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

Abstract

Let the 3-parameter family of vector fields given by $$ y\frac \partial {\partial x}+[x^2+\mu +y(\nu _0+\nu _1x+x^3)]\frac \partial {\partial y}\tag(A) $$ with $(x,y,\mu ,\nu _0,\nu _1)\in R^2\times R^3$ (\cite{DRS1}). We prove that if $\mu \rightarrow -\infty $ then (A) is $C^0$-equivalent to $$ \lbrack y-(bx+cx^2-4x^3+x^4)]\frac \partial {\partial x}+\varepsilon (x^2-2x)\frac \partial {\partial y}\tag(B) $$ for $\varepsilon \downarrow 0$, $b$, $c\in R$. We prove that there exists a Hopf bifurcation of codimension 1 when $b=0$ and also that, if $b=0$, $c=12$ and $\varepsilon > 0$ 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 $\varepsilon \downarrow 0$, $b=0$ 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.