A Global Analysis of the Bogdanov–Takens System

Source abstract

This paper studies the bifurcations that occur in the Bogdanov–Takens system $$\dot x = y,\qquad \dot y = x(x - 1) + \mu _1 y + \mu _2 xy,$$ both in the phase plane ${\bf R}^2$ and at infinity, i.e., on the Poincaré sphere ${\bf S}^2$, for all parameter values $(\mu _1 ,\mu _2 ) \in {\bf R}^2$. For all $\mu _2 \ne 0$, a unique limit cycle bifurcates from the origin at $\mu _1 = 0$, and it expands monotonically until it intersects the saddle at $(1, 0)$ and forms a homoclinic loop at $\mu _1 = h(\mu _2 )$. The function $h(\mu _2 )$ describing the homoclinic loop bifurcation curve in ${\bf R}^2$ is analytic for all $\mu _2 \in {\bf R},\, h(0) = 0,\,h'(0) = - 1/7$, and $h( \pm \infty ) = 0$. The only other bifurcation for this system is a saddle-saddle connection on ${\bf S}^2$, which occurs when $\mu _1 = - 1/\mu _2$. Even for $\mu _1 (\mu _1 \mu _2 + 1)(\mu _1 - h(\mu _2 )) \ne 0$, this system is not structurally stable on ${\bf S}^2$ due to the presence of nonhyperbolic critical points on the equator of ${\bf S}^2$; however, this system can be perturbed to a quadratic system that is structurally stable on ${\bf S}^2$, except on certain codimension-one bifurcation surfaces.