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 + [x2 + µ + y(ν0 + ν1x + x3)] ∂ ∂y with (x, y, µ, ν0, ν1) ∈ 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