Codimension-2 bifurcation: saddle-node meets Hopf — the full unfolding
Parameters (μ₁, μ₂)
Unfolding
ẋ = y
ẏ = μ₁ + μ₂y + x² + xy
Normal form of codim-2 BT singularity. The parameter space has 4 regions separated by:
Saddle-node curve: μ₁=0
Hopf curve: μ₁ = -μ₂²/4
Homoclinic orbit (Shilnikov)
BT point: origin (μ₁=μ₂=0)
Current Region
About BT Bifurcation
The Bogdanov-Takens point is a fixed point with a double-zero eigenvalue (nilpotent Jacobian). It is the organizing center for:
• Saddle-node bifurcation
• Hopf bifurcation
• Homoclinic bifurcation
All three are captured in this single 2-parameter unfolding — a beautiful example of how higher-codimension singularities govern lower-codimension behavior.