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  "body": "\n## Definition\n\nLet $M$ be a compact smooth manifold without boundary, such as a circle, a sphere or a torus.\nLet $\\mathfrak{X}^1(M)$ be the space of [[continuously differentiable|0VW90BBSJ4]] [[vector fields|0299VCW1TH]] on $M$, with the $C^1$ topology.\nTwo vector fields $X$ and $Y$ on $M$ are *topologically equivalent* if there is a [[homeomorphism|0QT00PB5G2]] $h \\colon M \\to M$ with two properties.\nIt maps each [[trajectory|0BVT6VSB3S]] of $X$ onto a trajectory of $Y$.\nIt preserves the direction of time along trajectories.\nA vector field $X \\in \\mathfrak{X}^1(M)$ is *structurally stable* if it has a neighbourhood $\\mathcal{N}$ in $\\mathfrak{X}^1(M)$ such that every $Y \\in \\mathcal{N}$ is topologically equivalent to $X$.\n\n## Remarks\n\nExample: on the circle $M = \\R / 2\\pi\\Z$, the field $X(\\theta) = \\sin \\theta$ has the two zeros $0$ and $\\pi$, with $X'(0) = 1$ and $X'(\\pi) = -1$.\nA field $Y$ that is $C^1$-close to $X$ still has exactly two zeros, near $0$ and $\\pi$, with derivatives of the same signs.\nSo $Y$ has one repelling and one attracting rest point joined by two arcs, and a homeomorphism of the circle carries the trajectories of $X$ to those of $Y$.\nHence $X$ is structurally stable.\n\nNon-example: $X(\\theta) = 1 - \\cos \\theta$ has the single zero $0$.\nFor $\\varepsilon \u003e 0$, the field $1 - \\cos \\theta + \\varepsilon$ has no zero, and it is $C^1$-close to $X$ for small $\\varepsilon$.\nA homeomorphism maps rest points to rest points, so these fields are not equivalent to $X$, and $X$ is not structurally stable.\n\nFor flows on a compact surface, structural stability is closely tied to hyperbolicity of the rest points and periodic orbits and to the absence of trajectories joining saddles.\nFor a region in the plane, the definition is adapted by asking the vector field to point inward on the boundary.\n",
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