C}}_{0}}\) has at least one of its eigenvalues on the imaginary axis, then we say that p is a nonhyperbolic point.
It is known from Fenichel’s theory [19, 20] that a compact and normally hyperbolic critical manifold \(\mathcal {S}_{0}\subseteq \mathcal {C}_{0}\) of Eq. 3 persists as a locally invariant slow manifold \({\mathcal {S}}_{{\varepsilon }}\) under sufficiently small perturbations. In other words, Fenichel’s theory guarantees that in a neighborhood of a normally hyperbolic critical manifold the dynamics of Eqs. 1–2 are well approximated by the limit systems Eq. 3.
Remark 1
Along this paper, we use the notation \(\mathcal {S}_{0}^{\text {a}}\) and \(\mathcal {S}_{0}^{\text {r}}\) to denote, depending on the eigenvalues of \(\text {D}_{x}f(x,y,0,\lambda )_{\mathcal {S}_{0}}\), the attracting an repelling parts of the (compact) critical manifold \(\mathcal {S}_{0}\). Accordingly, the corresponding slow manifolds are denoted as \(\mathcal {S}_{\varepsilon }^{\text {a}}\) and \(\mathcal {S}_{\varepsilon }^{\text {r}}\).
On the other hand, critical manifolds may lose normal hyperbolicity, for example, due to singularities of the layer equation, see Fig. 1. It is in fact due to loss of normal hyperbolicity that, as in this paper, some interesting and complicated dynamics may arise in seemingly simple fastslow systems. Fenichel’s theory, however, does not hold in the vicinity of nonhyperbolic points. In some cases, depending on the nature of the nonhyperbolicity, the blowup method [27] is a suitable technique to analyze the complicated dynamics that arise. In the forthcoming section, we introduce the particular type of orbits that we are concerned with and that arise due to loss of normalhyperbolicity of the critical manifold: the socalled canards.
Planar Folded Canards
In this section, we briefly describe folded canards and folded canard cycles in the plane. As we mention below, the adjective “folded” is due to a fold singularity. However, we remark that canards (and canard cycles) can be related to other types of singularities. The interested reader is refereed to, e.g. [15, 39, 57], references therein and, in particular, [41, Chapter 8] and [27, Section 3] for more detailed information.
Let us start by recalling that the canonical form of a canard point [39] is given by
where \((x,y)\in \mathbb {R}^{2}\), 0 < ε ≪ 1, and α is a parameter. Furthermore,
and h_{6} is smooth. For simplicity of notation, we rewrite Eq. 5 together with Eq. 6 as
where \(\widetilde f\) and \(\widetilde g\) denote the corresponding higher order terms, that is:
The critical manifold is locally (near the origin) a perturbed parabola and is given by
The (slow and fast) reduced flow corresponding to Eq. 7 is as shown in Fig. 1.
Remark 2
To fix ideas, consider for a moment Eq. 7 with zero higher order terms,^{Footnote 1} that is
Then, it is straightforward to check that, for ε > 0 and α = 0, the orbits of Eq. 10 are given by level sets of
Some orbits of Eq. 10 are shown in Fig. 2, and in fact it is known [39] that canard cycles exist for \(H\in (0,\frac {1}{4})\).
What is remarkable is that there are orbits that closely follow the unstable branch of the critical manifold for slow time of order \(\mathcal {O}(1)\). Such type of orbits are known as canards. There is a particular canard, which is called maximal canard and is given by \(\left \{ H=0\right \}\) that connects the attracting slow manifold \({\mathcal {S}}_{{\varepsilon }}^{{\text {a}}}\) with the repelling one \({\mathcal {S}}_{{\varepsilon }}^{{\text {r}}}\). More relevant to this paper are periodic orbits with canard portions, which called canard cycles.
In the following section, we design feedback controllers for Eq. 5 that render a particular canard cycle asymptotically stable. In other words, we consider the path following control problem where a canard orbit is the reference.
Controlling Folded Canards
We propose to study two control problems, namely
which we call the fast control problem and
to be referred to as the slow control problem. Recall that \(\widetilde f\) and \(\widetilde g\) stand for the higher order terms as in Eq. 8. The objective is to stabilize a certain reference canard cycle to be denoted by γ_{h}.
Remark 3

The choice of the above control problems is motivated by applications, especially in neuron models, see [16, 18, 26], where the input current appears in the fast (voltage) variable and regulates the distinct firing patterns. However, if one is interested in the fully actuated case, a combination of the techniques presented here shall also be useful.

Throughout this document, we assume that one has full knowledge of the functions \(\widetilde f\) and \(\widetilde g\). This means that for the fast (resp. slow) control problem, we assume \(\widetilde f=0\) (resp. \(\widetilde g=0\)). Otherwise, one considers a controller of the form \(u=\widetilde f+v\) (resp. \(u=\widetilde g+v\)) where now v is to be designed.
Notice that in the case of the fastcontrol problem Eq. 12, the controller changes the fast dynamics. This means that the controller can change the type of singularities the critical manifold may present. To be more precise, consider for a moment Eq. 12 with u = −kx, k > 0, a simple proportional feedback controller. The closedloop system then reads as
for which the origin is now normally hyperbolic. This means that the feedback controller has changed the type of singularity (at the origin) from a fold to a regular one. It is clear that these type of controllers are not compatible with our task. So, we shall design controllers that do not change the type of singularity of the openloop system. To formalize what we mean by “not changing the type of singularity”, let us first recall the following definition:
Definition 3 (kjet equivalence)
Let \(F:\mathbb {R}^{n}\to \mathbb {R}^{n}\) and \(G:\mathbb {R}^{n}\to \mathbb {R}^{n}\) be smooth maps. We say that F and G are (kjet) equivalent at \(p\in \mathbb {R}^{n}\) if F(p) = G(p) and \(F(x)G(x)=\mathcal {O}(xp^{k+1})\) as x → p. An equivalence class defined by the previous notion of equivalence is called the kjet of F at p, and shall be denoted by j^{k}F(p) [1].
Next, we have a formal definition of what we refer to as a compatible controller:
Definition 4 (Compatible controller)
Consider a control system
where \(\zeta \in \mathbb {R}^{n}\) is the state variable, \(\lambda \in \mathbb {R}^{p}\) denotes system parameters (possibly including 0 < ε ≪ 1), and \(u\in \mathbb {R}^{m}\) stands for the controller. Suppose that for the openloop system, that is when u = 0, the origin \(\zeta =0\in \mathbb {R}^{n}\) is a nilpotent equilibrium point of \(\dot \zeta = f(\zeta ,0,0)\) and that there is a \(k\in \mathbb {N}\) such that k is the smallest number so that j^{k}f(0)≠ 0. Let u be a statefeedback controller, that is u = u(ζ,λ,ℓ), where \(\ell \in \mathbb {R}^{m}\) stands for parameters of the controller such as controller gains, and denote by \(\dot \zeta =F(\zeta ,\lambda ,\ell )\) the closedloop system. We say that u is a compatible controller if the openloop vector field f(ζ,λ,0) and the closedloop vector field F(ζ,λ,ℓ) are kjet equivalent at the origin for λ = 0.
We emphasize that once one fixes coordinates on \(\mathbb {R}^{n}\), a kjet equivalence between two maps means that such maps coincide on their partial derivatives up to order k. As an example of the above definition, recall that a planar fastslow system with a generic fold at the origin is given by
with the defining conditions f(0,0,0) = 0, \(\frac {\partial f}{\partial x}(0,0,0)=0\), \(\frac {\partial ^{2} f}{\partial x^{2}}(0,0,0)\neq 0\), and the nondegeneracy condition \(\frac {\partial f}{\partial y}(0,0,0)\neq 0\). Next, let u = u(x,y,ε) be a statefeedback controller and suppose one considers the fastslow control system
Then, u is a compatible controller if the closedloop system verifies: F(0,0,0) = 0, \(\frac {\partial F}{\partial x}(0,0,0)=0\), \(\frac {\partial ^{2} F}{\partial x^{2}}(0,0,0)\neq 0\), and \(\frac {\partial F}{\partial y}(0,0,0)\neq 0\), which implies that the controller does not change the class of the singularity, since the origin is still a fold point of the closedloop system.
The Fast Control Problem
In this section, we study the control problem defined by
Due to the fact that the slow dynamics are not actuated, we are going to stabilize canards centered at (x,y) = (α,0). Then, it is convenient to define \({\hat {x}}=x\alpha \), which brings Eq. 18 into
where \({\hat {u}}({\hat {x}},y,\varepsilon ,\alpha )=u({\hat {x}}+\alpha ,y,\varepsilon ,\alpha )\) and similarly for \({\hat {g}}\), we have \({\hat {g}}({\hat {x}},y,\varepsilon ,\alpha )=g({\hat {x}}+\alpha ,y,\varepsilon ,\alpha )\).
Theorem 1
Consider Eq. 19 and let \(\hat H=H({\hat {x}},y,\varepsilon )\) be defined by Eq. 11. Then, the following hold:

1.
The compatible controller
$$ \hat u=2\alpha{\hat{x}}\alpha^{2}+c_{1}{\hat{x}}\varepsilon^{1/2}\exp(c_{2}y\varepsilon^{1})(\hat Hh), $$(20)where c_{1} > 0, \(c_{2}\in \mathbb {R}\) and \(h\leq \frac {1}{4}\), renders the canard orbit \(\hat {\gamma }_{h}=\left \{ ({\hat {x}},y)\in \mathbb {R}^{2}  \hat H=h \right \}\) locally asymptotically stable for ε > 0 sufficiently small.

2.
Let \(\hat {{\varGamma }}\subset \mathbb {R}^{2}\) be a neigborhood of \(\hat \gamma _{h}\) for \(h\in (0,\frac {1}{4})\). Suppose that, additionally to Eq. 8, \(\hat g\) is of the form \(\hat g={\hat {x}}\hat \phi ({\hat {x}},y,{\varepsilon },\alpha )\) for some function \(\hat \phi \), and that \(\hat \phi \neq 1\) for all \((\hat x,y)\in \hat {{\varGamma }}\). Then, the compatible controller
$$ \hat u=2\alpha{\hat{x}}\alpha^{2}+c_{1}{\hat{x}}\varepsilon^{1/2}(\hat Hh)\exp(c_{2}y\varepsilon^{1})(y{\hat{x}}^{2})\hat\phi, $$(21)where c_{1} > 0, \(c_{2}\in \mathbb {R}\) renders the canard orbit \(\hat {\gamma }_{h}=\left \{ ({\hat {x}},y)\in \mathbb {R}^{2}  \hat H=h \right \}\) locally (within \(\hat {{\varGamma }}\)) asymptotically stable.
Remark 4

The choice of the controller gain c_{2} in Theorem 1 has an important impact in numerical simulations due to the fact of it appearing as an argument of the exponential function. The choice c_{2} = 2 yields the better numerical results when stabilizing canard cycles, that is for \(h\in (0,\frac {1}{4})\). However, to stabilize the maximal canard (h = 0), it is necessary to choose c_{2} < 2 to ensure that the controller remains bounded as \(y\to \infty \). See more detail in Section 3.1.2.

We recall that although from Theorem 1 one is able to stabilize any canard (because \(h\leq \frac {1}{4}\)), canard cycles exist only for \(h\in (0,\frac {1}{4})\), see Fig. 2 and [39].

The second item of Theorem 1 holds for any ε > 0.
The proof of Theorem 1 follows from the forthcoming analysis and is summarized in Section 3.1.3. We show in Fig. 3 a simulation of the results contained in Theorem 1.
As already anticipated, the idea is to design the controller \({\hat {u}}\) in the blowup space. Therefore, let us consider a coordinate transformation defined by
where \((\bar {x},\bar {y},\bar {{\varepsilon }},\bar {\mu },\bar {\alpha })\in \mathbb {S}^{4}\), with \(\mathbb {S}^{4}\) denoting the 4sphere, that is \(\{\bar {x}^{2}+\bar {y}^{2}+\bar {{\varepsilon }}^{2}+\bar {\mu }^{2}+\bar {\alpha }^{2}=1\}\), and \(\bar {r}\in [0,\infty )\). As is usual with the blowup method [27], instead of working in spherical coordinates, we consider local coordinates in local charts. In our particular context, these local charts parametrize different hemispheres of \(\mathbb {S}^{4}\). Analogous to the analysis of the canard point in [39], we consider the charts \(K_{1}=\left \{ \bar {y}=1 \right \}\) and \(K_{2}=\left \{ \bar {{\varepsilon }}=1 \right \}\). To distinguish the local coordinates in these charts, we let (r_{1},x_{1},ε_{1},μ_{1},α_{1}) be local coordinates in K_{1}, and (r_{2},x_{2},y_{2},μ_{2},α_{2}) be local coordinates in K_{2}. In this way, these local coordinates are defined by:
In particular, it is worth noting that in chart K_{1} the coordinate r_{1} is a rescaling of the “original coordinate” y for y ≥ 0, while in chart K_{2}, the coordinate r_{2} is a rescaling of ε ≥ 0. Furtheremore, in a qualitative sense, in chart K_{1} one studies trajectories of Eq. 19 as they approach and leave a small neighborhood of the fold point in the positive y direction, while in chart K_{2} one investigates the trajectories of Eq. 19 within a sufficiently small neighborhood of the fold point.
The coordinates in the above charts are related by the transition maps:
for ε_{1} > 0 and
for y_{2} > 0.
Analysis in the Rescaling Chart K _{2}
The blownup (and desingularized) local vector field in this chart reads as
where g_{2} = g_{2}(r_{2},x_{2},y_{2},α_{2}) is smooth and defined by the blowup of \(\hat g\). More precisely, from Eq. 8 and keeping in mind the usual desingularization step, one has that
where \(\bar h_{6}\) is smooth. Then, it is clear that \(g_{2}\in \mathcal {O}({r}_{2})\). Similarly, μ_{2} = μ_{2}(x_{2},y_{2},r_{2},α_{2}) is the blownup statefeedback controller to be designed. Observe that, analogously to what is described in Remark 2, we have that for \({\bar {r}}={\bar {\alpha }}={\bar {\mu }}=0\) the orbits of Eq. 26 are given as level sets of the function
Having this in mind, we are going to design μ_{2} in such a way that for a trajectory (x_{2}(t_{2}),y_{2}(t_{2})) of Eq. 26 one has \(\lim _{t_{2}\to \infty } H_{2}({{x}_{2}}(t_{2}),{{y}_{2}}(t_{2}))= h\), where h defines the desired canard cycle and t_{2} denotes the timeparameter of Eq. 26.
We approach the design of μ_{2} as follows: we start by restricting to \(\left \{ \bar {r}=0 \right \}\) and define \(\widetilde {H}_{2}=H_{2}h\), where \(h\in (0,\frac {1}{4})\).^{Footnote 2} Next, we define a candidate Lyapunov function given by
and note that L_{2} > 0 for all \(\widetilde {H}_{2}\neq 0\) and that L_{2} = 0 if and only if \(\widetilde {H}_{2}=0\), if and only if (x_{2},y_{2}) ∈ γ_{h}, where by γ_{h} we denote the reference canard cycle, that is
It follows that
where \({\mu }_{2}^{0}={\mu }_{2}(0,{x}_{2},{y}_{2},\bar {\alpha })\). Naturally, we want to design \({\mu }_{2}^{0}\) such that \(L_{2}^{\prime }<0\), or at least \(L_{2}^{\prime }\leq 0\). We now see that a convenient choice of \({\mu }_{2}^{0}\) is
where c_{1} > 0 and \(c_{2}\in \mathbb {R}\) are the controller gains. Using Eq. 32 we have
Note that, because the exponential function is positive, the previous inequality holds for every value of \(c_{2}\in \mathbb {R}\); however, a particular choice of c_{2} may drastically change the performance of the controller, hence its inclusion in Eq. 32. This can be readily seen if we substitute \(\widetilde {H}_{2}\) in Eq. 32:
Let \(D\subset \mathbb {R}^{3}\) be a bounded domain. We see that \({\mu _{2}^{0}}\) is bounded for all \((\bar {\alpha },{x}_{2},{y}_{2})\in D\). However, since c_{2} appears inside the exponential, the upper bound of \({\mu _{2}^{0}}\) can vary widely depending on the choice of c_{2}. The relevance of c_{2} shall be detailed in Section 3.1.2.
By Lasalle’s invariance principle [44] we have that, under the controller Eq. 32 and r_{2} = 0, the trajectories of Eq. 26 eventually reach the largest invariant set contained in
Note, however, that \(\left \{ {x}_{2}=0 \right \}\) is generically not invariant for the closedloop dynamics Eq. 26. Indeed, the closedloop system Eq. 26 (restricted to r_{2} = 0) reads as
where setting x_{2} = 0 leads to \(({x}_{2}^{\prime },{y}_{2}^{\prime })=({y}_{2},0)\). Therefore, we now have that all trajectories of Eq. 26 eventually reach \(\mathcal {I}_{2}=\left \{ ({x}_{2},{y}_{2})=(0,0) \right \}\cup \left \{ \widetilde {H}_{2}=0 \right \}\). Since the origin is an equilibrium point of Eq. 36,^{Footnote 3} we have that every trajectory with initial conditions \(({x}_{2}(0),{y}_{2}(0))\in \mathbb {R}^{2}\backslash \left \{ (0,0) \right \}\) eventually reaches the set \(\left \{\widetilde {H}_{2}=0\right \}\) as \(t_{2}\to \infty \). With the previous analysis we have shown the following:
Proposition 1
Consider Eq. 26. Then, for r_{2} ≥ 0 sufficiently small a controller of the form
where c_{1} > 0 and \(c_{2}\in \mathbb {R}\) and with H_{2} is as in Eq. 28, renders the orbit γ_{h} locally asymptotically stable.
Proof
As described above, the stability of γ_{h} for Eq. 37 is equivalent to the stability of the zero solution of
Substituting Eqs. 32 in 38 we get
We have shown that for r_{2} = 0, the origin is locally asymptotically stable for Eq. 39. An apparent issue in Eq. 39 is the term \({x_{2}^{2}}\). However, we have also shown that \(\left \{ {x}_{2}=0 \right \}\) is not invariant. Therefore, Eq. 39 is a particular case of the nonautonomous scalar equation
where a(t_{2}) ≥ 0 for all t_{2} anda(t_{2}) > 0 for almost all t_{2} (here t_{2} is the time parameter in the chart K_{2}). The solution of the unperturbed Eq. 40 is \(H_{2}(t_{2})=k \exp \left ({\int \limits }_{t_{0}}^{t_{2}}a(s_{2})\text {d}s_{2} \right )\), for some \(k\in \mathbb {R}\). So, due to the properties of a(t_{2}), the trivial solution of Eq. 40, with r_{2} = 0, is asymptotically stable, which is preserved under sufficiently small perturbations \(\mathcal {O}({r}_{2})\) [11]. □
We show in Fig. 4 a simulation of the result postulated in Proposition 1.
Let us emphasize at this point that designing the controller in the rescaling chart justifies using H_{2} to define a convenient Lyapunov function, even if there are higher order terms in the original vector field Eq. 19. We also point out that the maximal canard becomes unbounded in this chart. Such a case shall be studied in chart K_{1} (see Section 3.1.2 below). Next we digress on how to deal with a certain class of higher order terms even if r_{2} (equivalently ε) is not small.
Lemma 1
Consider Eq. 26 with r_{2} > 0 fixed and let \({{\varGamma }}_{2}\subset \mathbb {R}^{2}\) be a neighbourhood of γ_{h}. Assume that the function g_{2} satisfies

1.
g_{2} = x_{2}ϕ_{2}(r_{2},x_{2},y_{2},α_{2}), where ϕ_{2} is smooth and vanishes at the origin.

2.
The function ϕ_{2} satisfies 1 + ϕ_{2}(r_{2},0,y_{2},α_{2})≠ 0 for all (0,y_{2}) ∈Γ_{2}.

3.
The function ϕ_{2} satisfies 1 + ϕ_{2}(r_{2},x_{2},y_{2},α_{2})≠ 0 for all (x_{2},y_{2}) ∈ γ_{h}.
Then, a controller of the form
where c_{1} > 0 and \(c_{2}\in \mathbb {R}\), renders γ_{h} locally asymptotically stable in Γ_{2}.
Proof
First, we recall that \(g_{2}\in \mathcal {O}({r}_{2})\), see Eq. 27. Therefore, under the assumptions of the Lemma we can write g_{2} = x_{2}ϕ_{2} = r_{2}x_{2}ψ_{2} for some function ψ_{2}. Next, and similar to the analysis performed above, we consider Eq. 26 but now with an extra \(\mathcal {O}({r}_{2})\)term in the controller, namely
where \({\mu }_{2}^{0}\) is as in Eq. 32 and now v_{2} = v_{2}(r_{2},x_{2},y_{2},α_{2}) is to be designed. Consider, as before, the candidate Lyapunov function Eq. 29. After substituting \({\mu }_{2}^{0}\) and g_{2} = r_{2}x_{2}ψ_{2}, we get
The above expression suggests to set \({v}_{2}=(y{x}_{2}^{2}){{{\psi _{2}}}}\). By doing so one gets Eq. 31 again and therefore, invoking again Lasalle’s invariance principle, we now take a look at the set \({\mathcal {I}}=\left \{ {{x}_{2}}=0 \right \}\cup \left \{ \widetilde {H}_{2}=0 \right \}\) related to the closedloop system. To be more precise we now focus on
where we have used r_{2}ψ_{2} = ϕ_{2}, and consider its dynamics restricted to \(\mathcal {I}\). On \(\left \{ {x}_{2}=0 \right \}\) one has \(({x}_{2}^{\prime },{y}_{2}^{\prime })=\left ({y}_{2}\left (1+{\phi _{2}}_{\left \{ {x}_{2}=0 \right \}}\right ),0\right )\). Therefore, to avoid \(\left \{ {x}_{2}=0 \right \}\) being invariant we impose the condition 1 + ϕ_{2}(r_{2},0,y_{2},α_{2})≠ 0. Note that the aforementioned condition would already suffice to show that trajectories converge towards \(\left \{ \widetilde {H}_{2}=0 \right \}\); however, there may still be a stable equilibrium point contained in \(\left \{ \widetilde {H}_{2}=0 \right \}\). The restriction of Eq. 44 to \(\left \{ \widetilde {H}_{2}=0 \right \}\) reads as
Now it suffices to give conditions on \({\phi _{2}}_{\left \{({x}_{2},{y}_{2})\in \gamma _{h}\right \}}\) such that Eq. 45 does not have equilibrium points (keep in mind that (0,0)∉γ_{h} for h ∈ (0,1/4)). Such a condition is simply 1 + ϕ_{2}(r_{2},x_{2},y_{2},α_{2})≠ 0 for all (x_{2},y_{2}) ∈ γ_{h}, completing the proof. □
Remark 5

If the third assumption of Lemma 1 does not hold, then trajectories converge to an equilibrium point contained in the set \(\left \{\widetilde {H}_{2}=0\right \}\).

A simpler to check and sufficient condition on ϕ_{2} satisfying the hypothesis of Lemma 1 is ϕ_{2}(r_{2},x_{2},y_{2},α_{2})≠ 1 for all (x_{2},y_{2}) ∈Γ_{2}. Also, if g = xϕ(x,y,ε,α) and g_{2} = x_{2}ϕ_{2} is its blownup version, then \({\phi _{2}}({{r}_{2}},{{x}_{2}},{{y}_{2}},{{\alpha }_{2}})=\phi ({{r}_{2}}{{x}_{2}},{{r}_{2}}^{2}{{y}_{2}},{{r}_{2}}^{2},{{r}_{2}}{{\alpha }_{2}})=\phi (x,y,{\varepsilon },\alpha )\). Therefore, ϕ_{2}≠ 1 implies ϕ≠ 1. These two arguments are the ones we use for Theorem 1.
We show in Fig. 5 a simulation regarding Lemma 1.
Analysis in the Directional Chart K1
We are now going to look at the controlled dynamics in the chart K_{1}. This serves two purposes: the first is of giving a more precise meaning to the constant c_{2} in the controller Eq. 37; the second is to corroborate that the controller designed previously is indeed able to also stabilize the (unbounded) maximal canard. Using the definition on K_{1} as in Eq. 23, we have that the dynamics in this chart read as
where, in particular, μ_{1} denotes the controller written in the local coordinates of this chart. Since we have already designed a controller in the chart K_{2}, see Eq. 37, we can use the transformation Eq. 25 to express μ_{1} as
where, analogous to what we have done in chart K_{2}, we define \(\widetilde {H}_{1}=H_{1}h\) with
Remark 6

μ_{1} is bounded along any reference canard \(\gamma _{h}=\left \{ \widetilde {H}_{1}=0\right \}\) with \(h\in (0,\frac {1}{4})\).

If h≠ 0, then μ_{1} becomes unbounded as ε_{1} → 0 unless \(\widetilde {H}_{1}=0\) (previous observation). This is to be expected as, in the limit ε_{1} → 0 the only canard orbit to stabilize is the maximal canard since \(\lim _{{{{\varepsilon }}_{1}}\to 0}H_{1}=0\). Therefore, we are going to study the closedloop dynamics Eq. 46 for the particular choice of h = 0 and for the limit ε_{1} → 0. Our goal is to refine the constant c_{2} so that μ_{1} remains bounded whenever h = 0 and ε_{1} → 0. Moreover, recalling that for this chart we have \({{{\varepsilon }}_{1}}=\frac {y}{{\varepsilon }}\), the limit ε_{1} → 0 corresponds to the limit \(y\to \infty \) for fixed ε > 0.
So from now on, we let h = 0, that is \(\widetilde {H}_{1}=H_{1}=\frac {1}{2}\exp \left (2{{\varepsilon }}_{1}^{1}\right )\left ({{\varepsilon }}_{1}^{1}{{\varepsilon }}_{1}^{1}{x}_{1}^{2}+\frac {1}{2}\right )\). We also restrict to \(\left \{{r}_{1}=0 \right \}\). In such a case we have
and the closed loop system reads as
It shall also be relevant to consider \(H_{1}^{\prime }\), namely
First of all, we note that \(\lim _{{{\varepsilon }}_{1}\to 0} H_{1}=0\), and \(\lim _{{{\varepsilon }}_{1}\to 0} H_{1}^{\prime }=0\) for c_{2} < 4. Next, we focus on Eq. 49 where we observe that in order for the controller to be bounded as ε_{1} → 0the constant c_{2} should be less than 2. To be more precise:
Lemma 2
Let (α_{1},x_{1}) be bounded and c_{1} > 0. Then, \(\lim _{{{\varepsilon }}_{1}\to 0}{\mu }_{1}<\infty \) if and only if c_{2} < 2.
Proof
Straightforward computations. □
From Lemma 2 we have that, to follow the maximal canard (h = 0) one must choose c_{2} < 2 to ensure that the controller is bounded. Although analytically any choice of c_{2} < 2 suffices, a particular choice may influence drastically numerical simulations since c_{2} appears in the exponential. For instance, we see from the first line of Eq. 51 that c_{2} < 2 but arbitrarily close to 2 reduces the contribution of the exponential term, which may induce issues in numerical simulations. For all other canards, \(c_{2}\in \mathbb {R}\) is sufficient. However, again from the computational perspective, c_{2} = 2 is the appropriate choice as it eliminates the exponential term in Eq. 49 and in Eq. 51, which is rather convenient for simulations. We remark that a completely analogous analysis, which we omit for brevity, follows for the chart \(K_{3}=\left \{ \bar x=1 \right \}\) where canards corresponding to h < 0 can be considered. The arguments and the conclusion are the same, namely, for h < 0 one should set c_{2} < 2 so that the controller remains bounded along the unbounded canards.
Proof of Theorem 1
To prove Theorem 1, we first blowdown the controller μ_{1}. To keep it simple, we shall blowdown Eq. 37, but of course the same holds for Eq. 41. So, recall from Eq. 37 that the blownup controller is
Next, from Eq. 23, we have
where \(\hat H=\hat H({\hat x},y,\varepsilon )=\frac {1}{2}\exp \left (\frac {2y}{\varepsilon }\right )\left (\frac {y}{\varepsilon }\frac {{\hat x}^{2}}{\varepsilon }+\frac {1}{2} \right )\) as stated in the first item of Theorem 1. Under Eq. 53, the closedloop system corresponding to Eq. 19 reads as
Next, it is important to observe that \(\lim _{\varepsilon \to 0}\hat H=0\). This means that for ε = 0 the only reference canard that is reachable is the maximal canard.^{Footnote 4} The maximal canard corresponds to h = 0. So, setting h = 0, and since one chooses c_{2} < 2 (recall Section 3.1.2), it follows that \(\lim _{{\varepsilon }\to 0}c_{1}{\hat x}{\varepsilon }^{1/2}\exp \left (c_{2}y{\varepsilon }^{1}\right )\hat H=0\), meaning that the layer equation for Eq. 54 is
which indeed has the same type of singularity at the origin as the openloop system, a fold. This shows that Eq. 53 is a compatible controller in the sense of Definition 4.
The Slow Control Problem
In this section, we consider the slowcontrol problem
where the objective is, as in Section 3.1, to stabilize a prescribed canard γ_{h}. Due to space constraints, and because the analysis is similar to the one performed in Section 3.1, we only state the relevant result.
Theorem 2
Consider Eq. 56 and let \(\hat H=H(x,y,\varepsilon )\) be defined by Eq. 11. Then, the compatible controller
where c_{1} > 0, \(c_{2}\in \mathbb {R}\) and \(h\leq \frac {1}{4}\) renders the canard orbit \(\gamma _{h}=\left \{ (x,y)\in \mathbb {R}^{2}  H=h \right \}\) locally asymptotically stable for ε > 0 sufficiently small. A convenient choice of controller gain c_{2} for the maximal canard is c_{2} < 2. By convenient we mean that such a choice ensures that the controller remains bounded as \(y\to \infty \).
In Fig. 6, we illustrate the statement of Theorem 2.
Controlling Canard Cycles for the Van Der Pol Oscillator
In this section, we are going to extend the ideas developed previously to control canard cycles in the van der Pol oscillator. The main idea is to adapt and extend the controller proposed in Theorem 1, and to use it to control canard cycles of the van der Pol oscillator. In this context, we distinguish two types of canard cycles: (a) canards with head and (b) canards without head. Canards with head refer to canard cycles with two fast segments, while canards without head have only one fast segment, see Fig. 8. Furthermore, due to its relationship with some neuron models, like the FitzhughNagumo model [21, 48], we shall consider that the controller acts on the fast variable only. The idea is that the controller represents input current. Thus, let us study
Remark 7
For simplicity, we have chosen to present the case α = 0. However, the case α≠ 0 follows straightforwardly from considering the arguments at the beginning of Section 3.1.
The corresponding critical manifold reads as,
The repelling and attracting parts of \(\mathcal {S}_{0}\) are denoted respectively by \(\mathcal {S}_{0}^{\text {r}}\) and \(\mathcal {S}_{0}^{\text {a}}\), and are given by
Furthermore, system Eq. 58 has two fold points, one at the origin and one at \((x,y)=(2,\frac {4}{3})\). In fact, the origin is a canard point and the singular limit of Eq. 58 is as shown in Fig. 7.
To state our main result, let \(\mathcal {N}_{1}\subset \mathbb {R}^{2}\) be a region containing a subset of the repelling critical manifold \(\mathcal {S}_{0}^{\text {r}}\) and \(\mathcal {N}_{2}\subset \mathbb {R}^{2}\) a small region containing a subset of \({\mathcal {S}}_{0}\) around the origin. Although it is not necessary to be precise on such regions, since several choices are possible, an example of \({\mathcal {N}}_{1}\) and \({\mathcal {N}}_{2}\) is as follows
where the defining positive constants are such that \(\mathcal {N}_{1}\) and \(\mathcal {N}_{2}\) have a nonempty intersection in the first quadrant, and \(0<y_{\text {min}}\in \mathcal {O}(\varepsilon )\) and \(y_{\min \limits }<y_{h}<\frac {4}{3}\). The precise meaning of these bounds is given in Sections 4.1 and 4.2, and is already sketched in Fig. 7.
Proposition 2
Consider Eq. 58, let ψ_{i} be a bump function with support \(\mathcal {N}_{i}\), and let the repelling slow manifold \(\mathcal {S}_{\varepsilon }^{\text {r}}\) be given by the graph of x = ϕ(y,ε). Then, one can choose \(\mathcal {N}_{i}\), positive constants c_{1} and k_{1}, and a small constant x^{∗}, x^{∗}≪ 1, such that the controller
where
and with
stabilizes a canard cycle with height y_{h}. Moreover, if x^{∗} < 0 then the canard is without head, while if x^{∗} > 0 then the canard is with head.
Proof Sketch of proof:
As before, all the analysis is carriedout in the blowup space. The overall idea is as follows: the controller to be designed acts only within a small neighbourhood of \(\left \{ 0 \right \}\cup {\mathcal {S}}_{{\varepsilon }}^{{\text {r}}}\), mainly because the rest of the slow manifold is already stable, so there is no need of stabilizing it. The desired height of the canard is regulated by the constant y_{h}. The controller u_{2} controls the trajectories near the canard point and therefore is given by Theorem 1, where we have made the choice h = 0 and c_{2} = 2. So, the new analysis is performed in the chart \(K_{1}=\left \{ \bar y=1 \right \}\), where the objective is to stabilize the (normally hyperbolic) repelling branch of the slow manifold \({\mathcal {S}}_{{\varepsilon }}^{{\text {r}}}\) resulting in the controller u_{1}. Later, in Section 4.2, we combine the two controllers and justify the form of the controller given in the Proposition. The most important feature of u_{1} is to control the location of the orbits relative to \({\mathcal {S}}_{{\varepsilon }}^{{\text {r}}}\) as it is precisely such location that determines the direction of the jump once the orbits reach the desired height. To avoid smoothness issues, the regions where the controllers are active are defined via bump functions. A schematic representation of this idea is provided in Fig. 7, while the details of the proof follows from Sections 4.1 and 4.2. □
In Fig. 8, we show some simulations using the proposed controller.
Before proceeding with the proof of Proposition 2, let us point out that it is straightforward to use the proposed controller to produce robust mixedmode oscillations (MMOs) [12]. One way to do this is as follows: first of all, we assume that we are able to count the number of small amplitude oscillations (SAOs) and of large amplitude oscillations (LAOs). Next, let us say that we start by following a canard without head, so we set the controller constant x^{∗} < 0 and y_{h} to the desired height. After the number of desired SAOs has been reached, we change the controller constant x^{∗} to x^{∗} > 0 and, if desired, y_{h} to a new height value. So, the controller will now steer the system to follow a canard with head. This process can be repeated to produce any other pattern allowed by the geometry of the van der Pol oscillator. We show in Fig. 9 an example of stable MMOs that are obtained using the controller of Proposition 2.
Analysis in the Directional Chart K1
Similar to the analysis in Section 3.1.2, we use a directional blowup defined by
Therefore, the local vector field associated to Eq. 58 reads as
To have a better idea of what we are going to achieve with the controller, it is worth to first look at the uncontrolled dynamics.
Let us define a domain
Lemma 3
Consider Eq. 66 with μ_{1} = 0. Then, one can choose constants ρ_{1} > 0 and δ_{1} > 0 such that the following properties hold within the domain D_{1}.

1.
There exist semihyperbolic equilibrium points p_{1,±} = (r_{1},ε_{1},x_{1}) = (0,0,± 1). The point p_{1,−} is attracting while p_{1,+} is repelling along the x_{1}axis.

2.
Let \({\mathscr{M}}_{1}\) be defined by
$$ \mathcal{M}_{1}=\left\{ ({r}_{1},{{\varepsilon}}_{1},{x}_{1})\in\mathbb{R}^{3}  {{\varepsilon}}_{1}=0, {r}_{1}=3\left( \frac{1}{{x}_{1}}\frac{1}{{x}_{1}^{3}}\right) \right\}. $$(68)The set \({\mathscr{M}}_{1}\) corresponds to the set of equilibrium points of Eq. 66 restricted to \(\left \{ {{\varepsilon }}_{1}=0 \right \}\). Moreover, let us denote the subsets
$$ \begin{array}{ll} \mathcal{M}_{1,} = \left\{ ({r}_{1},{{\varepsilon}}_{1},{x}_{1})\in\mathcal{M}_{1}  {x}_{1}<0 \right\},\\ \mathcal{M}_{1,+} = \left\{ ({r}_{1},{{\varepsilon}}_{1},{x}_{1})\in\mathcal{M}_{1}  {x}_{1}\geq1\right\}. \end{array} $$(69)The subset \({\mathscr{M}}_{1,}\) is attracting and the subset \({\mathscr{M}}_{1,+}\) can be partitioned as \({\mathscr{M}}_{1,+}={\mathscr{M}}_{1,+}^{\text {r}}\cup \left \{ \left (\frac {3\sqrt {3}}{\sqrt {3}},0,\sqrt {3}\right )\right \}\cup {\mathscr{M}}_{1,+}^{\text {a}}\) where
$$ \begin{array}{ll} \mathcal{M}_{1,+}^{\text{r}} = \left\{ ({r}_{1},{{\varepsilon}}_{1},{x}_{1})\in\mathcal{M}_{1,+}  1\leq{x}_{1}<\sqrt{3} \right\},\\ \mathcal{M}_{1,+}^{\text{a}} = \left\{ ({r}_{1},{{\varepsilon}}_{1},{x}_{1})\in\mathcal{M}_{1,+}  {x}_{1}>\sqrt{3} \right\} \end{array} $$(70)are the repelling and attracting branches of \({\mathscr{M}}_{1,+}\), respectively.

3.
Restricted to \(\left \{ {r}_{1}=0 \right \}\) there exist 1dimensional local center manifolds \(\mathcal {E}_{1,}\) and \(\mathcal {E}_{1,+}\) located, respectively, at the points p_{1,−} and p_{1,+}. Such manifolds are given by
$$ \mathcal{E}_{1,\pm} = \left\{ ({r}_{1},{{\varepsilon}}_{1},{x}_{1})\in\mathbb{R}^{3}  {r}_{1}=0, {x}_{1}=h_{1,\pm}({{\varepsilon}}_{1}) \right\}, $$(71)where
$$ h_{1,\pm}({{\varepsilon}}_{1})=\pm\left( 1+\frac{{{\varepsilon}}_{1}}{2}\right)^{1/2}. $$(72)The flow along \(\mathcal {E}_{1,}\) is directed away from the point p_{1,−} and the flow along \(\mathcal {E}_{1,+}\) is directed towards the point p_{1,+}. Furthermore, the center manifolds \(\mathcal {E}_{1,\pm }\) are unique.

4.
There exist 2dimensional local centre manifolds \(\mathcal {W}_{1,\pm }\) that contain, respectively, the point p_{1,±}, the branch of equilibrium points \({\mathscr{M}}_{1,\pm }\), and the centre manifold \({\mathcal {E}}_{1,\pm }\). These centre manifolds are unique and, moreover, \({\mathcal {W}}_{1,}\) is attracting and \({\mathcal {W}}_{1,+}\) is repelling.
Proof Sketch of the proof, see [ 40 ] for details
The first two items are obtained by straightforward computations. The expression of the centre manifolds follow from the fact that the restriction of Eq. 4.1 to \(\left \{ {r}_{1}=0 \right \}\) has the invariant (just as in the fold case) \(H_{1} = \frac {1}{2}\exp \left (2{{\varepsilon }}_{1}^{1}\right )\left ({{\varepsilon }}_{1}^{1}{{\varepsilon }}_{1}^{1}{x}_{1}^{2}+\frac {1}{2}\right )\). Therefore, the functions h_{1,±} are given by the solutions of H_{1} = 0. The flow on \(\mathcal {E}_{1,\pm }\) follows from the equation \({{\varepsilon }}_{1}^{\prime }={{\varepsilon }}_{1}^{2}{x}_{1}\). The uniqueness of \(\mathcal {E}_{1,\pm }\) is due to the fact that p_{1,±} is a semihyperbolic saddle of the dynamics of Eq. 66 restricted to \(\left \{{r}_{1}=0\right \}\). Finally, the existence and properties of \(\mathcal {W}_{1,\pm }\) follow from local analysis at p_{1,±}, centre manifold theory, the previous arguments, and by choosing \(\rho _{1}<\frac {3\sqrt {3}}{\sqrt {3}}\). The previous choice of ρ_{1} is particularly necessary for the stability property of \(\mathcal {W}_{1,+}\). □
Remark 8
\(\mathcal {W}_{1,+}\) is related, via the blowup map, to \(\mathcal {S}_{\varepsilon }^{\text {r}}\). Therefore, the task of the controller is going to be to stabilize the centre manifold \(\mathcal {W}_{1,+}\).
Remark 9
In what follows, we are going to define some geometric objects, in particular centre manifolds, for the closedloop dynamics. To make a clear distinction between their openloop counterparts, and to be able to compare them, we shall denote relevant geometric objects of the closedloop system by a cl superscript.
In this section, we are going to be interested only in \(({x}_{1},{{\varepsilon }}_{1})\in \mathbb {R}^{2}_{+}\). So, to simplify notation let
Furthermore, let us assume that the centre manifold \(\mathcal {W}_{1,+}\) (recall Lemma 3) is given by the graph of
Note that \(\phi _{1}(0,{{\varepsilon }}_{1})=h_1^{\text {cl}}({{\varepsilon }}_{1})\). Therefore, one can in fact write
for some coefficients \(\sigma _{ij}\in \mathbb {R}\). We now proceed with the following steps.

1.
Reverse the direction of the flow in the x_{1}direction: Define \(f_{1}({r}_{1},{{\varepsilon }}_{1},{x}_{1})=1+{x}_{1}^{2}\frac {1}{2}{x}_{1}^{2}{{\varepsilon }}_{1}\frac {1}{3}{r}_{1}{x}_{1}^{3}\) and let \({{\mu }_{1}}=f_{1}({{r}_{1}},{{{\varepsilon }}_{1}},{{x}_{1}})f_{1}({{r}_{1}},{{{\varepsilon }}_{1}},{{x}_{1}}{{x}_{1}}^{*})+{{v}_{2}}\), where \({{x}_{1}}^{*}\sim 0\) is a constant (the usefulness of \({{x}_{1}}^{*}\) will become evident below) and v_{2} = v_{2}(r_{1},ε_{1},x_{1}) is to be further designed. With this step we have that Eq. 66 now reads as
$$ \begin{array}{lll} {r}_{1}^{\prime} = \frac{1}{2}{r}_{1}{{\varepsilon}}_{1}{x}_{1} \\ {{\varepsilon}}_{1}^{\prime} = {{\varepsilon}}_{1}^{2}{x}_{1} \\ {x}_{1}^{\prime} = f_{1}({r}_{1},{{\varepsilon}}_{1},{x}_{1}{x}_{1}^{*})+{v}_{2}. \end{array} $$(76) 
2.
Design v_{2} so that Eq. 76 has \(\mathcal {W}_{1}^{\text {cl}}:= \left \{ ({r}_{1},{{\varepsilon }}_{1},{r}_{1})\in \mathbb {R}^{3}  {x}_{1}={x}_{1}^{*}+\phi _{1}({r}_{1},{{\varepsilon }}_{1})\right \}\) as a closedloop centre manifold: this step requires standard centre manifold computations. By performing them we find that
$$ {v}_{2} = \frac{2\phi_{1}+{x}_{1}^{*}}{\phi_{1}}\left( 1+{\phi_{1}^{2}}\frac{1}{2}{\phi_{1}^{2}}{{\varepsilon}}_{1}\frac{1}{3}{{r}_{1}\phi_{1}^{3}} \right). $$(77)Note that, if we restrict to \(\left \{ {r}_{1}=0 \right \}\), Eq. 76 now reads as
$$ \begin{array}{ll} {{\varepsilon}}_{1}^{\prime} = {{\varepsilon}}_{1}^{2}{x}_{1}\\ {x}_{1}^{\prime} = 1({x}_{1}{x}_{1}^{*})^{2} + \frac{1}{2}({x}_{1}{x}_{1}^{*})^{2}{{\varepsilon}}_{1} + {{\varepsilon}}_{1}^{2}\frac{2h_{1}^{\text{cl}}+{x}_{1}^{*}}{4h_{1}^{\text{cl}}}. \end{array} $$(78)We know that Eq. 78 has a centre manifold \(\mathcal {E}_{1}^{\text {cl}}:= \left \{ ({{\varepsilon }}_{1},{x}_{1})\in \mathbb {R}^{2}_{\geq 0}  {x}_{1}=\right .\) \(\left .{x}_{1}^{*}+h_{1}^{\text {cl}}({{\varepsilon }}_{1})\right \}\). Furthermore, it follows from straightforward computations that the equilibrium point \(p_{1}^{*}:= (0,0,1+{x}_{1}^{*})\) is attracting along the x_{1}axis. This means that \(\mathcal {E}_{1}^{\text {cl}}\), and also \(\mathcal {W}_{1}^{\text {cl}}\), are locally (near \(p_{1}^{*}\)) attracting. Next we improve such stability.

3.
Design a variational controller that renders \(\mathcal {W}_{1}^{\text {cl}}\) locally exponentially stable: For this, it is enough to take the x_{1}component of the variational equation. So, let \(z_{1}={{x}_{1}}\phi _{1}{{x}_{1}}^{*}\). The corresponding variational equation along \({\mathcal {W}}_{1}^{{\text {cl}}}\) is
$$ z_{1}^{\prime} = (2+{{\varepsilon}}_{1}+{r}_{1}\phi_{1})\phi_{1} z_{1}. $$(79)Recall from Eq. 75 that ϕ_{1} > 0 for r_{1} ≥ 0 sufficiently small. Then, we propose to introduce in Eq. 79 a variational controller w_{1}(ε_{1},z_{1}) of the form
$$ w_{1}=({{\varepsilon}}_{1}\phi_{1}+{{r}_{1}\phi_{1}^{2}}+k_{1})z_{1}, $$(80)where k_{1} ≥ 0. With w_{1} as above, the closedloop variational equation becomes
$$ z_{1}^{\prime} = \left( 2\phi_{1} + k_{1} \right)z_{1}, $$(81)and we readily see that, for r_{1} ≥ 0 sufficiently small, z_{1} → 0 exponentially as \(t_{1}\to \infty \) (where by t_{1} we denote the timeparameter in this chart). We also notice that the constant k_{1} helps to improve the contraction rate towards \(\mathcal {W}_{1}^{\text {cl}}\). Moreover, since w_{1} vanishes along \(\mathcal {W}_{1}^{\text {cl}}\), the variational controller does not change the closedloop centre manifold \(\mathcal {W}_{1}^{\text {cl}}\). Finally, observe that the role of the small constant \({x}_{1}^{*}\) is to shift the position of \(\mathcal {W}_{1}^{\text {cl}}\) relative to its openloop counterpart \(\mathcal {W}_{1,+}\). This is important in order to tune the direction along which the trajectory jumps once the controller is inactive.

4.
Restrict next to \(\left \{ {{\varepsilon }}_{1}=0\right \}\): Note that v_{2}(r_{1},0,x_{1}) = 0, then we have
$$ \begin{array}{lll} {r}_{1}^{\prime} = 0 \\ {{\varepsilon}}_{1}^{\prime} = 0 \\ {x}_{1}^{\prime} = f_{1}({r}_{1},0,{x}_{1}{x}_{1}^{*}). \end{array} $$(82)Similar to the previous step, the new line of equilibrium points is slightly shifted according to \({x}_{1}^{*}\). In fact, the relevant set of stable equilibrium points of Eq. 82 is given as
$$ \mathcal{M}_{1}^{\text{cl}}=3\left\{ ({r}_{1},{{\varepsilon}}_{1},{x}_{1})\in\mathbb{R}^{3}  {{\varepsilon}}_{1}=0, {r}_{1}=3\left( \frac{1}{{x}_{1}{x}_{1}^{*}}\frac{1}{({x}_{1}{x}_{1}^{*})^{3}}\right), {r}_{1}<\frac{2}{\sqrt{3}} \right\}. $$(83)Linearization of Eq. 82 along \({\mathscr{M}}_{1}^{\text {cl}}\) shows that \({\mathscr{M}}_{1}^{\text {cl}}\) is exponentially attracting in the x_{1}direction. Therefore, we can conclude that \(\mathcal {W}_{1}^{\text {cl}}\) is located at \({x}_{1}=1+{x}_{1}^{*}\), and that it contains the exponentially attracting centre manifold \(\mathcal {E}_{1}^{\text {cl}}\) and the curve of exponentially attracting equilibrium points \({\mathscr{M}}_{1}^{\text {cl}}\).

5.
Note that the flow along the centre manifold \(\mathcal {W}_{1}^{\text {cl}}\) has not changed and is given, up to smooth equivalence and away from its corner at \({x}_{1}=1+{x}_{1}^{*}\), by
$$ \begin{array}{ll} {r}_{1}^{\prime} = \frac{1}{2}{r}_{1}\\ {{\varepsilon}}_{1}^{\prime} = {{\varepsilon}}_{1}. \end{array} $$(84)In other words, the flow along \(\mathcal {W}_{1}^{\text {cl}}\) is of saddle type.
With the previous steps, we can now prove the main result of this section. For this, let us define the domain \(D_{1}^{+}=D_{1}_{{x}_{1}\geq 0}\) and the sections
Also, define a small rectangle
Proposition 3
Consider Eq. 66 with the controller
where
and the function ϕ_{1}(r_{1},ε_{1}) is defined by the openloop centre manifold \(\mathcal {W}_{1,+}\). Then, one can choose sufficiently small constants \((\delta _{1},\rho _{1},\sigma _{1},\tilde \rho _{1})\) such that the following hold for the closedloop system.

1.
\(D_{1}^{+}\) is forward invariant under the flow of Eq. 66.

2.
The centre manifold \(\mathcal {W}_{1}^{\text {cl}}\) is locally exponentially attracting for r_{1} ≥ 0 sufficiently small, ε_{1} ≥ 0 sufficiently small and for \({r}_{1}^{2}{{\varepsilon }}_{1}\geq 0\) sufficiently small.

3.
If \({x}_{1}^{*}=0\), the centre manifolds \(\mathcal {W}_{1}^{\text {cl}}\) and \(\mathcal {W}_{1,+}\) coincide. On the other hand, if \({x}_{1}^{*}<0\) (resp. if \({x}_{1}^{*}>0\)) then \(\mathcal {W}_{1}^{\text {cl}}\) is located “to the left” (resp. “to the right”) of \({\mathcal {W}}_{1,+}\) in the x_{1}direction.

4.
The image of R_{1} under the flow of Eq. 66 is a wedgelike region at \({\Sigma }_{1}^{\text {ex}}\cap {\mathscr{M}}_{1}^{\text {cl}}\).
Proof
The proof follows directly from our previous analysis. In particular, the second item is implied by the stability properties of \(\mathcal {W}_{1}^{\text {cl}}_{\left \{ {r}_{1}=0 \right \}}\), \(\mathcal {W}_{1}^{\text {cl}}_{\left \{ {{\varepsilon }}_{1}=0 \right \}}\), and the fact that \({r}_{1}^{2}{{\varepsilon }}_{1}=\varepsilon \). □
The closedloop dynamics corresponding to Eq. 66 under the controller Eq. 87 are as sketched in Fig. 10.
To finalize this section, we blowdown the controller of Proposition 3, as it will be used in the forthcoming section.
Lemma 4
Let u_{1} denote the blowdown of μ_{1}. Then,
where
and where ϕ = ϕ(y,ε) is defined by \(\mathcal {S}_{\varepsilon }^{\text {r}}\), that is by \(\mathcal {S}_{\varepsilon }^{\text {r}} = \left \{ x=\phi (y,\varepsilon ) \right \}\).
Proof
The expression of u_{1} follows from straightforward computations using Eqs. 65 in 87. To check ϕ is as stated, note that the blowdown induces the relation \(\left \{ x_{1}=\phi _{1} \right \} \leftrightarrow \left \{ x=\sqrt {y}{{\varPhi }}(\phi _{1})=\phi \right \}\), where by Φ(ϕ_{1}) we are denoting the blowdown of ϕ_{1}. □
Composite Controller and Proof of Proposition 2
In this section, we gather the controllers designed in the central chart K_{2} and in the directional chart K_{1} into a single one. Our arguments follow from the next general design methodology.

1.
Let us start with an openloop vector field \(X:\mathbb {R}^{N}\to \mathbb {R}^{N}\) such that X(0) = 0 (here possible parameters \(\lambda \in \mathbb {R}^{p}\) are already included in the vector field by the trivial equation \(\dot \lambda =0\)).

2.
Let \({\mathscr{B}}=\mathbb {S}^{N1}\times \mathcal {I}\) where \(\mathbb {S}\) is the unit sphere and \(\mathcal {I}\subseteq \mathbb {R}\) is an interval that contains the origin. Here, we shall be interested in \(\mathcal {I}=[0,r_{0}]\), r_{0} > 0. Recall that the blowup map is defined as \({{\varPhi }}:{{\mathscr{B}}}\to {\mathbb {R}}^{N}\). Moreover, the blowup transformation induces the socalled “blownup” vector field \(\bar X\), which is the vector field that makes the following diagram commute.
In other words, \(\bar X\) and X are related by the pushforward of \(\bar X\) by Φ, that is \({{\varPhi }}_{*}(\bar X)=X\), in the sense \(\text {D}{{\varPhi }}\circ \bar X=X\circ {{\varPhi }}\).^{Footnote 5}

3.
Let \(\mathcal {A}=\left \{ (K_{i},{{\varPhi }}_{i})\right \}\), with i = 1,…,M, be a smooth atlas of \({\mathscr{B}}\). This means that (K_{i},Φ_{i}) is a chart of \({\mathscr{B}}\), the open sets K_{i} cover \({{\mathscr{B}}}\), and \({{\varPhi }}_{i}:K_{i}\subset {{\mathscr{B}}}\to {\mathbb {R}}^{N}\) is a diffeomorphism. Then, there are local vector fields \(\bar X_{i}\) defined on K_{i} and given by \(\bar X_{i}={\text {D}} {{\varPhi }}_{i}^{1}\circ X\circ {{\varPhi }}_{i}\).

4.
On each chart K_{i}, let us introduce a local controller \(\bar u_{i}\), and define as \(\bar X_{i}^{\text {cl}}:=\bar X_{i} + \bar u_{i}\) the local closedloop vector field. Naturally, \(\bar u_{i}\) is a local vector field on K_{i}.

5.
Let \(\bar \psi _{i}:K_{i}\to \mathbb {R}\) be a bump function with compact support \(\bar {\mathcal {N}}_{i}\subset K_{i}\). We choose each \(\bar {\mathcal {N}}_{i}\) such that if K_{i} ∩ K_{j}≠∅ then \(\bar {\mathcal {N}}_{i}\cap \bar {\mathcal {N}}_{j}\neq \emptyset \) as well. Note that
$$ \bar\varphi_{i}:=\frac{\bar\psi_{i}}{{\sum}_{i=1}^{M}\bar\psi_{i}} $$(91)is a partition of unity subordinate to the open cover \(\left \{ K_{i}\right \}_{i=1}^{M}\).

6.
The sum
$$ \bar u:={\sum}_{1=1}^{M} \bar\varphi_{i} \bar u_{i} $$(92)is, by virtue of the partition of unity, well defined as a global controller on \({\mathscr{B}}\). Therefore, the global closedloop vector field \(\bar X^{\text {cl}}:=\bar X+\bar u\) is also well defined.

7.
Let us now blowdown \(\bar X^{\text {cl}}\). To be more precise, we now define the closedloop vector field X^{cl} on \(\mathbb {R}^{N}\) by \({{\varPhi }}_{*}(\bar X^{\text {cl}})=X^{\text {cl}}\). So, we have
$$ X^{\text{cl}} = {{\varPhi}}_{*}(\bar X^{\text{cl}}) = {{\varPhi}}_{*}(\bar X+\bar u ) = {{\varPhi}}_{*}(\bar X) + {{\varPhi}}_{*}(\bar u)=X+ {{\varPhi}}_{*}(\bar u), $$(93)where we have used the fact that the pushforward is linear [45]. Next we define \(u:={{\varPhi }}_{*}(\bar u)\) and compute
$$ \begin{array}{ll} u={{\varPhi}}_{*}(\bar u) ={{\varPhi}}_{*}\left( {\sum}_{i=1}^{M}\bar\varphi_{i}\bar u_{i} \right)={\sum}_{i=1}^{M}{{\varPhi}}_{*}(\bar\varphi_{i}\bar u_{i})={\sum}_{i=1}^{M}({{\varPhi}}_{i})_{*}(\bar\varphi_{i}\bar u_{i})\\ ~~~~~~~~~~~~~~~~~={\sum}_{i=1}^{M} \varphi_{i}\cdot({{\varPhi}}_{i})_{*}(\bar u_{i}), \end{array} $$(94)where \(\varphi _{i}:=\bar \varphi _{i}\circ {{\varPhi }}_{i}^{1}\) for i = 1,…,M, and it is clear from its definition that \(\left \{ \varphi _{i} \right \}\) is a partition of unity a neighborhood of the origin \(0\in \mathbb {R}^{N}\) subordinate to the open cover \(\left \{ {{\varPhi }}_{i}(K_{i}) \right \}\).
With the previous methodology, we define the controller that stabilizes canard cycles of the van der Pol oscillator as
where u_{1} is as given by Lemma 4 and u_{2} as in Theorem 1, and where ψ_{1} is a bump function with support \(\mathcal {N}_{1}\) containing the repelling branch \({\mathcal {S}}_{0}^{{\text {r}}}\) and \({\mathcal {N}}_{2}\) the parabola \(\left \{ y=x^{2} \right \}\) around the origin. Although several choices for these neighbourhoods are possible, we recall an example given at the beginning of Section 4:
with suitably chosen positive constants β_{1}, β_{2}, x_{min}, x_{max}, y_{min}, \(y_{h}<\frac {4}{3}\). We note that one must choose \(0<y_{\min \limits }\in {\mathcal {O}}({\varepsilon })\) in order to ensure that the slow manifold \({\mathcal {S}}_{{\varepsilon }}^{{\text {r}}}\) is within distance \({\mathcal {O}}({\varepsilon })\) of the critical manifold \({\mathcal {S}}_{0}^{{\text {r}}}\). Here, y_{h} controls the height of the desired canard cycle, therefore \(y_{h}<\frac {4}{3}\). The neighborhood \({\mathcal {N}}_{1}\) and \({\mathcal {N}}_{2}\) are sketched in Fig. 7.
With the controller as in Eq. 95, and given the analysis in Section 4.1, one has that orbits of Eq. 58 passing close to the origin follow closely the repelling branch of the slow manifold \({\mathcal {S}}_{{\varepsilon }}^{{\text {r}}}\) up to a height determined by y_{h}. Once orbits leave the neighborhood \({\mathcal {N}}_{1}\cup {\mathcal {N}}_{2}\), they are governed by the openloop dynamics. Finally, the controller of Proposition 2 is indeed Eq. 95. We have just dropped the subscript of the constant \(x_{1}^{*}\).
Conclusions and Outlook
In this paper, we have presented a methodology combining the blowup method with Lyapunovbased control techniques to design a controller that stabilizes canard cycles. The main idea is to use a first integral in the blowup space to regulate the canard cycle that the orbits are to follow. Later on, we have extended the previously developed method to control canard cycles in the van der Pol oscillator. Roughly speaking, this procedure follows two steps: first one needs a controller that stabilizes a folded maximal canard within a small neighborhood of the canard point. Next, one needs to stabilize the unstable branch of the openloop slow manifold and to tune the position of the closedloop orbits with respect to it. This is essential to determine whether the closedloop canard has a head or not. Finally, one combines such controllers by means of a partition of unity. We have further shown that the proposed controller can be used to produce stable MMOs.
Several new questions and possible extensions arise from our work, and we would like to finish this paper by briefly mentioning a couple of ideas. First of all, it becomes interesting to adapt the controllers designed here to neuron models such as the FitzHughNagumo, MorrisLecar, or HodgkinHuxley models. Another relevant extension is to develop optimal controllers to control canards. Although from a theoretical point of view one would be interested in arbitrary cost functionals, some particular choices might be more suitable for applications. For instance, one may one want to design minimal energy controllers. It is also not completely clear whether the strategy of combining the blowup method and control techniques still applies as the optimal controllers may be timedependent. Finally, the notion of controlling MMOs definitely requires further investigation, as here we have just given a simple sample of the possibilities. Thus, for example, extending the ideas of this paper to higherdimensional fastslow systems with nonhyperbolic points is a direction to be pursued in the future.
Notes
Refer to [39] for the much more complicated case that includes the higher order terms.
In principle, our analysis holds for \(h\leq \frac {1}{4}\), but only the considered interval provides canard cycles, which are our main focus. See also Section 3.1.2.
In fact it is straightforward to further show that the origin is an unstable equilibrium point of Eq. 36.
As it is expected, the controller becomes unbounded in the limit ε → 0 for any other canard, as they do not exist in such a limit.
Recall that \(\bar X\) is well defined: for r > 0 because of \({{\varPhi }}_{\left \{r>0\right \}}\) being a diffeomorphism, and on r = 0 due to continuous extension to the origin, see [41]. Moreover, if the origin is nilpotent, one defines the desingularized vector field \(\tilde X=\frac {1}{r^{k}}\bar X\) for some k > 0, which is smoothly equivalent to \(\bar X\) for r > 0, and all the forthcoming arguments hold equivalently for \(\tilde X\).
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Funding
Open access funding provided by the University of Groningen. HJK gratefully acknowledges support by a fellowship of the AlexandervonHumboldt Foundation. CK acknowledges partial support by the DFG via the SFB/TR109 Discretization in Geometry and Dynamics and support by a Lichtenberg Professorship of the VolkswagenFoundation.
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JardónKojakhmetov, H., Kuehn, C. Controlling Canard Cycles. J Dyn Control Syst 28, 517–544 (2022). https://doi.org/10.1007/s10883021095532
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DOI: https://doi.org/10.1007/s10883021095532
Keywords
 Canard cycles
 Singular perturbations
 Feedback control
Mathematics Subject Classification (2010)
 34E17
 93C70
 93D15