Singularly Perturbed Oscillators with Exponential Nonlinearities

Singular exponential nonlinearities of the form $e^{h(x)\epsilon^{-1}}$ with $\epsilon>0$ small occur in many different applications. These terms have essential singularities for $\epsilon=0$ leading to very different behaviour depending on the sign of $h$. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $\epsilon\rightarrow 0$ limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen [20], and prove - for both systems - existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.


Introduction
Exponential nonlinearities arise in many different areas of mathematical modelling. In electronic oscillators, for example, the Ebers-Moll model for an NPN transistor provides an exponential relationship between the 'emitter current' and the 'base-emitter voltage'. See [7,11]. Also, in chemical kinetics, the reaction rates are, by the Arrhenius equation, exponential functions of the temperature. Frequently, the temperature is assumed constant in such models, but in systems where large temperature variations occur (e.g. in explosions), the resulting exponential nonlinearity becomes important for the dynamics. Similar nonlinearities appear in other settings when the effect of temperature becomes important, see e.g. [2,8] for exponential nonlinearities in plastic deformation. In the related area of friction, exponential nonlinearities also play an important role, for example in rate-and-state friction laws, see [4,5,28,30] and [1,26,29] for dynamical studies of such models. Although these friction models were first derived from experiments, the exponential nonlinearities have later been connected to Arrhenius process resulting from breaking atomic bonds at the atomic level [27]. Sometimes modellers also introduce exponentials more heuristically, for example when regularizing a switch by a tanh-function.
All of the examples of exponential nonlinearities highlighted above, are also examples of (within relevant parameter regimes) singularly perturbed systems. Over the past decades, these type of systems have been successfully described by geometric singular perturbation theory (GSPT) and blow-up, see e.g. [6,9,14,21], but singular exponential nonlinearities like e h(x) −1 , with essential singularities along h(x) = 0 as → 0, have traditionally been seen as an obstacle to such analysis. The problem is two-fold. Firstly, such systems approach piecewise smooth systems (upon proper normalizations) as → 0, having very different behaviour for h(x) > 0 and h(x) < 0. This area is currently an active area of research, see e.g. [16,18,19].
Secondly, for systems with nonlinearities of the form e h(x) −1 the convergence of the smooth system to its nonsmooth counterpart happens at an exponential rate. Whereas this exponential convergence can be harmless in some cases, it can also lead to further degeneracies due to 'exponential loss of hyperbolicity'. We will demonstrate this through the study of two prototypical systems, which we introduce in the following. The usual blowup method [6,21] is adapted to algebraic loss of hyperbolicity, and cannot compensate for these exponential degeneracies. Nevertheless, recently in [20] it was shown how one can modify this approach (basically by extending the space dimension) to deal with these special degeneracies. We will use this modified blowup approach in the present paper.
1.1. Two prototypical oscillators: Hester and Le Corbeiller. The aim of our paper, is to shed further light on singularly perturbed systems with exponential nonlinearities. We will do so by considering two 'prototypical' singularly perturbed oscillators with exponential nonlinearities: The Hester system: ẋ = y, y = −x − 2γy + µ e y −1 − κe (1+α)y −1 , (1.1) with γ ∈ (0, 1), α > 0, µ > 0, κ > 0, and The Le Corbeiller system: ẋ = y + a, y = −x + by(2 − e y −1 ), (1.2) with b ∈ (0, 1), a > 0. In both systems, 0 < 1 is the singular perturbation parameter. The system in (1.1) is a model of a transistor oscillator, see [11], based on the Ebers-Moll large-signal approximation. In reference to [11], we will refer to (1.1) as the 'Hester' system. The constant −1 in the exponentials is given by e/kT where k is the Boltzmann constant, e is the magnitude of the electronic charge and T is the temperature in Kelvin. At room temperature this gives ≈ 10 −2 and the approximation 0 < 1 is therefore reasonable, we believe.  Fig. 1 and Fig. 2 are known as two-stroke relaxation oscillations, by reference to the two distinct components to the oscillation; similar oscillations have been considered in the context of GSPT in [13].  Regarding the motivation for the system in (1.2), we first point out that for a = 0 the can be scaled out by setting x = x 2 , such thaẗ x 2 + x 2 + µẋ 2 (eẋ 2 − 2) = 0, (1.4) when writing the system as a second order equation. This equation appears in [25,Eq. (25)] as an example of a simple system of 'electric-oscillator-type' exhibiting two-stroke oscillation.
Within the framework of electronic oscillators, we may therefore consider (1.2) with a > 0 as a forced version of (1.4) (by analogy with the 'forced van der Pol oscillator'). In reference to [25], we will refer to (1.2) as the 'Le Corbeiller' system. As in Fig. 1 and Fig. 2, the phase portrait and associated oscillations for (1.2) are shown in Fig. 3 and Fig. 4, for parameter values a = 1, b = 0.25, (1.5) = 0.1 in Fig. 3, and = 0.01 in Fig. 4. As with the Hester system (1.1), sharp transitions between distinct components of the oscillations indicates singular dynamics even for the 'large' value in Fig. 3. In contrast to the Hester system, however,ẋ = a > 0 along the (noninvariant) set defined by y = 0 for (1.2) and -as a result -the oscillations, spending a fraction of their time near this set, do not become slow-fast in kind with decreasing . Rather it appears that the period has a well-defined limit as → 0. Nevertheless, the 'singular' nature of the oscillations does become more pronounced as → 0, insofar as the transition between the two distinct components of the oscillation becomes sharper.  = 0.1. The attracting limit cycle is shown in red, the repelling equilibrium is marked by a black dot. Even for this large value of the system clearly displays a form of multi-scale dynamics, e.g. note the rather sharp corner of the limit cycle. In (b): x(t) and y(t) along the limit cycle shown in (a).

1.2.
Main results. We prove existence of limit cycles for (1.1) and (1.2) for all 0 < 1 using a combination of GSPT and the blow-up method adapted in [20] for the study of degeneracies caused by exponential nonlinearities. We present these results in the following, considering each system separately.
Remark 1.1. While systems (1.1) and (1.2) seem similar in nature, the level of difficulty to analyse them using the GSPT toolbox is remarkably different.
1.2.1. The Hester system. Due to the singular exponential nonlinearity e y −1 , system (1.1) has no limit for → 0 for y > 0. This problem can be circumvented by appealing to the notion of topological equivalence and applying a time transformation  which givesẋ = y 1 + e (1+α)y −1 , where with a slight abuse of notation the overdot denotes differentiation with respect to the new time t 1 . For any > 0, this corresponds to a smooth transformation of time; (1.1) and (1.7) therefore have the same orbits for any > 0.
Remark 1.2. Viewed less abstractly, the time transformation defined by (1.6) corresponds to a multiplication of the vector-field by the strictly positive function (1 + e (1+α)y −1 ) −1 . This leaves all orbits unchanged. Such space dependent rescalings together with the blow-up method are commonly used in GSPT, see e.g. [15,16,22] and will also be used frequently in the present paper.
Clearly, the time transformation defined by (1.6) is singular for = 0, but it has the advantage that the new system (1.7) has a well-defined pointwise limit as → 0 for any y = 0. In fact, in this limit we obtain the piecewise smooth (PWS) system: for y > 0 andẋ = y, y = −x − 2γy, (1.9) for y < 0, the dynamics of which we sketch in Fig. 5. The discontinuity set Σ = {y = 0} is called the switching manifold in the PWS literature [3]. Given γ ∈ (0, 1), system (1.9) for y < 0 has a stable focus at (x, y) = (0, 0) which is on the switching manifold Σ = {y = 0}.
In the PWS literature, this situation is known as a boundary focus, see [24]. Under the forward flow of (1.8) or (1.9), respectively, every point with y > 0 or y < 0 will reach y = 0 in finite time. The case γ ≥ 1 for which (0, 0) is a stable node is also interesting (though not considered in [11]); this is discussed in Section 7. Frequently, in PWS systems one prescribes a Filippov vector-field [3,10] on Σ to have a well-defined forward flow. However, sinceẋ = 0 on y = 0 for both (1.8) and (1.9), the Filippov system is completely degenerate on Σ, consisting entirely of (pseudo-)equilibria [3,24]. Our analysis of the Hester problem will reveal a slow flow near the switching manifold Σ for all 0 < 1 (see Lemma 1.4), and allow us to define a singular relaxation cycle Here Γ 1 is an orbit segment (a fast jump) of (1.9), obtained by flowing the uniquely identified jump-off point (x j , 0) ∈ Σ with forward until the first return to Σ at the drop point ( The switching manifold Σ = {y = 0} is shown in magenta. Under the conditions (1.11), the singular cycle Γ 0 = Γ 1 ∪ Γ 2 perturbs to a stable limit cycle by Theorem 1.3.
Our main result on the Hester system is the following theorem. Theorem 1.3. Consider the Hester system (1.1) and fix a large ball B r of radius r. Suppose that κ(1 + α) ∈ (0, 1), γ ∈ (0, 1), µ > 0. (1.11) Then there exists an 0 > 0 such that for all ∈ (0, 0 ), system (1.1) has a unique limit cycle Γ in B r . Furthermore, Γ is attracting and in Hausdorff-distance, as → 0 + . If κ(1 + α) / ∈ (0, 1] then no limit cycles exist for 0 < 1 in B r . In addition we are able to identify the following asymptotics of a corresponding locally invariant (slow) manifold: , with x j as in (1.10), be a closed interval. Then for (1.1), there exists an exponentially attracting locally invariant (slow) manifold given as a graph: with h smooth in both variables and 0 < 0 1.
1.2.2. The Le Corbeiller system. As in the 'Hester case', due to the exponential term e y −1 , system (1.2) does not have a limit as → 0 for y > 0. Again, we introduce a time transformation corresponding to multiplication of the vector-field by the strictly positive function (1 + e y −1 ) −1 , to obtainẋ = y + a 1 + e y −1 , (1.12) For this system, the pointwise limit as → 0 is well-defined for all y = 0, and gives the following PWS system:ẋ for y > 0 andẋ = y + a, y = −x + 2by, (1.14) for y < 0, with {y = 0} as switching manifold Σ. Some orbits of this limiting PWS system are shown in Fig. 6. Now, the y < 0 system (1.14) has an unstable focus for b ∈ (0, 1) at (x, y) = −a(2b, 1), but also a quadratic, visible fold tangency [3] on the switching manifold at (x, y) = (0, 0). On the other hand, the y > 0 system (1.13) has a line of equilibria along Σ. The Filippov system is therefore again completely degenerate along Σ.
Our analysis of the Corbeiller problem will reveal a reduced flow on an invariant manifold near the switching manifold Σ for all 0 < 1 (see Lemma 1.6). Anticipating this, we define a singular relaxation cycle where Γ 1 is the orbit segment of (1.14) obtained by flowing the tangency point (0, 0) forward until the first return (x d , 0), with x d < 0, to Σ, see Fig. 6. The set Γ 2 is defined as the segment (x d , 0) on the switching manifold Σ. Figure 6. Phaseportrait of the piecewise smooth system (1.13), (1.14) in black. The switching manifold Σ = {y = 0} is shown in magenta. Under the conditions (1.16), the singular cycle Γ 0 = Γ 1 ∪ Γ 2 perturbs to a stable limit cycle by Theorem 1.5.
Our main result on the Le Corbeiller system is: Consider the Le Corbeiller system (1.2) and fix a large ball B r of radius r.
Then there exists an 0 > 0 such that for all ∈ (0, 0 ), system (1.2) has a unique limit cycle Γ in B r . Furthermore, Γ is attracting and Again, we are also able to identify the following asymptotics of a corresponding locally invariant manifold with our methods: Lemma 1.6. For system (1.2), fix I ⊂ (−∞, 0). Then there exists an exponentially attracting locally invariant manifold given as a graph: for x ∈ I, ∈ (0, 0 ), with another smooth function h satisfying h(x, 0) = 2, and where W : Using the asymptotics of W for w → ∞, we realise that the invariant manifold (1.17) has the following leading order asymptotics Remark 1.7. The two oscillations in Fig. 2 and Fig. 4, and the singular versions in Fig. 5 and Fig. 6, look qualitatively similar. Also the statements in Theorem 1.3 and Theorem 1.5 are almost identical. However, their PWS versions have different degeneracies along Σ and we shall see that the systems are very different in the singular limit → 0, requiring different techniques for their analysis. For the Hester system (1.1) under the assumption (1.11) the exponentials do in fact not cause significant complications. In this respect the Le Corbeiller system (1.2) behaves very differently -the exponential terms lead to several complications which require a much more involved analysis.This is also reflected in the different asymptotics of their invariant manifolds presented in Lemma 1.4 and Lemma 1.6.

1.3.
Overview. In Section 2, we first study (1.1) and prove Theorem 1.3. The proof, based upon the blow-up method and GSPT, is fairly straightforward, in particular in comparison with the proof of Theorem 1.5, which makes up the rest of the paper, see Section 3. Obviously, an essential step in the proof of Theorem 1.5 will be to prove Lemma 1.6. This is done in Section 3.2 after having described our blow-up approach. Subsequently, we present two lemmas Lemma 3.9 and Lemma 3.10 that prove Theorem 1.5, see Section 3.4 and Section 3.5.
In Section 4 we then prove Lemma 3.9 before proving Lemma 3.10 in Section 5. Further details of the blow-up used to prove Lemma 3.10 are delayed to Section 6. In Section 7, we conclude our paper by presenting an outlook.
Remark 1.8. Throughout this paper, we will assume some familiarity with geometric singular perturbation theory and in particular with the blow-up method. The interested reader is referred to, e.g., [18,21,23] for background and more references. which provides a zoom into the dynamics close to the switching manifold Σ. This giveṡ x = y 2 , System (2.2) is a standard slow-fast system, which has the form x = 2 y 2 , on the fast time scale τ = t/ . By setting = 0 we obtain the corresponding layer problem x = 0, The set Fig. 7, is a manifold of equilibria, which in GSPT is called the critical manifold of system (2.2). The critical manifold C is the disjoint union of the following sets recall (1.10), and where y j := − 1 α log (κ(1 + α)) .
Here C a (C r ) is normally hyperbolic and attracting (repelling, respectively), whereas F is a regular fold point. Since C a is normally hyperbolic, any compact submanifold perturbs to an attracting locally invariant slow manifold for 0 < 1 by Fenichel's theory [9]. This proves the statement regarding the invariant manifold in Lemma 1.4.
The reduced problem on Cẋ = y 2 , is obtained by going to a (super) slow time and setting = 0. There is a unique equilibrium on C at y 2 = 0. For κ satisfying (1.11) the equilibrium is on the repelling branch C r and it is an unstable node. In this case the slow flow on C a is towards the right and approaches the fold point, where a fast downward jump along the unstable fiber of the layer problem occurs, see Fig. 7.
Remark 2.1. Notice that in Fig. 7, and in all subsequent figures, we are following some conventions which are commonly used in GSPT to distinguish between the dynamics of the different limiting problems, which need to be shown simultaneously in the same figure: Green segments indicate "fast" orbits (of layer problems) whereas blue indicates slow flow (on critical manifolds) that is obtained upon (further) desingularization (by speeding up time). Orbits which approach equilibria (in forward or backward time) in hyperbolic directions are highlighted by triple-headed arrows, whereas flows in slow and center directions are highlighted by single-headed arrows. Degenerate points, e.g. the fold point F , which need to be blownup, are given individual colors, which are then also used for the corresponding blown up higher-dimensional objects.
Summing up, we have shown that the upper half of the limit cycle in the Hester problem can be obtained by the GSPT analysis of system (2.2): For x < 0 there is fast dynamics towards C a along orbits of the layer problem, then there is slow flow along C a towards the fold point F , where a fast jump occurs with y 2 going to −∞. The x−coordinate of the fold point F is the value x j defining the right boundary of the segment Γ 2 of the singular cycle Γ 0 = Γ 1 ∪ Γ 2 , see Fig. 5.
However, we have to keep in mind that solutions of system (2.2) with y 2 = O(1) correspond to solutions of system (1.1) in a narrow strip y = O( ) around the switching surface Σ. To obtain the full limit cycle, we also have to consider the region y = O(1) < 0; this is not covered by the scaling (2.1). The analysis and the matching of these different regimes based on blow-up is carried out in the next subsection. Not surprisingly, we will see that the segment Γ 1 of the singular cycle Γ 0 is the orbit of the PWS system (1.9) starting at (x j , 0), see Fig. 5.
As always in the blow-up approach one has to consider the extended system in R 3 obtained from (1.7) written on the fast time defined by ( ) = ( ) by adding the trivial equation for . This extended system has the (x, y, 0)-plane as a set of equilibria. The line x ∈ R, (y, ) = (0, 0) is singular in the sense of lack of smoothness of the vector field (2.4) as → 0. Recall that this degenerate line is precisely the switching manifold Σ (embedded into R 3 ) of the piecewise smooth system defined by (1.8) for y > 0 and (1.9) for y < 0. We regain smoothness by applying the blow-up transformation blowing up the degenerate line to the cylinder R × S 1 with r = 0, see Fig. 8. Since we are only interested in ≥ 0, only the part of the cylinder with¯ ≥ 0 is relevant. The edges y = ±1,¯ = 0, r = 0 of this half cylinder will be important later.
Remark 2.2. Note the color-coding which will be used frequently: the switching manifold Σ shown in magenta in Fig. 5 is blown-up to the cylinder shown in magenta in Fig. 8.
The vector field (2.4) induces a vector field on the blown-up space. As always, the cylinder, corresponding to r = 0, and the plane¯ = 0 are invariant and capture the crucial dynamics, both corresponding to = 0. Notice, that the scaling (2.1) can be viewed as a directional chart (obtained by setting¯ = 1) of the blow-up transformation, which covers the side of the cylinder corresponding to¯ > 0. In contrast to the usual blow-up approach [6,21], we will not divide by r. Thus, we find that the slow-fast system (2.3) multiplied by the positive and smooth function 1 1 + e (1+α)y 2 , which does not change the orbits, describes the blown-up dynamics in the chart corresponding to¯ = 1. In particular, on the cylinder we recover the limiting dynamics shown in Fig. 7.
In addition blow-up provides a compactification of system (2.2) as y 2 → ±∞. Thus the unstable fiber of the fold point F limits now on a point on the edgeȳ = −1,¯ = 0 , see Fig. 8. Actually, the two edgesȳ = ±1,¯ = 0, r = 0 of the half cylinder are lines of equilibria of the blown-up system, which must be studied in directional charts corresponding toȳ = ±1. In these charts we recover the PWS system (with improved hyperbolicity properties) within = 0 after dividing out factors of¯ , respectively. We illustrate our findings in Fig. 8. The required analysis -to establish this rigorously -is standard, see e.g. [15,17,18,19]. Also, very similar computations are carried out in detail for the (more complicated) Le Corbeiller system below. We therefore only summarise the results for the Hester system. Along the edgesȳ = ±1,¯ = 0 we find lines of equilibria (magenta in Fig. 8) having a , corresponding to r = 0, and in the plane (black)¯ = 0. We show a view from the top, i.e. from¯ > 0, with the orientation of the x, y, axis being indicated by black arrows labeled with x, y, . The edgesȳ = ±1,¯ = 0, r = 0 of the half cylinder are lines of equilibria, which are hyperbolic except at the point (orange) withȳ = −1, x = 0. In the plane¯ = 0 we recover the PWS dynamics (black) of system (1.8), (1.9). On the cylinder we recover the layer problem (green) and the reduced problem (blue) of system (2.2). This allows to define an improved singular cycle Γ 0 = Γ 1 ∪ Γ 2,1 ∪ Γ 2,2 ∪ Γ 2,3 which perturbs to a true cycle Γ for 1.
hyperbolic saddle-structure, except for x = 0,ȳ = −1,¯ = 0 (orange circle) which is fully non-hyperbolic. This structure provides an improved singular cycle (thick closed curve in Fig. 8), having good hyperbolicity properties except at the fold F of the critical manifold C (blue). Therefore, it is easy to perturb this singular cycle into an actual limit cycle for 0 < 1 by first considering a return mapping to the section {y = −δ} near x j , for example, and then applying e.g. [21] to the passage near fold F (working in the scaled coordinates (2.1)) to show that the Poincaré map is a strong contraction. We leave out the details because they are standard. See again [15] for a related system where more details are provided.
Motivated by the similarity of the two systems and by the success of the scaling approach for the Hester system, we begin our analysis by considering the original Le Corbeiller system (1.2) by rescaling y = y 2 , i.e. by zooming into the switching manifold Σ. This produces the following systeṁ System (3.2) is a slow-fast system in standard form, with layer problem which has a degenerate, non-hyperbolic line L 2 = {(x, y 2 ) : x = 0} of equilibria. Away from x = 0 the flow is trivial and regular (upward for x < 0, downward for x > 0), see Fig. 9.  in Section 2.1 the slow-fast dynamics of the the rescaled system (3.2) is quite degenerate, e.g. there exists no normally hyperbolic critical manifold. At this stage the rescaled system seems to capture very little of the observed limit cycle. Nevertheless, the flow defined by the rescaled system (3.2) and in particular the nonhyperbolic line L 2 of equilibria, will play an important role in a refined analysis of the limit cycle based on blow-up. More precisely, the flow of the rescaled system (3.2) will be recovered as the flow on the blow-up of the switching manifold Σ to a cylinder, see Fig. 10. However, the full resolution of the Le Corbeiller model will require more than just one cylindrical blow-up due to its more singular dependence on .
To obtain a full resolution and to connect (3.2) with the PWS system (1.13) and (1.14) we study again the extended system obtained by transforming (1.12) to the fast time scale τ = t/ end adding the trivial equation for . The set defined by (x, y, 0) is then a plane of equilibria, with the subset given by y = 0 being extra singular due to the lack of smoothness there. We gain smoothness by applying the blow-up transformation are necessary to analyze the dynamics on the blown-up space. We will make use of the usual subscript notation to specify coordinates in each chart, defining chart-specific coordinates as follows: Transition maps between a chart K i and K j (i = j) will be denoted by κ ij , and are given here by (3.9) We will also adopt the convention of denoting a set γ by γ i when viewed in a particular coordinate chart K i .

Remark 3.2.
Notice again that the scaling (3.1) corresponds to the directional chart K 2 , which is hence customarily referred to as a "scaling chart". In the following we will call the chart K 1 an "entry chart", because in this chart the singular flow relevant for the limit cycle returns to the edge (ȳ,¯ ) = (−1, 0) of the cylinder and continues from there on the cylinder, see see Fig. 10. Note that in charts K i , i = 1, 3 the cylinder corresponds to r i = 0. These charts provide a compactification of the variable y 2 from the scaling chart K 2 , i.e. y 2 → −∞ in the plane r 2 = 0 corresponds to 1 → 0 in the plane r 1 = 0, and y 2 → ∞ in the plane Since it is the easiest chart to analyze we start with the scaling chart K 2 .
Chart K 2 . The governing equations in chart K 2 are which is system (3.2) written on the fast time scale and multiplied by the positive and smooth factor 1 1 + e y 2 . Thus for r 2 =0, i.e. on the cylinder, we recover the layer problem and in particular the fully nonhyperbolic line L 2 corresponding to x = 0, see Fig. 10.
Chart K 1 . We now discuss the dynamics in the entry chart K 1 . The governing equations are x = r 1 (a − r 1 ), These equations are obtained by transforming to the new coordinates followed by a desingularization of the vector field corresponding to dividing out a factor of 1 (1 + e − −1 1 ) −1 , which is smooth for 1 ≥ 0. Note, that it is this division that allows us to recover the PWS system within 1 = 0, see Lemma 3.3. In particular, there is an isolated equilibrium at p 1 := (−2ba, a, 0), and two lines of equilibria The following lemma summarizes important dynamical features of this system, which are illustrated in Fig. 10. (ii) The line of steady states L 1 is fully non-hyperbolic, and coincides (where domains overlap) with the line L 2 observed in the K 2 chart. (iii) The plane 1 = 0 is invariant. Within this plane points (x, 0, 0) ∈ l s,1 with x > 0 are hyperbolic repelling, respectively hyperbolic attracting for x < 0. The flow in this plane in r 1 > 0 is precisely the flow of the y < 0 system (1.14) multiplied by a factor r 1 , through the identification y = −r 1 . The equilibrium p 1 = (−2ba, a, 0) is an unstable focus within the invariant 1 = 0 plane for any b ∈ (0, 1). Furthermore, there is a quadratic tangency between the flow of (3.11)| 1 =0 and l s,1 at q 1 , corresponding to the tangency observed in Section 1.2.2. The equilibrium p 1 extends to 1 > 0 as a line of equilibria which coincides upon coordinate change back to the original variables with the true equilibrium of the system identified in Section 1.
are hyperbolic attracting, respectively hyperbolic repelling for x b < 0, with stable, reps. unstable manifolds given by Proof. The statement (i) follows immediately from the linearization along l s,1 , and the simple form of the equations when restricted to the invariant plane r 1 = 0: The statement (ii) follows by an application of the transition map κ 12 given in (3.9), and the statements (iii) and (iv) follow by simple calculations made using the system governing the dynamics in the invariant plane 1 = 0: x = r 1 (a − r 1 ), Chart K 3 . Dynamics in chart K 3 are governed by after desingularization through division of the right hand side by 3 In particular, there are two lines of equilibria: Restricted to the invariant planes r 3 = 0 and 3 = 0, we obtain the following equations and x = 0, r 3 = −br 2 3 , respectively. It is clear that any point in either l e,3 and L 3 is fully non-hyperbolic. We notice in particular the 'exponential loss of hyperbolicity' in (3.15) as 3 → 0. The following lemma summarizes the important features of this system, which are illustrated in Fig. 10.
Lemma 3.4. The following hold for system (3.14): (i) The line of steady states l e,3 is fully non-hyperbolic.
(ii) The line of steady states L 3 is fully non-hyperbolic, and coincides where domains overlap with the non-hyperbolic line L 2 observed in the K 2 chart. (iii) The plane 3 = 0 is invariant. The flow in this plane in r 3 > 0 is precisely the y > 0 system (1.13) multiplied by a factor r 3 , through the identification y = r 3 . The flow in 3 = 0, r 3 > 0 is parallel to the r 3 −axis and toward l e,3 . (iv) The plane r 3 = 0 is invariant. The flow in this plane for 3 > 0 is parallel to the x−axis, and toward (away from) the line l e,3 for x < 0 (x > 0).

Proof. Straightforward.
We briefly sum up the results obtained by the above analysis, which are illustrated in in Fig. 10. We have achieved a certain desingularization of (3.4) by means of the blow-up transformation (3.5). On the cylinder r = 0 we have recovered the layer problem of (3.2) in particular the nonhyperbolic line L. In the invariant plane¯ = 0 we have recovered the PWS system (1.14) and (1.13) forȳ < 0 respectivelyȳ > 0. At the invariant line l s of saddle type we have gained hyperbolicity away from the nonhyperbolic point q. The invariant lines l e and L are still fully degenerate and will be treated by further blow-ups.
3.1. Blow-up in the exponential regime. To deal with the (exponential) loss of normal hyperbolicity associated with the line l e,3 in system (3.15), we use the approach put forward in [20]: introduce the auxiliary variable and extend the phase space by including an equation for q . For improved readability (in subsequent blow-up transformations) we drop the subscripts in the coordinates of chart K 3 in the following. Thus, the following system is obtained after a further multiplication of the right hand side by ; we therefore basically undo the division by used to obtain (3.14). It is worth keeping in mind that is invariant. We will often use this fact, utilizing it when it helps in the analysis. In system (3.16), the non-hyperbolic line of equilibria L 3 in (3.14) shows up as the Qrestricted subset: of the non-hyperbolic plane of equilibria The subscript 'e' (for 'exponential') in L e signifies that we are considering the object in the extended, four-dimensional system (3.16). Similarly, the line of equilibria can be viewed as an improved version of the line of degenerate equilibria l e,3 . It is the intersection of two separate degenerate planes: Although the extended system (3.16) is clearly quite degenerate, the main advantage of this system is that the loss of normal hyperbolicity is now algebraic, and blow-up methods are applicable. Therefore, we introduce a blow-up of the plane of equilibria P e of system (3.16) via the map We are primarily interested in the dynamics observable in the entry chart which has chart specific coordinates In this chart, we obtain the system after a suitable desingularization (division by ρ 1 ). Note that the set is invariant under the flow induced by (3.22). The non-hyperbolic plane P defined in (3.18) becomes a plane of equilibria for system (3.22), constitutes a line of equilibria for the system (3.22). We also have which will be important for our analysis.
Lemma 3.5. The following hold for the system (3.22): (i) Within = 0, there exists a two-dimensional manifold of equilibria given by The linearization about any point in C 1 has a single non-trivial eigenvalue where β 1 > 0 is chosen sufficiently small for the inequality to hold. Accordingly, C 1 is normally hyperbolic and attracting within = 0 for x < 0, and non-hyperbolic along the line (ii) Within ρ 1 = 0, there exists a two-dimensional manifold of equilibria given by The linearization about any point in S 1 has a single non-trivial eigenvalue Accordingly, S 1 is normally hyperbolic and attracting within within ρ 1 = 0 for x < 0, and non-hyperbolic along the line The manifolds C 1 and S 1 intersect the invariant domain of interest Q 1 along a onedimensional manifold of equilibria C 1 satisfying Considered within the invariant plane = ρ 1 = 0, the manifold C 1 is normally hyperbolic and attracting for x < 0, being non-hyperbolic at the point P : (0, 0,0, 0). (iv) The plane P 1 is fully non-hyperbolic, and L e,1 coincides with the image of the nonhyperbolic line L 3 observed in system (3.14) (recall Lemma 3.4). (v) The eigenvalues associated with the linearization along l e,1 are given by implying that l e,1 is a line of partially hyperbolic saddles for x = 0, and non-hyperbolic for x = 0.
Proof. The statement (i) follows after linearization of the system obtained by restricting to the invariant hyperplane = 0: Similarly, the statement (ii) is obtained after linearization of the system obtained by restricting to the invariant hyperplane ρ 1 = 0: (3.25) Statement (iii) follows immediately from (i)-(ii) and the observation that Statement (iv) follows after linearization of the system (3.22), together with an application of the blow-down transformation.
Statement (v) also follows after linearization of the system (3.22).
The most important properties of system (3.22) described in Lemma 3.5 are sketched in Fig. 11. Figure 11. Dynamics upon blow-up (3.20) of l e on the cylinder (orange), corresponding to ρ = 0. We show a view from the top, i.e. from¯ > 0, with the orientation of the x, y, axis being indicated by black arrows labeled with x, y, . Fast flow on the orange cylinder is shown in green. In comparison with Fig. 10 we have gained hyperbolicity at the line of equilibria l e and we are able to identify a normally hyperbolic critical manifold C (blue). The line (halfcircle) L (brown) of equilibria and, in particular, the equilibrium P (purple) are still degenerate and will require additional blow-up transformations.
Remark 3.6. Clearly, not all properties of the four-dimensional system (3.22) are visible in Fig. 11, which focuses on explaining how the second blow-up improves the situation shown in Fig. 10. Keep in mind, that the second blow-up (3.20) affects only a small neighborhood of the line l e in Fig. 10. The degenerate line l e is blown up to the orange cylinder, corresponding to ρ = 0, i.e. ρ 1 = 0 in chart (3.21). In this chart r 1 > 0 means "going up" on the cylinder, which due to y = r 3 = ρ 1 r 1 justifies our use of the y-axis in the figure.
The preceding observations are sufficient to prove the following result, which will in turn allow for a proof of Lemma 1.6.  Proof. Follows from center manifold theory and the statements in Lemma 3.5.

3.2.
Proof of Lemma 1.6. To prove Lemma 1.6 and the expression in (1.17) we simply restrict M 1 in (3.26) to the invariant set Q 1 and blow down to the (x, y)-coordinates. Using (3.8), (3.21) and by returning the subscript 3 on in (3.26), we obtain the following equations and = e − −1 where 0 < 1 is the original small parameter. In these expressions we have also introduced a newm obtained by expanding the first term in (3.26) about ρ 1 = 0 and setting −x/b > 0 outside a bracket. It is straightforward to show thatm(x, 0, 0) = 2. The expression in (1.17) is the result of solving these equations for y as a function of . Let for any s > 0, where W is the Lambert-W function satisfying W (t)e W (t) = t for every t > 0. Then and Z has a continuous extension to s = 0 with value Z(0) = 0. We then solve (3.28) by introducing the auxiliary variable u through the expression Notice that • u = 0 in this expression by (3.30) gives the 'leading order solution' of (3.28) obtained by settingm ≡ 0.
after canceling out a common factor on both sides obtained from (3.30) with s = b/(−x). This equation is smooth in u, x and z ≥ 0. In particular, u = z = 0 is a solution for any x ∈ I and the partial derivative of the right hand side with respect to u gives −1 for u = z = 0, x ∈ I. Therefore, by applying the implicit function theorem, we obtain a locally unique solution u =h(x, z) of (3.34) withh smooth, satisfyingh(x, 0) = 0 for all x ∈ I, and where z ∈ [0, β 3 ] for β 3 > 0 sufficiently small. A simple computation shows thath is exponentially small with respect to z:h for some smooth h satisfying h(x, 0) =m(x, 0, 0) = 2. Inserting (3.33) into this expression produces, by (3.31) and (3.30), the following locally unique solution of (3.28) Finally, by inserting this into (3.32) we obtain the expression which by (3.29) gives (1.17).
3.3. Improved singular cycle and preliminary results. The analysis thus far is sufficient for the construction of an improved singular orbit and corresponding Poincaré map. The singular orbit can now be viewed as a union of distinct orbit segments Γ i , i = 1, 2, 3, 4, and the line L. The segment Γ 1 is already defined as the part of the singular cycle contained in the half plane y < 0, and x d < 0 is the x-coordinate of the first intersection of the trajectory flowed forward from (0, 0) with the line y = 0 in system (1.14); see the discussion leading to the expression (1.15) in Section 1.2.2. The remaining segments Γ i are are listed below, together with the coordinate chart used to define them. (3.35) Remark 3.8. Note that we are permitting a slight abuse of notation here by allowing Γ 2 to refer to a different segment as in Section 1.2.2, expression (1.15). In fact, it is Γ 2 ∪Γ 3 ∪Γ 4 ∪L in (3.35) that upon blowing down to the (x, y)-plane becomes Γ 2 in (1.15).
We can now define an improved singular cycle Γ 0 by see Figure 12. Although this cycle has improved hyperbolicity properties, it is still completely degenerate along L. A fully nondegenerate singular cycle, resolving the dynamics around L, will be presented in Section 5.1, relying on details in Section 6 once the remaining prerequisite blow-ups have been defined.
3.4. The Poincaré map Π = Π 2 • Π 1 . Also shown in Figure 12 are two sections Σ 1 and Σ 2 , both transversal to Γ 0 . We define these as follows:  To obtain the last two components of Π 1 , we have used (3.8), (3.21) and the invariance of the set Q.
Lemma 3.9. The following holds for α > 0 and 0 > 0 sufficiently small: (i) Π 1 is well-defined and at least C 1 with respect to x.
where M is the center manifold in Lemma 3.7. (iii) In particular, the restricted map x → r 1 1 (x, ) for any ∈ (0, 0 ] is a strong contraction, i.e.
Given the form of the map in (3.37) and Lemma 3.9, we write Π 2 as follows for any r 1 ∈ I, ∈ [0, δ]. Here I is a small neighborhood of η/b, recall (3.38). The form in (3.39) restricts Π 2 to the relevant subset of Σ 2 defined by invariance of and the set Q, and allows for composition with Π 1 , see (3.37).
Lemma 3.10. The following hold for R, β > 0 and 0 sufficiently small: (i) Π 2 is well-defined and at least C 1 with respect to r 1 .
3.5. Proof of Theorem 1.5. Taken together, Lemma 3.9 and Lemma 3.10 show that Π is a contraction. Theorem 1.5 and the existence of a perturbed limit cycle Γ for all 0 < 1 is then a consequence of the contraction mapping theorem. Lemma 3.9 is proved in Section 4. A rigorous proof of Lemma 3.10 requires additional blow-ups. We summarise these in in Section 5, see Section 5.1, allowing for the proof of Lemma 3.10 to be outlined in Section 5.2. The blow-up analysis outlined for the purposes of proving Lemma 3.10 are given in greater detail in Section 6.
The mappings Π 1,4 and Π 1,5 are less standard because they occur in the exponential regime. We therefore include some more details in the following.

4.2.
The local transition map Π 1,4 : Σ 1,4 → Σ 1,5 . We work in chart K 1 , for which   Here the order of O(r log r −1 ) does not change by differentiation with respect to x.
Proof. Consider a solution of (4.2) satisfying The system decouples from (4.2), and can be integrated directly to give One can use these equations to obtain the following expression for the transition time T : By the above we obtain (These expressions also follow from the conservation of the original and the set Q, see (3.8) and (3.21): r in δe −δ −1 = R 5 (T )e − (T ) −1 .) Furthermore, which is C 1 −smooth, guaranteeing that the leading order is well behaved with respect to integration. Hencex The result then follows from (4.1) and using the asymptotics (1.18) of W .
Proof. Trajectories with initial conditions in Σ 1,5 are strongly attracted to their base-points on the invariant center manifold M 1 in Lemma 3.7, and track the slow flow after reaching a local tubular neighborhood of M 1 . To leading order, the flow on M 1 is determined by x ρ 1 (1 + O( + ρ 1 ))), (4.4) so that x > 0 and Π 1,5 is therefore well-defined. Center manifold theory implies exponential contraction (e −ct ) along one-dimensional fibers with base points on M 1 . Since • r 1 (t) ≥ ν > 0 for some ν > 0 sufficiently small during the transition from Σ 1,4 to Σ 1,5 ; • r 1 ρ 1 = const., see

The map Π 2 : Proof of Lemma 3.10
The analysis of the map Π 2 requires good control of the flow close to the line L of degenerate equilibria L, see Figure 12. Additional blow-ups are necessary in order to prove Lemma 3.10. In this section we summarise the blow-up transformations and dynamical features leading to the construction of the final nondegenerate singular cycle Γ 0 as shown in Fig. 14. The cycle Γ 0 has a total of eight distinct segments Γ i , i ∈ {1, . . . , 8}. We have already identified the segments Γ i for i ∈ {1, 2, 3, 4} in (3.35); the segments Γ i for i ∈ {5, 6, 7, 8} will be defined by three additional blow-ups needed to resolve the degeneracies of the line L. This section is included here for expository purposes: a more detailed technical presentation is given in Section 6.
By looking at Fig. 12 and by extrapolating from what we did so far, it is natural to expect that a straightforward (cylindrical) blow-up of the non-hyperbolic line (circle) L will be necessary for the construction of the final singular cycle Γ 0 . However, it turns out that additional difficulties arise close to the point P which is the endpoint of L on the (orange) cylinder, corresponding to ρ = 0. This is somewhat expected, given that this transition corresponds dynamically to a transition between regimes in which exponential terms are dominant (the upper horizontal cylinder) to a regime in which algebraic terms are dominant (the vertical cylinder obtained after blow-up of L). The analysis of this transition requires two additional spherical blow-ups shown in Fig. 14, which are necessary for resolving the degeneracies and 'connecting' the two regimes. 5.1. Successive blow-ups and the fully nondegenerate singular cycle Γ 0 . The required sequence of blow-up transformations is sketched in Fig. 15, and outlined in the following.
By Lemma 3.5, Section 3.1, the manifold C terminates in a non-hyperbolic point P (purple) at the endpoint of the non-hyperbolic curve L e (the origin in chart K 1 coordinates) corresponding to the endpoint of the vertical non-hyperbolic line L (brown); see Fig. 15a. In order to resolve this degeneracy, this point is blown-up to a sphere as in Fig. 15b; this is done in Section 6.1.1 as part of a larger 'cylinder of spheres' blow-up. Note that this sphere is the 'outer sphere' (also purple) in Fig. 14. The manifold C then terminates in a partially hyperbolic and attracting point P L on the equator of this sphere. One identifies the extension of the singular cycle as a normally hyperbolic attracting curve of equlibria N (blue) on this sphere, terminating in another non-hyperbolic point P O (cyan), as shown in Fig. 15b.
The non-hyperbolic point P O is analyzed in Section 6.1.3 by means of a second spherical blow-up. This is the 'inner sphere' (also cyan) sketched in Fig. 14 and Fig. 15c. We identify an attracting, partially hyperbolic point p l (cyan) on the equator of the sphere as the endpoint of N . The relevant thing for the construction of Γ 0 is the existence of a unique center manifold W (cyan) emanating from p l , which we can identify with Γ 6 ; compare Fig. 14 and Fig. 15c. Phase plane arguments (the system is no longer slow-fast) are given in Section 6.1.3 (see Lemma 6.6 in particular) in order to show that W terminates in yet another non-hyperbolic point p O (brown), from which the non-hyperbolic line L (also brown) emanates.
All remaining non-hyperbolicity is resolved by means of cylindrical blow-up of L in Section 6.1.4 and Section 6.2, leading to the fully resolved scenario sketched in Fig. 14, see also Fig. 15d. In Section 6.1.4, a cylindrical blow-up applied locally near the inner blow-up sphere leads to the identification of a partially hyperbolic saddle point p s (brown). The orbit W terminates at p s , and the next component of Γ 0 can be identified with the unstable manifold W u (p s ), which is shown in Lemma 6.7 to lie along the outside 'edge' of the cylinder, denoted Figure 15. Sequence of blow-up transformations needed to desingularize the dynamics near the line of degenerate equilibria L and to connect the exponential regime and the critical manifold C with the orbit Γ 1 (black) in the planē = 0. In (a) we show again Fig. 11. The degenerate equilibrium P (purple) is blown up to a sphere (purple) in (b), as described in Section 6.1.1. On this sphere another degenerate equilibrium P O (cyan) is identified, which is blown-up to another sphere (cyan) shown in (c), as detailed in Section 6.1.3. Finally, the still remaining line of degenerate equilibria L (brown) is blown-up to a cylinder (brown) shown in (d) and analyzed in Section 6.1.4 and Section 6.2.
by H (also brown) in Fig. 15d. The remaining analysis is carried out in Section 6.2, after it is shown in Lemma 6.8 that the dynamics in the local cylindrical blow-up described above can be mapped to those obtained in a cylindrical blow-up of the line L in the algebraic regime, i.e. in charts K i , i = 1, 2, 3, subsequent to the (first) cylindrical blow-up leading to the lower horizontal cylinder in Fig. 14. The analysis leads to the identification of the final two cycle segments Γ 7 and Γ 8 in Fig. 14. The first can be identified with H, which is shown in Lemma 6.13 to be invariant and terminating in a partially hyperbolic singularity at q s . The second can be identified with the unstable manifold W u (q s ), which is invariant, and shown (also in Lemma 6.13) to terminate in a hyperbolic (resonant) saddle singularity q o . Although the analysis here is to some extent standard, it is frequently complicated by the fact that transition across different coordinate charts becomes non-trivial after the application of multiple blow-up transformations.
To summarise, we define the fully nondegenerate singular cycle Γ 0 by where Γ i for i ∈ {1, 2, 3, 4} have already been defined in (3.35) and where Regarding Γ 8 , the coordinates specified by the chart K 11 are defined in Section 6.2.6, see also (6.40). Note also that the union Γ 5 ∪ Γ 6 ∪ Γ 7 ∪ Γ 8 of the segments in (5.1) becomes L in (3.35) upon blowing down. Each segment has improved hyperbolicity properties that allow us to describe the mapping Π 2 and prove Lemma 3.10.

5.2.
Proof of Lemma 3.10: A summary. Similar to our analysis of the map Π 1 , the map Π 2 can be analyzed by defining additional transversal sections Σ 2,i , i ∈ {2, . . . , 9} as in Fig. 16, and considering the composition Here Π 2,1 : Σ 2 → Σ 2,2 , Π 2,9 : Σ 2,9 → Σ 1 , and Π 2,i : Σ 2,i → Σ 2,i+1 for i ∈ {2, . . . 8} denote transition maps induced by the flow. To prove Lemma 3.10, we have to consider the maps Π 2,i in turn. However, each mapping is standard so we just focus on explaining in what coordinates the mappings are described, making references to equations, sections and lemmas to come below, and summarise the findings as follows: The transition map Π 2,1 is described in the coordinates specified by the chart K 1 , see (6.3). By center manifold theory and Lemma 6.2 below the mapping is well-defined and exponentially contracting in the direction transverse to W.
The transition map Π 2,2 is described in the coordinates specified by the chart K 2 , see (6.3). The properties of this map are similar to Π 2,1 since the line of partially hyperbolic equilibria N extends all the way down to P O . However, P O is non-hyperbolic and it is important to highlight that in K 2 we are leaving the 'exponential regime'. We can therefore reduce the phase space dimension again, see (6.7). This produces a 3-dimensional system. The transition map Π 2,3 is described in the coordinates specified by the chart K 1 , see (6.16). Center manifold theory and Lemma 6.5 below show that the mapping is well-defined and exponentially contracting in the direction transverse to W.
The transition map Π 2,4 is a diffeomorphism by regular perturbation theory applied in the coordinates specified by the chart K 2 , see (6.16).
The transition map Π 2,5 is described in the coordinates specified by the chart K 31 , see (6.23) and the equations of motion in (6.24), associated with the blow-up of L. In these coordinates we gain hyperbolicity, as indicated in Fig. 15d, see also the local picture in Fig. 21. Within the center manifold, the point p s (see Fig. 15d) is a hyperbolic saddle upon further desingularization, having H as an unstable manifold. The mapping Π 2,5 is therefore well-defined and (algebraically) contracting.
The transition map Π 2,6 . By Lemma 6.8 below, we can transform the result on Π 2,5 into the chart K 21 , see (6.28), associated with blowing up L in the original scaling chart K 2 , recall (3.7). Here we can also describe the mapping Π 2,6 as a diffeomorphism by regular perturbation theory using the invariance of the line H. See also Lemma 6.11.
The transition map Π 2,7 is described in the coordinates specified by the chart K 11 , see (6.27), as a local passage near the semi-hyperbolic saddle q s . The mapping is well-defined and non-expanding; the details being similar to those in [15,Theorem 4.2].
The transition map Π 2,8 is described in the coordinates specified by the chart K 12 , see (6.27), as passage near the resonant hyperbolic saddle q o with a one dimensional unstable manifold Γ 1 . The details are similar to those in [17, lemma 3.6].
Each map is one-dimensional and at least C 1 upon restriction to 0 < 1 and the set Q. This completes the (sketched) proof of Lemma 3.10 upon composition. Further details can be found below and in the forthcoming PhD-thesis [12].
6. Blow-up analysis for the map Π 2 The blow-up transformations and main features in the analysis for the proof of Lemma 3.10 are presented in this section. Our analysis divides into two parts: (i) an understanding of the transition between the exponential and algebraic regimes, and (ii) a blow-up analysis describing the dynamics near the non-hyperbolic line L in the algebraic regime. Part (i) is considered in Section 6.1, and focuses (among other things) on the spherical blow-ups shown in Fig. 14. Part (ii) is considered in Section 6.2.
6.1. Exiting the exponential regime. Here we consider the transition out of the 'exponential regime'. In terms of Fig. 14, our aim is to understand the manner in which trajectories move from the upper horizontal cylinder (exponential regime), to the vertical cylinder (algebraic regime).
(6.1) By Lemma 3.5, the one-dimensional manifold C identified with the cycle segment Γ 4 terminates at (0, 0, 0, 0) ∈ P, where P : x = r = 0 constitutes an entire plane of non-hyperbolic fixed points for (6.1). One could proceed by blowing up the entire plane P, however only the dynamics near the curve L e ⊂ P will be relevant for the transition (recall that L e and L 3 coincide where domains overlap; see again Lemma 3.5). This motivates a blow-up of (x, r, ρ) = (0, 0, 0) in the following form We introduce the coordinate charts for which we have chart-specific coordinates given by (II) Considered in terms of its parameterization (3.23), L e is flat at (0, 0, 0, 0), and thus flat with respect to the line L as → 0; see Fig. 17. Geometrically, it seems more natural to blow-up L e . To do this one would rectify L e and apply the cylindrical blow-up transformation (6.2) along (the transformed) L e . However, our approach avoids this unnecessary coordinate transformation. Besides (a) only the sphere CS ∩ { = 0}, which is the same for both approaches (recall (6.5)), will be relevant, and (b) once we leave the exponential regime and enterρ > 0 of the sphere (6.2), we will apply a subsequent blow-up transformation that effectively blows up L e (or L). Figure 17. Illustration within {x = 0} of the lines L (magenta) and L e (brown) as subsets of the plane P : x = r = 0 (shaded magenta).
We will adopt the following notational convention: given an object γ identified in chart K 1 , we denote its image under the blow-up transformation (6.2) by γ , and it's image in a particular coordinate chart K i by γ i (this will help to avoid confusion given the earlier dropping of subscripts etc).
The transition maps between coordinates in charts K i , i = 1, 2 are given by Recall also that the set Q = (x, r, , ρ) : ρ = e − −1 is invariant for the system (6.1). In chart K i coordinates, then, the following are invariant: K 1 Chart. The dynamics in chart K 1 are governed by after a suitable desingularization (division by ν 1 ). The manifold of equilibria C 1 in Lemma 3.5 shows up as a manifold of equilibria C 1 for the system in the invariant hyperplane = 0 given by 9) and the manifold of equilbira S 1 in Lemma 3.5 shows up as a manifold of equilibria S 1 for the system in the invariant hyperplane ρ 1 = 0, given by We are interested in the intersection C 1 ∩ S 1 , since this is contained within the domain of interest Q 1 . In particular, this intersection constitutes a line of equilbria in the = ρ 1 = 0 plane. The line C 1 terminates at the point P L : (b −1 , 0, 0, 0), which sits on the equator of the invariant sphere segment ν 1 = = 0. In fact, within the invariant plane ν 1 = = 0 we have which has a line of equilibria emanating from P L . Finally, we identify a line of equilibria along the positive ν 1 −axis, i.e. along l e,1 = {(0, 0, 0, ν 1 ) : ν 1 ≥ 0} .
Lemma 6.2. The following hold for system (6.8): (i) The point P L is partially hyperbolic with a single non-zero eigenvalue λ = −1, and there exists a corresponding three-dimensional center manifold M 1 tangent to the center eigenspace E c (P L ), which can be chosen to be the extension of the manifold M 1 .
Locally, M 1 contains the two-dimensional manifolds of equilbria C 1 and S 1 as restrictions and the one-dimensional manifolds of equilbria C 1 and N 1 as restrictions The variables , ρ 1 (r 1 ) are increasing (decreasing) along M 1 \ (C 1 ∪ S 1 ∪ N 1 ). (ii) The eigenvalues associated with the linearization along l e,1 are given by λ = 1, 0, −1, 0, implying that l e,1 is a line of partially hyperbolic saddles.
Proof. Assertion (i) is standard center manifold theory. Assertion (ii) is a direct calculation. 6.1.2. K 2 Chart. The system in chart K 2 is given by after a suitable desingularization (division by ν 2 ). At this point we note that the analysis may be simplified by restricting to the invariant set Q 2 . By doing so we reduce the dimension by 1 after eliminating ν 2 , and consider the reduced system on this set (6.13) Note that the system obtained by restricting to = 0 in (6.13) is equivalent to the system obtained by restricting to ν 2 = = 0 in system (6.12), and given by (6.14) Here we identify the line of equilbria corresponding to the extension of N in chart K 2 , as well as a line of equilibria (iii) The line L e,2 is non-hyperbolic, and coincides where domains overlap with the nonhyperbolic line L 3 identified in chart K 3 coordinates.
Proof. Statement (i) follows immediately after linearization of the system (6.14) along N 2 . The statement (ii) follows from center manifold theory, together with uniqueness of the manifold M 1 described in Lemma 6.2 and an application of the transition map κ 12 in (6.6).
The statement (iii) follows after linearization of the system (6.13) and an application of the blow-down transformation. Figure 18. Illustration of the blowup of the degenerate equilibrium P (purple) to the 'outer sphere' (purple) within¯ = 0. On the sphere we find the attracting critical manifold N (blue) with reduced flow connecting P L to the degenerate equilibrium P O (cyan). The fast flow on the sphere is shown in green.
The situation is illustrated in Fig. 18. Fig. 18 shows dynamics not visible in charts K 1 or K 2 . To gain a complete picture of the dynamics one must look in the additional coordinate charts K 3 :r = 1 and K 4 :x = 1. In particular, a hyperbolic saddle P R is identified at (x,r,ρ) = (0, 1, 0), = 0 in chart K 3 , and a partially hyperbolic line of saddle-type equlibria is identified along the positivex−axis in chart K 4 . We omit the details here for expository reasons. We are primarily interested in the dynamics observable in the phase directional charts for which we introduce the chart specific coordinates (6.16) We adopt the following notational convention: given an object γ identified in chart K 2 , we denote its image under the blow-up transformation (6.15) byγ, and it's image in a particular coordinate chart K i byγ i . The transition map between charts K 1 and K 2 is given bỹ The dynamics in chart K 1 are governed by after a suitable desingularization (division by σ 1 ). We identify an equilibrium for the system (6.18) at p r : (0, 0, 0), as well as a line of equilibria corresponding to the extension of the line of equilbria N 2 observed in chart K 2 . Note that N 1 terminates at p l : (−b, 0, 0).
Lemma 6.5. The following hold for system (6.18): (i) The point p l : (−b, 0, 0) is partially hyperbolic with a single non-zero eigenvalue λ = −b < 0. There exists a corresponding two-dimensional center manifold M 1 tangent to the center eigenspace E c (p l ), which can be chosen to be the extension of the manifold M 1 . The manifold M 1 | 1 =0 contains the one-dimensional manifold N 1 as a normally hyperbolic and attracting manifold of equilbria. Moreover, M 1 | σ 1 =0 contains a unique one-dimensional center manifold W 1 , and the slow flow on W 1 is increasing in the (ii) The equilibrium p r : (0, 0, 0) is a hyperbolic saddle with eigenvalues The stable manifold W s (p r ) is contained in x 1 = 1 = 0, which is invariant.
Proof. The statement (i) follows after linearization of the system (6.18) and an application of the center manifold theorem. In particular, one obtains the following graph expression for W 1 via the usual matching approach: Hence the dynamics on W 1 are governed by from which assertion (i) follows. The statement (ii) follows after linearization of the system (6.18), and the observation that restriction to x 1 = 1 = 0 gives K 2 Chart. The dynamics in chart K 2 are governed by Proof. In order to prove the assertion (i) we consider the system in the invariant plane σ 2 = 0: and choose c 2 > 0 sufficiently large so that by Lemma 6.5, W 2 = κ 12 ( W 1 ) enters V transversally through r 2 = c 2 . Since V is compact and forward invariant, the Poincaré-Bendixon theorem applies, and W 2 is forward asymptotic to p o at (0, 0). To see that W 2 approaches (0, 0) tangent to the positive x 2 −axis, notice that trajectories in V \{(0, 0)} reach the forward invariant region  Finally, in this subsection, we consider a cylindrical blow-up of the non-hyperbolic line L e,2 identified in equation (6.20). This is done in a neighborhood of the point p 0 covered by chart K 2 . In Section 6.2.1 we will carry out a similar cylindrical blow-up of the line L within the first cylindrical blow-up as defined in (3.5), which needs to be carried out in coordinate charts K i , i = 1, 2, 3 defined in (3.6), (3.7), and (3.8). There, we will also show that these two cylindrical blow-ups match up. We start with system (6.19), drop the subscripts, introduce a hat notation (needed in the process of matching the results obtained here with the results obtained in Section 6.2.1), and define a weighted blow-up transformation by the map We are primarily interested in the dynamics observable in coordinate charts K 31 :x = 1, K 32 :r = 1, for which we introduce chart specific coordinates K 31 :x =ŝ 1 ,r =ŝ 2 1r 1 , K 32 :x =ŝ 2x2 ,r =ŝ 2 2 . (6.23) The transition map between charts K 31 and K 32 is given bŷ The subscript notation, although a little cumbersome, will be helpful in when considering the dynamics in coordinate charts covering the lower portion of the blown-up line (circle) L.
K 31 Chart. The equations in the K 31 chart are given bŷ after a suitable desingularization (division byŝ 1 ). The system (6.24) has a single equilibrium at p s : (0, 0, 0).
K 32 Chart. We omit the details in chart K 32 for the sake of brevity, simply noting that calculations reveal no equilibria and an invariant flow along the 'equator'ŝ 2 = σ = 0, as indicated in Fig. 21. 6.2. Blow-up of L in the algebraic regime. In order to obtain the fully nondegenerate singular cycle, it remains to blow-up the vertical non-hyperbolic line (circle) L in the algebraic regime. We return to the dynamics observable after the (first) cylindrical blow-up, as defined in (3.5), i.e. the problem considered in coordinate charts K i , i = 1, 2, 3. In Section 6.2.1 we introduce the blow-up of L, and show that the dynamics observed in Section 6.1.4 can be related to the dynamics observed in this blow-up in an overlapping domain. In Section 6.2.2 we derive the qualitative properties of the dynamics associated with the 'lower part' of the vertical cylinder in Fig. 14. 6.2.1. Blow-up of L. We consider the dynamics near the lower portion of the non-hyperbolic line (circle) L. We introduce a secondary weighted blow-up defined via the transformation (x, r, (ȳ,¯ )) → (s, (x,r) , (ȳ,¯ )) , where s ≥ 0, (x,r) ∈ S 1 → x = sx, r = s 2r . (6.25) Composing this with the map (3.5), we obtain Geometrically, the transformation (6.26) blows up the circle of non-hyperbolic points L to the torus {s = 0} × S 1 × S 1 , for which only the subset defined byr ≥ 0 and¯ ≥ 0 is relevant. In total, six coordinate charts are necessary for an understanding of the main dynamical features: K 11 :ȳ = −1,x = 1, K 21 :¯ = 1,x = 1, K 31 :ȳ = 1,x = 1, K 12 :ȳ = −1,r = 1, K 22 :¯ = 1,r = 1, K 32 :ȳ = 1,r = 1. (6.27) In particularx = 1 in charts K i1 andr = 1 in charts K i2 , for i = 1, 2, 3, and the subscript i signifies the 'visible region' of the first (horizontal) cylinder defined by the blow-up transformation (3.5). For charts K 1j covering the region visible inȳ = −1 we have chart specific coordinates K 11 : x = s 1 , y = −s 2 1 r 11 , = s 2 1 r 11 1 , K 12 : x = s 2 x 2 , y = −s 2 2 , = s 2 2 1 . For charts K 2j covering the region visible in¯ = 1 we have chart specific coordinates (6.28) For charts K 3j covering the region visible inȳ = 1 we have chart specific coordinates The transition maps between overlapping charts are given by κ 1121 : r 11 = −r 21 y 2 , 1 = −y −1 2 , y 2 < 0, κ 1221 : s 2 = (−r 21 y 2 ) 1/2 , 1 = −y −1 2 , x 2 = (−r 21 y 2 ) −1/2 , y 2 < 0, r 21 > 0, κ 2131 : r 21 = r 31 3 , and their inverses can be computed directly using these expressions if necessary.
We will focus in this section on the dynamics observable in charts K ij , i = 1, 2, j = 1, 2. The dynamics in charts K ij with i = 3 have already been considered in Section 6.1.4, as is shown in the following result.
Lemma 6.8. Coordinates (r 31 , 3 , s 1 ) in chart K 31 are related to coordinates (r 1 , σ,ŝ 1 ) in chart K 31 via Coordinates (x 2 , 3 , s 2 ) in chart K 32 are related to coordinates (x 2 , σ,ŝ 2 ) in chart K 32 via Proof. The expressions given for the coordinates (r 31 , 3 , s 1 ) in chart K 31 are obtained by composing blow-up maps (3.8), (3.21), (6.16) and the coordinates for K 31 in (6.23). In the notation below, we avoid dropping subscripts and append them at each coordinate change (except where the 'hat notation' suffices). Explicitly, the first three compositions give and subsequent restriction to ν 2 = e − −1 1 . Dropping the subscript in σ 2 and comparing with the K 31 coordinates given in (6.29) yields the result.
We obtain the following corollary. Corollary 6.9. The (invariant) σ−axis in system (6.24) is mapped to under the transformation defined by the equations (6.30), which is invariant for the system obtained in chart K 31 coordinates. Dynamics on H 31 are governed by 6.2.2. Blow-up for the lower part of L. Lemma 6.8 shows how the dynamics in the transitional regime can be related to the dynamics in the algebraic regime, after application of the blow-up transformation (6.26). Moreover, Lemma 6.7 and Corollary 6.9 are sufficient for an understanding of the main dynamical features and in particular, the construction of Γ 0 . Hence, we restrict attention here to dynamics in charts K 11 , K 12 , K 21 , K 22 only in this section.
6.2.3. K 22 Chart. We can determine the equations in the chart K 22 by considering the system (3.2) on the fast time scale with = r 2 , i.e.
Lemma 6.10. The system (6.34) is a regular perturbation problem, with leading order dynamics on compact domains determined by the dynamics of the limiting system Proof. This follows by direct integration of the equations (6.35).
6.2.4. K 21 Chart. The equations in chart K 21 can be determined by considering the system (6.32) and applying the secondary transformation defined by the equations x = s 1 , r 2 = s 2 1 r 21 . We obtain the following system, In all three subspaces y 2 is the only dynamic variable, with dynamics governed by Proof. Straightforward restriction.
6.2.5. K 12 Chart. The equations in the chart K 12 can be determined by considering the system (3.11) and applying the secondary transformation x = s 2 x 2 , r 1 = s 2 2 . One obtains the system Lemma 6.12. The following holds for the system (6.38): (i) The equilbria q i and q o are hyperbolic saddles with eigenvalues and respectively. There is a strong resonance in each case due to the relation λ 1,i/o = λ 2,i/o + λ 3,i/o . (ii) The equilibrium p 12 is an unstable focus within the invariant 1 = 0 plane for any b > 0, a ∈ (0, 2), and coincides upon coordinate change with the true equilibrium of the system p. Proof. Statements (i) and (ii) are immediate upon linearization of the system (6.38) and an application of the blow-down map, respectively. To prove the statement (iii), consider the system in the invariant plane s 2 = 0: (6.39) It is easy to verify that this system has invariant lines along x 2 = ± √ 2a, 1 ≥ 0.
6.2.6. K 11 Chart. The equations in the chart K 11 can be determined by considering the system (3.11) and applying the secondary transformation x = s 1 , r 1 = s 2 1 r 11 . Proof. The statement (i) follows immediately by a linearization of (6.41) and an application of the transition map κ 1112 . To prove the statement (ii), consider the system in the invariant plane s 1 = 0: r 11 = r 11 (1 − 2ar 11 ) , 1 = − 1 . (6.42) The equations decouple, and the lines along r 11 = 1/2a, 1 ≥ 0 and r 11 = 0, 1 ≥ 0 are invariant with dynamics governed in each case by Taken together, Lemma 6.10, Lemma 6.11, Lemma 6.12 and Lemma 6.13 imply the main dynamical features associated with the lower portion of the blow-up of L, as sketched in Fig. 22. Note the resemblance to the regular fold when viewed 'from below' (as one might expect due to the presence of the quadratic tangency in the PWS system). For further details on the regular fold see [17]. This resemblance is further exemplified by the fact that the invariant lines G ± observed in the K 1j charts connect in the region of the cylinder visible in the K 2j charts in Fig. 22; this is shown below.
Proposition 6.14. The invariant lines G ± observed in charts K 1j connect along the invariant parabola given in chart K 22 coordinates by , r 2 = 0.
Proof. Starting from chart K 11 and applying the relevant blow-down transformation, we may parameterize the G + in chart K 1 coordinates as Applying the κ 12 transition map in (3.9) gives the following parameterization for G + in chart K 2 , G +,2 = x, y 2 , − x 2 2ay 2 : y 2 ≤ 0 , and applying the secondary blow-up transformation (6.33) and expressing G +,2 in chart K 22 coordinates gives A similar argument applies for G − , and the result follows.

Summary and outlook
In this paper, we have proved existence of limit cycles in two prototypical examples with singular exponential nonlinearities. Our approach was geometric and consisted of the following: Firstly, we applied a proper normalization by dividing the right hand side by a suitable factor, producing a PWS limit as → 0. Secondly, we applied a modification of the blow-up approach following e.g. [17,18] to deal with degeneracies of this type, recall Section 2.2. For the Hester system, this basically led to a system with desirable hyperbolicity properties allowing us to perturb away from a singular cycle. Under the assumptions (1.11), the exponential nonlinearities did therefore not provide any obstacles for this result. However, if (1.11) is violated by γ ≥ 1, then (0, 0) is a stable node for (1.9) and Γ 1 is therefore asymptotic to this point. This leads to a more degenerate singular cycle in Fig. 8, with the orbit within y < 0 connecting to the orange circle at x = 0. This point is extra singular due to the exponentials and studying limit cycles in this case would require use of the same machinery used to describe the oscillations in the Le Corbeiller problem.
For the Le Corbeiller problem, we did not recover the essential geometric structure, recall Fig. 7, by a simple scaling. Instead the 'slow manifold' was hidden within a separate 'exponential' blow-up, see C in Fig. 11. This also led to more complicated asymptotics, recall Lemma 1.6, which we were able to capture using the method in [20].
In conclusion, our geometric approach for studying the Hester and the Le Corbeiller systems is general and we therefore expect that it can be applied to different problems of this kind, including the ones discussed in the introduction. This will be part of future work in the area. We also expect that it is possible to use these methods to explain the 'canard explosion' phenomena that occurs in both systems when small Hopf cycles grow to the 'relaxation-type' oscillations described in our main results.