Graph Transform and Blow–up in Singular Perturbations

Graph Transform and Blow–up in Singular Perturbations

Kaspar Nipp

, Daniel Stoffer

and Peter Szmolyan

∗Seminar for Applied Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland

†Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland

∗∗Angewandte und Numerische Mathematik, TU Wien, A-1040 Wien, Austria

Abstract. We show that the discrete dynamical system obtained by applying the Euler method to the van der Pol relaxation oscillator admits a closed attractive invariant manifold and that this curve is close to the limit cycle of the ODE.

Keywords: Singular perturbations, van der Pol equation, limit cycle, blow–up, graph transform, invariant manifold, Euler method PACS: <02.30Hq>

INTRODUCTION

Singularly perturbed systems are characterized by the occurrence of slow and fast motions. Van der Pol’s relaxation oscillator is a simple but nontrivial example nicely exhibiting these phenomena:

dx/dτ = x^′ = f(x,y,ε) := ε³(1+y)

dy/dτ = y^′ = g(x,y,ε) := −x−y²−y³/3 (1) with x,y∈R,ε∈(0,δ0]. Here,τis the fast time as usual in singular perturbations. It is well known that the van der Pol equation (1) admits a limit cycle of relaxation type, cf. Figure 1. The existence of such a closed orbit can be shown using Poincaré-Bendixson theory, see, e.g., [1]. The properties of the limit cycle have been discussed in the context of matched asymptotic expansions, e.g., in [2], [3], [4]. Szmolyan et al. [5], [6] investigate the transition from slow to fast by a so called blow–up approach establishing appropriate local coordinates and by using a Poincaré return map.

We consider the discrete dynamical system obtained by applying a numerical integration scheme to (1) and ask if there is a closed curve also for the discrete system. For this problem a Poincaré return map is not a suitable approach.

Instead, we use a graph transform approach. We start with a closed curve near the limit cycle of the ODE (1) and apply the map induced by the integration scheme. In this way we obtain a sequence of closed curves converging to an invariant limit curve. To keep things as simple as possible we choose the explicit Euler method as numerical integration scheme. We show that the discrete dynamical system obtained by applying the Euler method to the van der Pol equation (1) admits an attractive invariant manifold for all step sizes h≤ν0and all parameter valuesε≤δ0.

Since the dynamics of (1) strongly varies along the limit cycle it makes sense to describe the objects of the graph transform, i.e., the closed curves, in different charts a sketched in Figure 2. In these charts a closed curve near the limit cycle is described in coordinates relative to the limit cycle. Hence, we need to describe the limit cycle itself in the charts. Since in the literature the existence of the limit cycle is shown by qualitative arguments the limit cycle is not explicitly described as a graph of smooth functions. We therefore first derive the existence of the limit cycle of (1) together with its description in each chart as the graph of a smooth function.

As a simple example of a graph transform approach we show the existence of an invariant manifold close to the upper branch y=s⁰₊(x)of the reduced manifoldγ, cf. Figure 1. We introduce new coordinates(x0,y₀)as follows:

x=x₀, y=s⁰₊(x0) +y₀

for x₀∈Ξ:= (−∞,−ξ0]and y₀∈H := (−η0,−η0),ξ0,η0>0. In the new coordinates the ODE (1) is transformed to x^′₀ = f₀(x0,y₀,ε) := ε³(1+s⁰₊(x0) +y₀)

y^′₀ = g₀(x0,y₀,ε) := −y₀

s⁰₊(x0) (2+s⁰₊(x0)) +y₀(1+s⁰₊(x0) +y₀/3)

−ε^3s0+^′(x0)(1+s⁰₊(x0) +y₀) (2) We state an invariant manifold theorem applicable to this situation. The assumptions are as follows.

Assumption HD

• The flow of (2) is in-flowing with respect to H and out-flowing with respect toΞ.

• The estimates

|f_0,y| ≤ ℓ12, |f_0,ε| ≤ ℓ13

|g_0,x| ≤ ℓ21, |g_0,ε| ≤ ℓ23

µ(−f_0,x) ≤ −γ11, µ(g0,y)≤ℓ22 < 0 for some appropriate constants whereµdenotes the logarithmic norm.

Condition CD 2√

ℓ12ℓ21<γ11−ℓ22

Condition CDA ℓ22+ℓ12α<0 whereα is chosen such that

ℓ12α²−(γ11−ℓ22)α+ℓ21<0 and 2ℓ12α<γ11−ℓ22

Condition CDA(2) ℓ22+ℓ12α<2(γ11−ℓ12α)

Note that under Condition CD there isαsatisfying Condition CDA.

Theorem 1 Assume that Assumption HD and Conditions CD, CDA, CDA(2) are satisfied for an ODE of type (2) with vector field(f0,g₀). Then the following assertions hold.

i) The time–T map of the ODE, T>0, induces a mapF_T mapping the function space

C_α,ν:={σ∈C⁰(Ξ×(0,δ0],H)|σ^isα–Lipschitz continuous with respect to x andβ–Lipschitz continuous with respect toε^withβ≥(ℓ23+ℓ13α)/(ℓ22+ℓ12α)}

into itself and is a contraction.

ii) The mapF_T has a unique fixed point s₀∈C_λ,β⊂C_α,β with

λ = 2ℓ21

γ11−ℓ22+p

(γ11−ℓ22)²−4ℓ12ℓ21

being the smallest possible Lipschitz constantα. The set M_ε={(x0,y0)|x₀∈Ξ,y₀=s₀(x0,ε)}is an out-flowing invariant manifold of the ODE.

It is easily verified that Theorem 1 applies to (2) with ℓ12=ε^3,ℓ13=O(ε²), ℓ21=O(η0),ℓ23=0, γ11=O(ε³), ℓ22=O(p

−ξ0)<0 and withα=O(η0),β=η0.

THE INVARIANT MANIFOLD FOR THE ODE

Theorem 1 cannot be applied directly to prove the existence of a limit cycle of (1). We describe a closed curveΓin the vicinity of the limit cycle in the following way. We consider the charts

Φi:



 x yε



∈W_i⊂R³7→

u_i v_i

∈K_i=U_i×V_i⊂R²×R, i=1, . . . ,16,

mapping the piece of curveΓi=Γ∩W_ito K_i. An important feature of this approach is that the K_iare independent of the small parameterε, in contrast to the matched asymptotic expansions approach. In K_ithis piece of curve is described as a graph

{(ui,v_i)|v_i=σi(ui)}

whereσi∈C⁰_b(αi,βi)is a bounded function with Lipschitz constantαiwith respect to(ui)1and with Lipschitz constant βi with respect to(ui)2. Note thatαi+8=αi,βi+8=βifor i=1, . . . ,8, since the corresponding charts are reflected.

Since W_i,i+1:=Wi∩W_i+1is chosen to be non–empty one has

Γi∩W_i,i+1=Γi+1∩W_i,i+16=/0

which relatesσitoσi+1(withσ17=σ1) in the following way, cf. Figure 3. Define K_i,i+1=Φi(Wi,i+1) ⊂ K_i, K_i+1,i=Φi+1(Wi,i+1) ⊂ K_i+1

The piece of curveΓ∩W_i,i+1has the two representations v_i=σi(ui)in K_i,i+1and v_i+1=σi+1(ui+1)in K_i+1,i. We thus have

Φi+1◦Φ⁻_i ¹ σi(ui) u_i

= σi+1(ui+1) u_i+1

(3) We introduce the function spaceΣ={σ= (σ1, . . . ,σ16)|σi∈C_b⁰(αi,βi)}equipped with the norm

kσk=max

i |σi|i

where the norms| · |iin C⁰_b(αi,βi)}are given as

|σi|i=N_isup

ui∈Ui

|σi(ui)| with positive norm factors N_i.

We introduce the time-T map PT of the ODE (1) for small T >0 such that PT(Wi\W_i,i+1)⊂Wi. Note that this condition replaces the assumption in HD that the flow of (2) is out-flowing with respect to Ξ. We show that the Lipschitz constantsαi,βimay be chosen such that the map P_T induces a mapF_T :Σ→Σ. Applying Assertion i) of Theorem 1 in each chart the mapF_T is shown to be a contraction for an appropriate choice of the norm factors Ni. This implies the existence of a unique fixed point s∈Σdescribing a unique invariant curve of the ODE (1). If the right-hand sides f , g of (1) are of class C² and if in every chart Condition CDA(2) is satisfied then the function s is also of class C². Stepping from chartΦ1toΦ2toΦ3, etc., the Lipschitz constantsαi,βibecome typically larger and larger. Indeed, in chartΦ7we haveα7=O(ε⁻²)andβ7=O(ε⁻¹). Hence, by means of (3) the functionσ8has also Lipschitz constants of these orders inε. Since chartΦ1and its reflected chartΦ9have the same small Lipschitz constantsα1=α9,β1=β9, the Lipschitz constants ofσ8at the end of chartΦ8have to be correspondingly small. To achieve this we use the so called elephant trunk argument. The independent variable u₈is essentially 1–dimensional (one component is the parameterε) and the flow in direction of u₈is monotone. We therefore subdivide the u₈–interval in many small subintervals with Lipschitz constants on these intervals decreasing geometrically.

We introduce the chartΦ1with its domain K₁. Away from the fold point(0,0)close to y=s⁰₊we define the local coordinates x₁,y₁,ε1by

Φ1: x=x1, y=s⁰₊(x1) +y1, ε=ε1

and define the first chart as (x₁,y1,ε1)∈K₁= [−2/3−ξ1,−ξ1]×[−η1,η1]×(0,δ1]withξ1,η1,δ1being small. The dynamics in chartΦ1is described by the ODE (2) with x₀,y₀,ε0replaced by x₁,y₁,ε1. Assertion i) of Theorem 1 holds as seen before.

The blow-up approach, cf. [5], is a good means to magnify the critical vicinity of the fold point and thus allows to introduce appropriate charts with suitable local coordinates. As in [5] we consider the blow-up transformation

x=er²xe, y=erey, ε=ereε

as a map from B :=S²×[0,R]intoR³withex,y,eeε∈S². In order to define charts near the fold point we consider half spheres of S²with centersex=−1,eε=1 andye=−1, respectively. The choiceex=−1 leads to chartΦ2,eε=1 leads to the chartsΦ3,Φ4andye=−1 yields chartΦ5, cf. Figure 4. We discuss chartΦ5in more detail. For the half sphere with centerey=−1 we useex andeεas local coordinates. In order to simplify the coordinate transformation even more we setey=−1 and introduce the coordinates x₅,ε5, r₅inR³for the fifth chart as follows:

Φ5: x=r²₅x₅, y=−r₅, ε=r₅ε5

with(x5,r₅,ε5)∈K₅= [0,ξ5]×[0,ρ5]×(0,δ5]. The ODE (1) is transformed to ˙r₅

ε˙5

= f₅(r5,ε5,x₅) = r₅

−ε5

˙

x₅ = g₅(r5,ε5,x₅) = −2x₅+ε₅³(1−r₅)/[1+x₅−r₅/3]

(4)

where ˙ denotes the derivative with respect to the new time t withτ=r₅(1+x₅−r₅/3)t. This time scaling desingular- izes the ODE yielding a nontrivial flow for r₅=0. In K₅the pieceΓ5of a closed curveΓis described as the graph of x₅=σ5(r5,ε5)with Lipschitz constantsα5andβ5=α5(note that the independent variable is 2–dimensional and that there is no parameter). To apply Theorem 1 i) we verify the assumptions:

HD: ℓ12=0, ℓ21=O(ε₅²), γ11=−1, ℓ22=−2

It follows immediately that Conditions CD, CDA, CDA(2) are satisfied forα=O(ε₅²) =:α5. At the end of chartΦ5

we are away from the fold point(y=−ρ5). It therefore seems natural to choose the original variables x, y,ε ^for chartΦ6with y₆∈[−3+η6,−ρ5]. Here,Γ6is described as x=σ6(y,ε). However, the ODE (1) is not contracting in x–direction. In a first step we modify chartΦ6as follows:

Φb6: x=xb₆e^kby6, y=−by₆, ε=bε6

where k has to be chosen sufficiently large such that the contraction in x–direction is strong enough. Theorem 1 i) can still not be applied since there is also contraction in y–direction violatingγ11−ℓ22>0 of Condition CD. Therefore, we make the flow in y–direction almost constant by the transformation

b

y₆=Y(r6) (5)

with Y defined by dY/dr6=Y²(1−Y/3). This yields the chart

Φ6: x=x₆e^kY(r6), y=−Y(r6), ε=ε6

with(x6,r₆,ε6)∈K₆= [0,ξ6]×[ρ₆,ρ6]×(0,δ6]. In these local variables the ODE (1) has the form r₆^′

ε₆^′

= f₆(x6,r₆,ε6) := 1+x₆e^kY/[Y²(1−Y/3)]

0

x^′₆ = g₆(x6,r₆,ε6) := −kx₆

Y²(1−Y/3) +x₆e^kY

+ε₆^3e−kY(1−Y) We verify the assumptions of Theorem 1 i):

HD: ℓ12=O(1), ℓ13=0, ℓ21=O(1), ℓ23=O(ε₆²), γ11=O(1), ℓ22=O(k)<0

Thus, for k sufficiently large Conditions CD, CDA, CDA(2) are satisfied for α =O(1/k) =:α6. For β ^{one gets} β=O(ε₆²) =:β6.

We show that x₅=σ5(r5,ε5)in K₅restricted to K_5,6is given as x₆=σ6(r6,ε6)in K₆restricted to K_6,5, cf. (3). We denote this restriction byσe6. Transforming x₅=σ5(r5,ε5)withΦ⁻₆¹◦Φ5we obtain

x₆=σe6(r6,ε6) =Y²(r6)e^−kY(r6)σ5(Y(r6),ε6/Y(r6)) with Lipschitz constants

αe6=O(e^−kY(ρ5)), βe6=O(α5e^−kY(ρ5)) For k sufficiently largeαe6≤α6,βe6≤β6holds implyingσe6∈C_b⁰(α6,β6).

We determine the norm factor N₆. Given two functionsσ₅^1,σ₅²we show that the corresponding functionsσe₆^1,σe₆²

satisfy

eσ6¹−σe6²

6≤σ5¹−σ5²

5 (6)

We estimate

eσ₆¹−σe₆²₆=N₆ sup

(r₆,ε6)

eσ₆¹(r6,ε6)−σe₆²(r6,ε6)≤N₆

N₅Y²(ρ₅)e^−kY(ρ5)|σ₅¹−σ₅²|

implying that (6) holds if N₆≤N₅e^kY(ρ5)/Y²(ρ₅). An analogous computation has to be done in each chartΦi,i= 1, . . . ,8. ChartΦ8is connected toΦ9which is the reflected chartΦ1with Lipschitz constantsα9=α1,β9=β1and norm factor N₉=N₁. Summarizing, we have shown the following result.

Theorem 2 The time–T map of the van der Pol equation (1) induces a mapF_T :Σ→Σwhere Σ={σ= (σ1, . . . ,σ16)|σi∈C_b⁰(αi,βi)}.

F_T is a contraction and therefore has a unique fixed point s= (s1, . . . ,s₁₆)implying the existence of a limit cycle of the ODE (1). Moreover, s is of class C².

Remark The differentiability of s is implied by the smoothness of the vector field and by Condition CDA(2) in each

chart. ⊳

THE INVARIANT MANIFOLD FOR THE EULER METHOD

We apply the explicit Euler method with fixed step size h to the van der Pol equation (1). This generates the discrete dynamical system

x = x+hε³(1+y) y = y+h(−x−y²−y³/3)

x = ε

x = h

(7)

We show that this system admits a closed attractive invariant manifold in the vicinity of the limit cycle of (1). We use 16 charts

Ψi:



 x yε h



∈W_i⊂R⁴7→

u_i v_i

∈K_i=U_i×V_i⊂R³×R, i=1, . . . ,16,

to describe the dynamical system in an analogous way as for the differential equation. Here, u_icontains an additional variable corresponding to the step size h and v_i measures the distance to the limit cycle of the ODE (1). In order to be able to expand the map about the limit cycle s in each chart it is mandatory that s is of class C². In the blow–up charts it was necessary to desingularize the ODE by an appropriate time scaling with factor r_i, i=2, . . . ,5. For the discrete system this leads to a scaling of the step size h forΨi, i=2, . . . ,5. In chartsΨ2andΨ5the transformation ε=r_iεi, i=2,5, gives rise to a slowly varying step size variable hi. We discuss the procedure in more detail for chart Ψ5. Subsequently, we describe the elephant trunk approach in chartΨ8in more detail.

ChartΨ5is given by

Ψ5:

x = r²₅(s5(r5,ε5) +x₅) y = −r₅

ε = r₅ε5

h = h₅/[r5(1+s₅(r5,ε5) +x₅−r₅/3)]

with(x5,r5,ε5,h₅)∈K₅= [−ξ5,ξ5]×[0,ρ5]×(0,δ5]×(0,ν5]. We express the map (7) in the local coordinates of chart Ψ5. To simplify the formulas we omit the index 5.

h = h+h²E(h;(r,ε),x) = h+h²

1−(s+x) 2+h

(1+h)(1+s+x−r/3)

+ r

3(1+s+x−r/3) + ε³(1−r) (1+h)(1+s+x−r/3)

r ε

= r ε

+h F(h;(r,ε),x) = r(1+h) ε/(1+h)

= r ε

+h r

−ε/(1+h)

(8)

x = −(s−s) + h (1+h)²

−2s+ ε³(1−r) 1+s−r/3−hs

+xh

1+ h

(1+h)²

−2−h− ε³(1−r)

(1+s−r/3)(1+s+x−r/3)

i (9)

where s is an abbreviation for s(r,ε). Using the mean value theorem we get

−(s−s) =− Z 1

0

s_r(r+τ(r−r),ε+τ(ε−ε))dτ(r−r)− Z 1

0

s_ε(r+τ(r−r),ε+τ(ε−ε))dτ(ε−ε) (10) Note that r−r=hr andε−ε=−hε/(1+h). From (4) we obtain with ˙s=s_r˙r+s_εε˙the invariance equation for s

s_rr−s_εε=−2s+ ε³(1−r) 1+s−r/3

In (9) we thus may replace−2s+ε³(1−r)/(1+s−r/3)by s_rr−s_εε. The terms of (8) hr

(1+h)²s_r and − hε (1+h)²s_ε, respectively, are written as

hr Z ₁

0

s_r(r,ε)dτ+O(h²r) and − hε 1+h

Z ₁

0

s_ε(r,ε)dτ+O(h²ε), respectively. These integrals are combined with the integrals in (10) as follows:

−hrh^{Z 1}

0

s_r(r+τ(r−r),ε+τ(ε−ε))dτ− Z 1

0

s_r(r,ε)dτⁱ =

= −hrh^{Z 1}

0 Z 1

0 τ^srr(r+τσ(r−r),ε+τσ(ε−ε))dσ^dτ(r−r) + +

Z ₁

0 Z ₁

0 τ^srε(r+τσ(r−r),ε+τσ(ε−ε))dσ^dτ(ε−ε)i

=:−h²r²I_rr+h²rε 1+hI_rε

and a similar formula for the terms s_ε. Note: It is crucial that the function s is of class C². Summarizing, we obtain for the x–equation (9)

x= [1+hB(h;(r,ε),x)]x+h²G(h;(r,ε),x) (11) with

B(h;(r,ε),x) = − 1 (1+h)²

h

2+h+ ε³(1−r)

(1+s−r/3)(1+s+x−r/3) i

G(h;(r,ε),x) = O(r+ε+s)

We formulate a graph transform result similar to Theorem 1 for a map of the form h = h+h²E(h; u,v) ≤ ν0

u = u+hF(h; u,v) (12)

v = [1+hB(h; u,v)]v+h²G(h; u,v) ∈V_d

with h∈(0,ν0], u∈Rⁿ, v∈V_d={v| |v| ≤d}. We assume that the following assumptions hold:

Assumption HMH

Lip_hE=L₀₀, Lip_uE=L₀₁, Lip_vE=L₀₂ Lip_hF=L₁₀, Lip_uF=L₁₁, Lip_vF=L₁₂ Lip_hB=ℓ20, Lip_uB=ℓ21, Lip_vB=ℓ22

Lip_hG=L₂₀, Lip_uG=L₂₁, Lip_vG=L₂₂ Condition CMH There are constants a,b,c∈R, b>0, independent of h, such that

|1+2hE| ≥1−hc, lip_u(u+hF)≥1−ha, |1+hB| ≥1−hb holds where lip_udenotes the "lower Lipschitz constant" with respect to u.

Theorem 3 Assume that the map (12) satisfies assumption HMH and Condition CMH. Then there are positive constants d,ν0, K such that the following assertions hold.

i) The map (12) maps the graph of a function

σh∈C_α,γ:={σ∈C⁰((0,ν0]×Rⁿ,Vd)|σhisα–Lipschitz continuous with respect to u andγ–Lipschitz continu- ous with respect to h}

with

α = 2(ℓ21d+hL₂₁+K_α¹) [b−a−ℓ22d−hL₂₂−h K_α⁰] +p

[···]²−4L₁₂(ℓ21d+hL₂₁+K_α¹) γ = (|B|+hℓ20)d+h(2|G|+hL20) + (|F|+hL10)α

h(b−c−ℓ22d−hL₂₂−L₁₂α−K_γ) where the nonnegative quantities K_α⁰, K_α¹, K_γ satisfy

K_α⁰≤K(L01+L₀₂), K_α¹≤K_α⁰ d+h²(|G|+hL₂₁+hL₂₀)

, K_γ≤K(L01+L₀₂+L₀₀) (d+h) to the graph of a functionσh∈C_α,γ. The induced mapF_h:σh7→σhis a contraction.

ii) The mapF_hhas a unique fixed point s_h∈C_α,γ. The graph of s_his an out-flowing invariant manifold of the map (12).

For simplicity we have formulated Theorem 3 for one chart only. In our context, however, where we have several charts Ψ1, . . . ,Ψ16the maps of the form (12) are not overflowing with respect to u_i∈U_i, i=1, . . . ,16. Hence, Assertion ii) of Theorem 3 does not hold in a single chart. For the global picture we again have to modify the procedure as done in the ODE case relating the functionsσh,itoσh,i+1. This yields a result analogous to Theorem 2 for the Euler map (7) obtained by applying the Euler method to the van der Pol equation (1).

Assumption HMH and Condition CMH hold for the map (8), (11) of chartΨ5with L₀₀=O(1), L₀₁=O(1), L₀₂=O(1), L₁₀=ε5 L₁₁=1, L₁₂=0, ℓ20=O(1), ℓ21=O(ε₅²), ℓ22=O(ε₅³), L₂₀=O(1), L₂₁=O(1), L₂₂=O(1), and

b=5/3, a=4/3, c0=0.

Theorem 3 i) implies the existence of an invariant space C_α5,γ5 for the graph transform with Lipschitz constants α5=O(ξ5),γ5=O(ξ5)/h5.

Similar to the ODE case the Lipschitz constants in the graph transform become large at the end of chartΨ7 (of the order of a negative power ofε). In the elephant trunk approach (cf. the sketch in Figure 5) we consider graphs of functions y=σh(x,ε)with large Lipschitz constants at the beginning of chartΨ8,α0with respect to x,β0with respect toε^,γ0with respect to h (as given by chartΨ7) and Lipschitz constantsαi,βi,γidecreasing geometrically toε^-small Lipschitz constants (tending to 0 asε→0)αN,βN,γN, N=O(h⁻¹ε⁻³), at the end of chartΨ8, being there d_N–close to the limit cycle s of the ODE (1). And, we apply a graph transform result similar to Theorem 3 i).

SinceΨ8=Ψ9in W_8,9=W₈∩W₉it follows that forσeh,9, i.e.,σh,8related to chartΨ9, the equalityσh,8=σeh,9=σh,9

holds in the common domain. The Lipschitz constants ofσh,8at the end of chartΨ8tend to 0 asε→0. Hence, for sufficiently smallεit follows that in the common domain ofΨ8,Ψ9the functionσeh,9lies in C_α9,β9,γ9 withα9=α1, β9=β1,γ9=γ1being independent ofε^.

Summarizing, we have shown the existence of an invariant curve of the Euler map (7) being d_N=O(hε⁶)–close to the limit cycle s of the ODE (1) at the end of chartΨ8. (This is remarkable: Note that s is O(ε³)–close to the lower branch s⁰₋ofγat the end of chartΦ8(cf. Figures 1 and 2)).

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FIGURE 1. The limit cycle of the van der Pol equation (1) and the reduced manifoldγdefined by g(x,y,0) =0 with the upper branch y=s⁰₊and the lower branch y=s⁰₋. FIGURE 2. Sketch of the charts used in the graph transform approach.

The atlas for the closed curves consists of the chartsΦ1, . . . ,Φ16,Φi: Ki→Wi, whereΦ9, . . . ,Φ16are the reflections at the point (−2/3,−1)of the chartsΦ1, . . . ,Φ8. The figure shows sketches of the images W_i=Φi(Ki).

FIGURE 3. The chartsΦiandΦi+1. FIGURE 4. The blow-up of the fold point.

FIGURE 5. One step of the graph transform in the elephant trunk approach.