Canard cycles and homoclinic bifurcation in a 3 parameter family of vector fields on the plane

Publicacions Matem`atiques, Vol 43 (1999), 163–189. CANARD CYCLES AND HOMOCLINIC BIFURCATION IN A 3 PARAMETER FAMILY OF VECTOR FIELDS ON THE PLANE Paulo Ricardo da Silva Abstract Let the 3-parameter family of vector fields given by (A) y∂ ∂x +[x2+µ+y(ν0+ν1x+x3)] ∂ ∂y with (x, y, µ, ν0,ν 1)∈R2×R3([DRS1]). We prove that if µ→ −∞ then (A) is C0-equivalent to (B) [y−(bx +cx2−4x3+x4)] ∂ ∂x +ε(x2−2x)∂ ∂y for ε↓0, b,c∈R. We prove that there exists a Hopf bifurcation of codimension 1 when b= 0 and also that, if b=0,c= 12 and ε>0 then there exists a Hopf bifurcation of codimension 2. We study the “Canard Phenomenon” and the homoclinic bifurcation in the family (B). We show that when ε↓0, b= 0 and c=12 the attracting limit cycle, which appears in a Hopf bifurcation of codimension 2, stays with “small size” and changes to a “big size” very quickly, in a sense made precise here. 1. Introduction Let χthe set of C∞vector fields on the plane and χ0⊂χ, the set of vector fields with a singularity at (0,0). (1) χ0=P(x, y)∂ ∂x +Q(x, y)∂ ∂y |P, Q ∈C∞,P(0,0) = Q(0,0) = 0. Consider the equivalences introduced by the following definitions. Definition 1.1. a) X,Y∈χare called C0-equivalent if there exists a homeomorphism h:R2→R2sending X-orbits to Y-orbits in a sense preserving way. b) X,Y∈χ0are called C∞-equivalent if there exists a diffeomorphism g:R2→R2, fixing (0,0) ∈R2, such that Y(p)= [g(p)]−1·X(p), p∈R2. 164 P. R. da Silva Definition 1.2. a) A k-parameter family of vector fields on R2,Xλ, with λ∈Rk denoting the parameter, is defined to be a vector field a(m, λ)∂ ∂x + b(m, λ)∂ ∂y ,m=(x, y)∈R2, where the coefficient functions aand bare C∞with respect to (m, λ)∈R2×Rk. We say that Xλis a k-parameter unfolding of X0. b) Xλand Yµ,µ,λ∈Rk, are (fibre) C0-equivalent if there exist homeomorphisms µ=φ(λ) and hλ:R2→R2such that hλis a topological equivalence between Xλand Yφ(λ). Let X0∈χ0. Consider the Jordan form JX0of ∂P ∂x ∂P ∂y ∂Q ∂x ∂Q ∂y at (0,0) ∈R2 ([M]). (2) T11 :ab 0ca=0,c= 0 and b=0ora=c= 0 and b=1 T12 :ab −ba a=0,b=0 T2:a0 00 a=0 T3:0−a a0a=0 T4:01 00  T5:00 00 . Let Wi={X0∈χ0|JX0∈Ti}where T1=T11 ∪T12.Thus χ0=W1∪W2∪W3∪W4∪W5.W1is the set of vector fields with hyperbolic singularity at (0,0). One obtains for generic k-parameter families (with any k) the so called generalized saddle-node bifurcation of codimension kwhich is an unfolding of X0∈W2but whose restriction to center manifold starts with non-zero terms of order k. For generic k-parameter families (k≥2) one finds the generalized Hopf bifurcation where such Hopf bifurcation of codimension kis a k-parameter unfolding of X0∈W3and whose radial component of the normal form in polar coordinates starts with non-zero terms of order 2k+1. The study of the unfoldings of X0∈W4starts with the BogdanovTakens bifurcation ([RW]). If X0∈W4then, according [DRS1], j2X0 Canard cycles and homoclinic bifurcation 165 is C∞-equivalent to (3) y∂ ∂x +(ax2+bxy)∂ ∂y. If a= 0 we say that X0∈W4has singularity of the kind cusp. For the case a= 0, see [DRS2]. Generically ab = 0 and j2X0is C∞-equivalent to ([D]) (4) y∂ ∂x +(x2±xy)∂ ∂y. Bogdanov and Takens showed that, generically, any local 2 parameter unfolding of (4) is fibre C0-equivalent to (5) Xµ,ν =y∂ ∂x +[x2+µ+y(ν±x)] ∂ ∂y. Dumortier, Roussarie and Sotomayor ([DRS1]) studied X0∈W4with j2X0C∞-equivalent to (6) y∂ ∂x +ax2∂ ∂y. They showed that, generically, j4X0is C∞-equivalent to (7) y∂ ∂x +(x2±x3y)∂ ∂y. Generically, any local 3 parameter unfolding of (7) is fibre C0-equivalent to (8) Xµ,ν0,ν1=P∂ ∂x +Q∂ ∂y =y∂ ∂x +[x2+µ+y(ν0+ν1x±x3)] ∂ ∂y. The bifurcation set of Xµ,ν0,ν1is the smallest closed subset ⊂ R3(µ, ν0,ν 1) such that the topological type of Xµ,ν0,ν1for (µ, ν0,ν 1)∈ R3(µ, ν0,ν 1)\is locally constant. In Figure 1 we have the bifurcation set of (8+). It is composed of 9 surfaces and 5 curves. Figure 1 shows the intersection of the bifurcation set with the sphere Sδ={(µ, ν0,ν 1)|µ2+ν2 0+ν2 1=δ2,µ<0}, with δ>0 small. The parameter values for which ∂P ∂x +∂Q ∂y (−√−µ, 0) = 0 give the surface H(Hopf bifurcation). His composed of two parts separated by the 166 P. R. da Silva curve H2(Hopf bifurcation of codimension 2). In one of them, when the parameter value crosses H, the focus changes the stability and we have the appearance of one repelling limit cycle. In the other part, the change of the stability gives the appearance of one attracting limit cycle. When the parameter value crosses H2we have the appearance of two limit cycles, the inner one is attracting and the other is repelling. If (µ, ν0,ν 1)∈L(homoclinic bifurcation) then Xµ,ν0,ν1has a homoclinic orbit. The parameter values on Lfor which ∂P ∂x +∂Q ∂y (√−µ, 0)=0 give the curve L2.Lis composed of two parts separated by the curve L2 (homoclinic bifurcation of codimension 2). In one of them, the attracting limit cycles which appear along Hdisappear along L. In the other part, the repelling limit cycles which appear along Hdisappear along L. The cycles which appear along H2disappear along L2. There exists a surface Cwhere the attracting limit cycle and the repelling limit cycle coalesce. If (µ, ν0,ν 1)∈Cthen Xµ,ν0,ν1has a semistable cycle. For the parameter values on the region limited by H,Land C,Xµ,ν0,ν1has two limit cycles. On b1(δ) and b2(δ) we have the Bogdanov-Takens bifurcations corresponding to the cases (5+) and (5−). For the parameter values on H∩Lwe have simultaneous Hopf bifurcation and homoclinic bifurcation. ν1 µ ν0 b1 LH H2 C L2 b2 Figure 1. The Bifurcation Set on Sδ. In this work we study the surfaces Hand Lfor µ→−∞. In section 2 we make some coordinate changes, such as the Li´enard transformation, and prove that if µ→−∞then (8+) is (fibre) C0-equivalent to (9) [y−Fb,c(x)] ∂ ∂x +ε(x2−2x)∂ ∂y for ε↓0, b,c∈R. Canard cycles and homoclinic bifurcation 167 We denote Fb,c the function given by (10) Fb,c(x)=bx +cx2−4x3+x4. In section 2 we study the bifurcation set of the functions defined by (10). It is illustrated in Figure 2. b c6 4 Figure 2. The Bifurcation Set of Fb,c. 168 P. R. da Silva In section 3 we study the singularities of the family (9). If ε= 0 then the singularities, with (b, c) fixed, are on the curve (11) Lb,c ={(x, y)|y=Fb,c(x)}. Except for the critical points, all the points on Lb,c are normally hyperbolic singular points. The phase portrait is illustrated in Figure 3. Figure 3. Phase Portrait of (9) for (ε, b, c)=(0,0,12) and for (ε, b, c)=(0,0,4). For ε>0, (9) has two singularities: (0,0) and (2,F b,c(2)). 0<ε≤b2/8ε>b 2/8 (0,0) b>0 Attracting Node Attracting Focus b<0 Repelling Node Repelling Focus (2,F b,c(2)) Saddle Saddle Table I. Singularities of (9) for ε>0. We have a special care to classify the singularity (0,0) when b=0 because in this case the associated eigenvalues have real part equal zero. We prove in section 3 the existence of Hopf bifurcation on H={b=0,c=12}and Hopf bifurcation of codimension 2 on H2= {b=0,c=12}. The cycles are so that inner one is repelling and the other is attracting. We study the behaviour of the cycles for ε↓0. Definition 1.3. a) Let A,B⊂R2compact sets. The Hausdorff distance between A and Bis d(A, B) = max x∈A, y∈B{d(x, B),d(y,A)}. b) Γb,c ⊂R2is a “Limit Periodic Set” of Xε,b,c if there exist (εn,b n,c n)→(0,b,c) and Γn⊂R2, limit cycle (or homoclinic orbit) of Xεn,bn,cn, such that Γn→Γb,c, for the Hausdorff distance between the compact sets in R2. Canard cycles and homoclinic bifurcation 169 H H2 12 bc Figure 4. Hopf Bifurcation. For (ε, b, c)→(0,0,12) the possible limit periodic sets are those for which yis constant or y=Fb,c(x). The limit periodic sets are known by “Canard Cycles”. We use the term “canard” because the shape of the limit periodic sets of the Van der Pol’s equation is like a duck ([E]). 2 y32 Γ0,4Γy 0,12 Γ32 0,12 22 Figure 5. Limit Periodic Sets. Using the method introduced in [DR1], which consists in global desingularization and center manifolds, we prove Theorem 1.1. Let the family given by (9) and Γ32 0,12 the compact set illustrated in Figure 5. There exist ε0>0,δ>0and a surface S={(ε, b, c)|b=ϕ(√ε, c),ϕ∈C∞,ϕ(0,c)=0,0≤ε<ε 0,12 −δ< c<12 + δ}such that if (ε, b, c)∈Sand ε>0then two separatrices of the saddle point (2,F b,c(2)) give a homoclinic orbit which approaches Γ32 0,12 for the Hausdorff distance. With the same method one can find surfaces Sy,0≤y≤32, such that if (ε, b, c)∈Sythen the phase portrait of (9) has a limit cycle Γy ε,b,c and Γy ε,b,c →Γy 0,12, for the Hausdorff distance, for (ε, b, c)→(0,0,12). 170 P. R. da Silva The divergence at (2,F b,c(2)) determines the stability of the homoclinic orbit. We have (12) div Xε,b,c(2,F b,c(2)) = 0 ⇔b=−4c+16. We say that the homoclinic orbit is non-generic if the divergence at (2,F b,c(2)) is zero. Theorem 1.2. Let the family given by (9) and Γ0,4the compact set illustrated in Figure 5. There exist ε0>0,δ>0and a surface L= {(ε, b, c)|b=ψ(√ε, c),ψ∈C∞,ψ(0,c)=0,0≤ε<ε 0,4−δ<c< 4+δ}such that if (ε, b, c)∈Land ε>0then two separatrices of the saddle point (2,F b,c(2)) give a homoclinic orbit Lε,b,c which approaches Γ0,4. Besides the limit of the parameter values for which Xε,b,c has nongeneric homoclinic orbits for ε↓0is (0,0,4). The surface Lis such that for b<−4c+ 16, Lε,b,c is a repelling homoclinic orbit and for b>−4c+ 16, Lε,b,c is an attracting homoclinic orbit. According [ALGM], if b=−4c+ 16 and (ε, b, c)∈Lthen for any δ1>0 and δ2>0, there exists a perturbation  Xε,b,c of Xε,b,c,δ1-closed to Xε,b,c, such that  Xε,b,c has at least two limit cycles, δ2-closed to Lε,b,c. We prove the Theorems 1.1 and 1.2 in sections 5 and 6. In section 4 we prove that the parameter values for which Xε,b,c has limit cycles approache {ε=b=0}, for ε↓0. 2. The Li´enard Transformation Let Xµ,ν0,ν1given by (8). We start with the transformation (13) x=t2x1ν0=t6ν 0µ=−t4 y=t3y1ν1=t4ν 1T=t5. Divide by tto get (14) y1 ∂ ∂x1 +[x2 1−1+Ty1ν 0+ν 1x1+x3 1]∂ ∂y1 . Next, use the Li´enard transformation (15) Ty2=T(ν 0+ν 1x1+x3 1)dx1−y1 x2=x1. Canard cycles and homoclinic bifurcation 171 Thus we have (16) Tν 0x2+ν 1 2x2+x4 2 4−Ty2∂ ∂x2−1 Tx2 2−1∂ ∂y2 . The Li´enard transformation (15) is a diffeomorphism from (x1,y 1)- plane to (x2,y 2)-plane. Multiply (16) by −1 Tand denote ε=1 T2 (17) y2−1 44ν 0+2ν 1x2 2+x4 2∂ ∂x2 +εx2 2−1∂ ∂y2 . Multiply by 4 and put y3=4y2,ν 0=4ν 0,ν 1=2ν 1,ε=1 16 ε (18) y3−ν 0x3+ν 1x2 3+x4 3 ∂ ∂x3 +εx2 3−1∂ ∂y3 . To simplify the desingularization, make the change (19) x=x3+1 b=ν 0−2ν 1 y=y3−1+ν 0−ν 1c=ν 1−6 and denote xand yagain to get (9). Proposition 2.1. Fb,c :R→Rgiven by (10) satisfies: a) If b=0then Fb,c has one zero for c>4,2zeroes for c=4or c=0and 3zeroes for c<0or 0<c<4. b) The equation (20) 3c−16 93 +−36c+ 128 −27b 54 2 =0 defines a cusp on the (b, c)-plane which is tangent to {b=0} at (b, c)=(0,0), crosses {b=0}at (b, c)=(0,4) and crosses {c=0}at (b, c)=(9.48,... ,0). The cusp and {b=0}divide the (b, c)-plane defining the number of zeroes of Fb,c. Proof: The equation (20) is D= 0 with Dthe discriminant of Fb,c(x) x=0. 178 P. R. da Silva There exists  Xon [0,T]×Ksuch that ϕ∗( X)=Xε,b,c. Dividing  X by uthe vector field resulting Xwill be “the desingularized vector field” in [0,T]×K. (48) Xu,b,c =(y−bx−ucx2−12x2+4ux3−u2x4)∂ ∂x +ε(ux2−2x)∂ ∂y . Let M0={(x, y)}×{(ε, b, c)}and T1=[0,T]×K. We define M1= (T1∪M0−{0})/(m∼ϕ(m)). Let φ:M1→M0such that φ|(K×{0})= ϕand φ|M0−{0}= id. Consider π:M0→{(ε, b, c)}given by π(x, y, ε, b, c)=(ε, b, c) and π:M1→{(ε, b, c)}by π(x, y, ε, b, c)= π(φ(x, y, ε, b, c)). Let λ0=(ε0, b0, c0)∈S2,lλ0={(u2ε0,ub0,uc0)|u>0}and Pλ0= π−1(lλ0). We have that Pλ0is a 3-dimensional space. b c 1 S1×R vu D Figure 6. Foliation of M1. Define  Xon M1by (49)  X=Xε,b,c on M1−{0} Xon T1. Canard cycles and homoclinic bifurcation 179 On S1×Rwe have ε=b=c= 0. Make the change (50) x=ucos θy=u2sin θ and denote c= cos θand s= sin θto get (51) uc c2+2s2s−12c2+4uc2−u2c4∂ ∂u −2s c2+2s2s−12c2+4uc2−u2c4∂ ∂θ. The singularities of (51) are (0,0), (π,0), (θ1,0) and (θ2,0) with θ1 and θ2solutions of s=12c2. We have θ1∈(0,π 2) and θ2∈(π 2,π). Multiplying (51) by (c2+2s2)wehave (52) X(θ,u)=−2s(s−12c2+4uc2−u2c4)∂ ∂θ +uc s−12c2+4uc2−u2c4∂ ∂u DX(θ,0) = −4sc +24c3−48s2c−8c2s 0sc −12c3 DX(0,0) = 24 0 0−12 DX(π,0) = −24 0 012 .(53) For (θ,u)=(0,0) the angular eigenvalue is positive and the radial eigenvalue is negative. For (θ,u)=(π,0) the angular eigenvalue is negative and the radial eigenvalue is positive. If θ=θ1(θ=θ2) the angular eigenvalue −4sc +24c3−48s2c= −2sc −48s2cis negative (positive) and the radial eigenvalue is zero. Thus we have (54) DX(θ1,0) = <0=0 00 DX(θ2,0) = >0=0 00 . 180 P. R. da Silva When ε= 1 and u= 0 in (48) we have (55) Xb,c =(y−bx −12x2)∂ ∂x −2x∂ ∂y Figure 7. Phase Portraits of  Xfor b>0, b= 0 and b<0. Xb,cindepends of cand when b=0,Xb,c has a center in (x, y)=(0,0). (56) X0=(y−12x2)∂ ∂x −2x∂ ∂y . If we take ε=1,b=b0>0, u=0orε=1,b=b1<0, u= 0 the phase portraits are illustrated in Figure 7. 5.2. The desingularization at (x, y, ε, b, c)=(2,0,0,0,4).In this case F0,4(2) = 0. We start with the change (57) x1=x−2b1=−b−4c+16 y1=y+2b−4c+16 c1=c−4. Denote x,y,b,cto get (58) (y+bx −cx2−4x2−4x3−x4)∂ ∂x +ε(x2+2x)∂ ∂y. Consider the map given by (47) and ε=1 (59) (y+bx −4x2−4ux3−u2x4)∂ ∂x +(2 x+ux2)∂ ∂y . If b=c=u= 0 then (60) (y−4x2)∂ ∂x +2 x∂ ∂y Canard cycles and homoclinic bifurcation 181 has a singular point (0,0) and (61) DX(0,0) = 01 20 . Thus (0,0) is a saddle point. Now we use the same steps used in (50), (51), (52), (53) and (54): −2us c2+2s2(s−4c2−4uc3−u2c4)∂ ∂θ (62) +u2c c2+2s2(s−4c2−4uc3−u2c4)∂ ∂u −2s(s−4c2−4uc3−u2c4)∂ ∂θ (63) +uc(s−4c2−4uc3−u2c4)∂ ∂u DX(θ,u)=−4sc −16s2c+8c38sc3 0sc −4c3 (64) DX(π,0) = −80 04 DX(0,0) = 80 0−4 (65) DX(ϕ1,0) = <0=0 00 DX(ϕ2,0) = >0=0 00  (66) for ϕ1and ϕ2solutions of sin ϕ= 4 cos ϕ. 5.3. The desingularization at (x, y, ε, b, c)=(1,1,0,0,4).We start with the change (67) x1=x−1b1=−b−2c+8 y1=y−b−c+3 c1=c−4. Denote x,y,b,cto get (68) (y+bx −cx2+2x2−x4)∂ ∂x +ε(x2−1) ∂ ∂y. Consider the map δ:[0,T]×S4 +→R5given by (69) x=uxb=ubε=u3ε y=u2yc=uc. 182 P. R. da Silva Make X=1 uXand take ε=1 (70) (y+bx −ucx2+2x2−u2x4)∂ ∂x +(u2x2−1) ∂ ∂y . For b=c=u=0wehave (71) (y+2x2)∂ ∂x −∂ ∂y . It has no singular point. Using again the same steps we have −2s (c2+2s2)(s+2c2−u2c4)∂ ∂θ (72) +uc (c2+2s2)(s+2c2−u2c4)∂ ∂u −2s(s+2c2−u2c4)∂ ∂θ +uc(s+2c2−u2c4)∂ ∂u (73) DX(θ,0) = −4sc −4c3+8cs20 0sc +2c3 (74) DX(0,0) = −40 02 DX(π,0) = 40 0−2 (75) DX(θ1,0) = <00 00 DX(θ2,0) = >00 00  (76) for θ1and θ2solutions of s+2c2=0. Figure 8. The Desingularization at (1,1,0,0,4) and at (2,0,0,0,4). Canard cycles and homoclinic bifurcation 183 Figure 9. Simultaneous Desingularization. 6. The Canard Phenomenon 6.1. Proof of Theorem 1.1. To simplify the calculations we use the change (45). First we prove that there exists a hamiltonian Hwith integrating factor Ksuch that KX0=dH,X0given by (56). Let the change (77) X=xY=y−12x2. Thus (56) becomes (78) Y∂ ∂X +(−2X−24XY)∂ ∂Y . Let F(X,Y ) given by (79) F(X,Y ) = 144X2+(1+12Y)−ln(1 + 12Y). We have that F(X, Y ) is a first integral of (78). In fact (80) (FX,F Y)(Y,−2X−24XY )=0. Let G(x, y) and H(x, y) given by (81) G(x, y)=(1+12y)−ln(1 + 12y−144x2) H(x, y) = exp(1 −12y)(1+12y−144x2). 184 P. R. da Silva We have (82) Hx=−(144 exp(1 −12y))(2x) Hy=−(144 exp(1 −12y))(y−12x2). Thus (83) K(x, y)=−144 exp(1 −12y) is the integrating factor and the hamiltonian His given by (81). Now we consider P(1,0,0) defined in section 5. We have that P(1,0,0) is a 3-dimensional space and we can look a point in P(1,0,0) with coordinates (u, v, θ), indicates in the Figure 6. c+ b,c c− b,c Figure 10. Center Manifold CN. For u= 0, we denote Dthe set associated to v= 0. Let Ra rectangle on P(1,0,0) with one side ron Dand such that Ris transversal to ∂P.We take Rsuch that the connection cjoing (θ1,0) and (θ2,0) is transversal to rat its middle point. Let Sa subrectangle in Rwith one side s on Dand such that c∩ris the middle of s. Let N={(2,F b,c(2)) | b, c ∈R, 0≤ε≤ε0}composed by saddle points of Xε,b,c. We take CN, the sature of N, that is, the closure of the union of segments of orbits of X(u,v,θ)through the points on Nand taken between the first intersection of this trajectory with Sin negative time and with Rin positive time. Let wu,b,c the dual 1-form associated to (48) (84) wu,b,c =y−bx −ucx2−12x2+4ux3−u2x4dy −ux2−2xdx. Canard cycles and homoclinic bifurcation 185 We have (85) −144e(1−12y)wu,b,c =dH(x, y)−M(x, y) M(x, y)=144e(1−12y)[−bx−ucx2+4ux3−u2x4dy−ux2dx]. Consider the section [α, β] joing the center α=(0,0) and β=(0,−1 12 ) in c. We parametrize [α, β] by the value of the hamiltonian H. Let Ch=H−1(h) for h∈(0,e)(H(0,0) = e,H(0,−1 12 ) = 0). The assymptotic development of Pu,b,c(Poincar´e Mapping), using the Perturbation Lemma ([DRS1]), is given by (86) Pu,b,c(h)=h+bI1(h)+cI2(h)+uI3(h)+o(u, b, c) with (87) I1(h) = 144  Ch e(1−12y)xdy I2(h) = 144u Ch e(1−12y)x2dy =0 I3(h)=−576  Ch e(1−12y)x3dy + 144  Ch e(1−12y)x2dx. If u=h= 0 the development of the separation between C+ b,c and C− b,c is given by (88) ∆(b, c)=bI1(0). If {h=f(u, b, c)}and {h=g(u, b, c)}represent the intersections of CN(b, c) with Rthen we can extend (88) (89) f(u, b, c)−g(u, b, c)=bI1(0) + uI3(0) + o(u, b, c). Let ∆(u, b, c) given by (90) ∆(u, b, c)=f(u, b, c)−g(u, b, c). For h= 0 we have K(x, y)=−144e(2−144x2)and dy =24xdx.By integrating by parts we have I1(0) =0,I 2(0) = 0 and I3(0) = 0. Thus the equations (91) ∆ = 0 ∂∆ ∂b =0 define S={b=ϕ(c, u)}, the Canard Surface. 186 P. R. da Silva The homoclinic orbits γassociated to the parameter values on Sare attracting because the parameter is near of (0,0,12) (see (12)). In order to study the stability of the limit cycles γassociated to the parameter values on S32 we must to compute γdiv Xε,b,c. We aproach the integral (92) γ div Xε,b,c ≃ 2  −1.3013 (b+2cx −12x2+4x3)(1 + (24x−12x2+4x3)2+1)dx. The computation, using maple, gives (93) S1= subs(b=0.0001,c=12.0001) ≃−518.12 S2= subs(b=0.0001,c=11.9999) ≃−523.55 S3= subs(b=−0.0001,c=11.9999) ≃−537.06 S4= subs(b=−0.0001,c=12.0001) ≃−531.63. The values given by (93) attest that the limit cycles are attracting for the parameter values (ε, b, c)∈S32. Remark 6.1. The “Canard phenomenon” consists in a rapid variation of the shape of the periodic orbits in function of the variation of the parameter. According Theorems 1.1 and 3.1 one can find Xε1,b1,c1and Xε2,b2,c2two perturbations of X0,0,12 such that: a) The phase portrait of Xε1,b1,c1has two limit cycles contained in a small neighbourhood of (0,0), the inner one is repelling and the other is attracting. b) The phase portrait of Xε2,b2,c2has an attracting limit cycle contained in a small neighbourhood of Γ32 0,12 (see Figure 5). Proof of Theorem 1.2. The surface Lis obtained implicitly of the same way that canard surface. To make it we consider γ= {(2,F b,c(2)) |b, c ∈R, 0≤ε≤ε0}composed by saddle points of Xε,b,c. Saturing γby the flow of Xε,b,c we have that the intersections of the manifolds define the homoclinic bifurcation surface. Let l(ε, b, c), the homoclinic loop associated to the parameters (ε, b, c). There is not limit of l(ε, b, c), for ε↓0 and c<4, for the Hausdorff distance. In fact, the unstable separatrice does not touch the slow manifold for ε↓0. Canard cycles and homoclinic bifurcation 187 2 Figure 11. Non Existence of Limit Homoclinic Loops in {b=0,c<4}. The parameter values for which Xε,b,c has limit cycles aproache {b=0} for ε↓0. Thus, the surface Laproaches {b=0}, for ε↓0, and the parameter values for which Xε,b,c has non-generic homoclinic loops aproach (ε, b, c)=(0,0,4), for ε↓0. We need to precise the codimension of the non-generic homoclinic loops. We start with the desingularization of Xε,b,c in (x, y, ε, b, c)= (0,0,0,0,4). Let the change (94) c1=c−4. Denote cand make the change (47) to get (95) (y−bx −4x2−ucx2+4ux3−u2x4)∂ ∂x +(ux2−2x)∂ ∂y . H(x, y) and K(x, y) given by (96) H(x, y) = (exp(1 −4y))(1 + 4y−16x2)K(x, y) = 16 exp(1 −4y) satisfy (97) KX0=−Hy ∂ ∂x +Hx ∂ ∂y . Let the map (98) R(h)=bI1(h)+ucI2(h)+uI3(h)+o(u, b, c) with I1,I2and I3given by (99) I1(h)=−ch Kxdy I2(h)=−ch Kx2dy =0 I3(h)=ch 4Kx3dy −ch Kx2dx.