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  "body": "\n## Definition\n\nLet $f \\colon \\R^2 \\times \\R^2 \\to \\R^2$ be [[smooth|0V33BNKZ9S]], and consider the family of planar systems $x' = f(x, \\mu)$ with a parameter $\\mu \\in \\R^2$.\nLet $x_0$ be an [[equilibrium point|0G4X36AE0X]] of the system with $\\mu = \\mu_0$, and let $A$ be the [[Jacobian matrix|047NDTEPCV]] of $x \\mapsto f(x, \\mu_0)$ at $x_0$.\nThe family has a *Bogdanov–Takens bifurcation* at $(x_0, \\mu_0)$ if:\n\n1. $0$ is an [[eigenvalue|0BQAA2BK3Q]] of $A$ of algebraic multiplicity $2$, that is, $\\tr A = 0$ and $\\det A = 0$;\n2. $A \\ne 0$, so that $A$ is similar to $\\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$;\n3. two nondegeneracy conditions on the quadratic terms of $f(\\cdot, \\mu_0)$ at $x_0$ hold;\n4. the map from $\\mu$ to the unfolding parameters of the normal form is a local diffeomorphism at $\\mu_0$.\n\n## Remarks\n\nUnder these conditions, the family is locally topologically equivalent, near $(x_0, \\mu_0)$, to the normal form\n$$y_1' = y_2, \\qquad y_2' = \\beta_1 + \\beta_2 y_1 + y_1^2 + s\\, y_1 y_2, \\qquad s = \\pm 1,$$\nwith parameters $(\\beta_1, \\beta_2)$ near $0$.\nIn the $(\\beta_1, \\beta_2)$-plane, a saddle-node curve, a Hopf curve and a homoclinic bifurcation curve start at the origin.\n\nExample: the normal form itself at $\\beta = 0$ has the equilibrium $0$ with Jacobian $\\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$, so clauses 1 and 2 hold.\n\nNon-example: if $A = 0$, then $0$ is a double eigenvalue but clause 2 fails; this degenerate case needs more parameters to unfold.\nIn higher dimensions, the same definition applies to the system reduced to a two-dimensional centre manifold.\n",
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