Ordinary differential equations Definition

Bogdanov–Takens bifurcation

Builds on Equilibrium point Jacobian matrix Eigenvalue Smooth function

Definition

Let $f \colon \R^2 \times \R^2 \to \R^2$ be smooth, and consider the family of planar systems $x' = f(x, \mu)$ with a parameter $\mu \in \R^2$ . Let $x_0$ be an equilibrium point of the system with $\mu = \mu_0$ , and let $A$ be the Jacobian matrix of $x \mapsto f(x, \mu_0)$ at $x_0$ . The family has a Bogdanov–Takens bifurcation at $(x_0, \mu_0)$ if:

$0$ is an eigenvalue of $A$ of algebraic multiplicity $2$ , that is, $\tr A = 0$ and $\det A = 0$ ;
$A \ne 0$ , so that $A$ is similar to $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ ;
two nondegeneracy conditions on the quadratic terms of $f(\cdot, \mu_0)$ at $x_0$ hold;
the map from $\mu$ to the unfolding parameters of the normal form is a local diffeomorphism at $\mu_0$ .

Remarks

Under these conditions, the family is locally topologically equivalent, near $(x_0, \mu_0)$ , to the normal form $y_1' = y_2, \qquad y_2' = \beta_1 + \beta_2 y_1 + y_1^2 + s\, y_1 y_2, \qquad s = \pm 1,$ with parameters $(\beta_1, \beta_2)$ near $0$ . In the $(\beta_1, \beta_2)$ -plane, a saddle-node curve, a Hopf curve and a homoclinic bifurcation curve start at the origin.

Example: the normal form itself at $\beta = 0$ has the equilibrium $0$ with Jacobian $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ , so clauses 1 and 2 hold.

Non-example: if $A = 0$ , then $0$ is a double eigenvalue but clause 2 fails; this degenerate case needs more parameters to unfold. In higher dimensions, the same definition applies to the system reduced to a two-dimensional centre manifold.

Tag	`H3GWX277EF`
URI	`m2://H3GWX277EF`
Kind	definition
Writing	W0 stub
MSC	34C23, 37G10
Defines	Bogdanov–Takens bifurcation
Uses	Equilibrium point, Jacobian matrix, Eigenvalue, Smooth function
Changed	2 November 2026

Bogdanov–Takens bifurcation

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