CANARD CYCLES AND HOMOCLINIC BIFURCATION IN A 3 PARAMETER FAMILY OF VECTOR FIELDS ON THE PLANE

Let the 3-parameter family of vector fields given by (A) y ∂ ∂x + [x + μ + y(ν0 + ν1x + x )] ∂ ∂y with (x, y, μ, ν0, ν1) ∈ R2 × R3 ([DRS1]). We prove that if μ → −∞ then (A) is C0-equivalent to (B) [y − (bx + cx − 4x + x)] ∂ ∂x + ε(x − 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.


Introduction
Let χ the set of C ∞ vector fields on the plane and χ 0 ⊂ χ, the set of vector fields with a singularity at (0, 0).
Consider the equivalences introduced by the following definitions.
a) X, Y ∈ χ are called C 0 -equivalent if there exists a homeomorphism h : R 2 → R 2 sending X-orbits to Y -orbits in a sense preserving way.b) X, Y ∈ χ 0 are called C ∞ -equivalent if there exists a diffeomorphism g : R 2 → R 2 , fixing (0, 0 Definition 1.2. a) A k-parameter family of vector fields on R 2 , X λ , with λ ∈ R k denoting the parameter, is defined to be a vector field a(m, λ) ∂ ∂x + b(m, λ) ∂ ∂y , m = (x, y) ∈ R 2 , where the coefficient functions a and b are C ∞ with respect to (m, λ) ∈ R 2 × R k .We say that X λ is a k-parameter unfolding of X 0 .b) X λ and Y µ , µ, λ ∈ R k , are (fibre) C 0 -equivalent if there exist homeomorphisms µ = φ(λ) and h λ : R 2 → R 2 such that h λ is a topological equivalence between X λ and Y φ(λ) .
. W 1 is 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 k which is an unfolding of X 0 ∈ W 2 but 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 k is a k-parameter unfolding of X 0 ∈ W 3 and 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 X 0 ∈ W 4 starts with the Bogdanov-Takens bifurcation ( [RW]).If X 0 ∈ W 4 then, according [DRS1], If a = 0 we say that X 0 ∈ W 4 has singularity of the kind cusp.For the case a = 0, see [DRS2].
Figure 1 shows the intersection of the bifurcation set with the sphere H is composed of two parts separated by the curve H 2 (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 H 2 we have the appearance of two limit cycles, the inner one is attracting and the other is repelling.If (µ, ν 0 , ν 1 ) ∈ L (homoclinic bifurcation) then X µ,ν0,ν1 has a homoclinic orbit.The parameter values on L for which ∂P ∂x + ∂Q ∂y ( √ −µ, 0) = 0 give the curve L 2 .L is composed of two parts separated by the curve L 2 (homoclinic bifurcation of codimension 2).In one of them, the attracting limit cycles which appear along H disappear along L. In the other part, the repelling limit cycles which appear along H disappear along L.
The cycles which appear along H 2 disappear along L 2 .There exists a surface C where the attracting limit cycle and the repelling limit cycle coalesce.If (µ, ν 0 , ν 1 ) ∈ C then X µ,ν0,ν1 has a semistable cycle.For the parameter values on the region limited by H, L and C, X µ,ν0,ν1 has two limit cycles.On b 1 (δ) and b 2 (δ) we have the Bogdanov-Takens bifurcations corresponding to the cases (5 + ) and (5 − ).For the parameter values on H ∩ L we have simultaneous Hopf bifurcation and homoclinic bifurcation.
In this work we study the surfaces H and L for µ → −∞.
In section 2 we make some coordinate changes, such as the Liénard transformation, and prove that if µ → −∞ then (8 We denote F b,c the function given by ( 10) In section 2 we study the bifurcation set of the functions defined by (10).It is illustrated in Figure 2. 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 Except for the critical points, all the points on L b,c are normally hyperbolic singular points.The phase portrait is illustrated in Figure 3. and for (ε, b, c) = (0, 0, 4).
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 H 2 = {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.
) and Γ n ⊂ R 2 , limit cycle (or homoclinic orbit) of X εn,bn,cn , such that Γ n → Γ b,c , for the Hausdorff distance between the compact sets in R 2 .For (ε, b, c) → (0, 0, 12) the possible limit periodic sets are those for which y is constant or y = F b,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]).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 Γ 32 0,12 the compact set illustrated in Figure 5.There exist ε 0 > 0, δ > 0 and a surface give a homoclinic orbit which approaches Γ 32 0,12 for the Hausdorff distance.
The divergence at (2, F b,c (2)) determines the stability of the homoclinic orbit.We have 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 Γ 0,4 the compact set illustrated in Figure 5.There exist give a homoclinic orbit L ε,b,c which approaches Γ 0,4 .Besides the limit of the parameter values for which X ε,b,c has nongeneric homoclinic orbits for ε ↓ 0 is (0, 0, 4).
The surface L is such that for b < −4c + 16, L ε,b,c is a repelling homoclinic orbit and for b > −4c + 16, L ε,b,c is an attracting homoclinic orbit.
Multiply ( 16) by −1 T and denote ε = 1 Multiply by 4 and put To simplify the desingularization, make the change and denote x and y again to get (9).Proof: The equation ( 21) is Figure 2 shows two sets of bifurcation: one defined by the equality of critical points and other by the equality of zeroes.

The Singularities of (9)
If ε = 0, the singularities of the vector field X ε,b,c , given by ( 9), are the points (0, 0) and ( 2 The equation of the eigenvalues is For ε > 0 the singularity (2, 2b + 4c − 16) is a saddle.The saddle is a critical point of F b,c for the parameter values in the plane given by (12).Theorem 3.1.Let X ε,b,c given by ( 9).We have: Proof: Make the change and denote y = y 1 to get Make the change The phase portraits of ( 30) are the graphics of We have Thus, according [ALGM], we get (36) Define the returning map π : R + → R by ( 37) The Lyapunov coefficients are given by ( 38) The computation, using maple, gives When b = 0 we have V 1 = V 2 = 0 and V 3 = 0 if and only if c = 12.Besides V 3 > 0 for c > 12 and V 3 < 0 for c < 12. Thus X ε,b,c has a weakly attracting focus for c > 12 and a weakly repelling focus for c < 12.For c = 12 we have (40) Thus, (0, 0) is a weakly attracting focus.

The Phase Portrait of X ε,b,c
First we are going to study the phase portrait of X ε,b,c given by ( 9) when ε = 0.In this case we have For b and c fixed the singularities of ( 41) are the points (x, y) such that y = F b,c (x).Except for the the critical points, all the points are normally hyperbolic singular points.Figure 3 shows the phase portrait for some values of (b, c).
For ε = 0, when x = 0 and x = 2 cross transversally the graphic of y = F b,c (x) in non-critical points we can use the R. Lutz and M. Goze lemma ( [LG]) to sketch the phase portrait of X ε,b,c .The trajectories are so that ( 42) Thus the vector field is almost horizontal for ε ↓ 0.
The main difficulty to obtain the whole phase portrait is the number of limit cycles.

Lemma 4.1 (Copell). Let the vector field given by
Then there is no limit cycle in ξ 0 < x < β.
The proof of the Copell's Lemma uses a symmetry in the line where the divergence is zero ( [DR2]).
Consider b = 0 and c ∈ R and suppose that X ε,b,c has a limit cycle Γ ε , then either the limit cycle disappear or the limit periodic set will be (0, 0) for ε ↓ 0. In fact, if b = 0 the singularity (0, 0) is not a critical point.

The Desingularization of
given by ( 47) There exists X on [0, T ] × K such that ϕ * ( X) = X ε,b,c .Dividing X by u the vector field resulting X will be "the desingularized vector field" in [0, T ] × K.

The desingularization at
Consider the map δ : [0, T ] × S 4 + → R 5 given by ( 69) For b = c = u = 0 we have It has no singular point.Using again the same steps we have for θ 1 and θ 2 solutions of s + 2c 2 = 0.
Let the change Thus ( 56) becomes Let F (X, Y ) given by ( 79) We have that F (X, Y ) is a first integral of (78).In fact Let G(x, y) and H(x, y) given by ( 81) We have ( 82) is the integrating factor and the hamiltonian H is 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.For u = 0, we denote D the set associated to v = 0. Let R a rectangle on P (1,0,0) with one side r on D and such that R is transversal to ∂P .We take R such that the connection c joing (θ 1 , 0) and (θ 2 , 0) is transversal to r at its middle point.Let S a subrectangle in R with one side s on D and such that c ∩ r is the middle of s.
. We take C N , the sature of N , that is, the closure of the union of segments of orbits of X (u,v,θ) through the points on N and taken between the first intersection of this trajectory with S in negative time and with R in positive time.
Let w u,b,c the dual 1-form associated to (48 The homoclinic orbits γ associated to the parameter values on S are attracting because the parameter is near of (0, 0, 12) (see ( 12)).In order to study the stability of the limit cycles γ associated to the parameter values on S 32 we must to compute γ div X ε,b,c .
We aproach the integral (92) The computation, using maple, gives The values given by (93) attest that the limit cycles are attracting for the parameter values (ε, b, c) ∈ S 32 .
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 X ε1,b1,c1 and X ε2,b2,c2 two perturbations of X 0,0,12 such that: a) The phase portrait of X ε1,b1,c1 has 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 X ε2,b2,c2 has an attracting limit cycle contained in a small neighbourhood of Γ 32 0,12 (see Figure 5).
Proof of Theorem 1.2.The surface L is 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 ≤ ε 0 } composed by saddle points of X ε,b,c .Saturing γ by the flow of X ε,b,c we have that the intersections of the manifolds define the homoclinic bifurcation surface.
We need to precise the codimension of the non-generic homoclinic loops.We start with the desingularization of X ε,b,c in (x, y, ε, b, c) = (0, 0, 0, 0, 4).Let the change The equation R(h) = 0 defines the surface L and besides R (h) is finite if and only if the divergence at the saddle point is zero [R].Thus R (h) is finite if and only if b = −4c.To prove that the homoclinic loop is of codimension 2 we need to prove that R (h) = 0. We use the map R(h) = b + u I3(h)  I1(h) and we have (100) R (h) = u I 3 (h) I 1 (h) .
With a similar argument used in [CW], one can prove that (101) I 3 (h) I 1 (h) < 0 for h < e.It follows that R (h) < 0.
c = 0 and b = 0 or a = c = 0 and b = 1

Figure 2 .
Figure 2. The Bifurcation Set of F b,c .