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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be open, let $f \\colon U \\to \\R^n$ be continuously differentiable, and let $\\varphi$ be the [[flow|02903SSA48]] of the system $x' = f(x)$.\nLet $p$ and $q$ be two distinct [[equilibrium points|0G4X36AE0X]] of the system.\nA *heteroclinic orbit* from $p$ to $q$ is the [[trajectory|0BVT6VSB3S]] $\\set{\\varphi_t(x) : t \\in \\R}$ of a point $x \\in U$ such that:\n\n1. the solution through $x$ exists for all $t \\in \\R$;\n2. $\\varphi_t(x) \\to p$ as $t \\to -\\infty$;\n3. $\\varphi_t(x) \\to q$ as $t \\to +\\infty$.\n\n## Remarks\n\nExample: for $x' = 1 - x^2$ on $\\R$, the equilibria are $-1$ and $1$.\nThe solution with $x(0) = 0$ is $x(t) = \\tanh t$, since $(\\tanh t)' = 1 - \\tanh^2 t$.\nIt exists for all $t$, tends to $-1$ as $t \\to -\\infty$ and tends to $1$ as $t \\to +\\infty$.\nSo $(-1, 1)$ is a heteroclinic orbit from $-1$ to $1$.\n\nNon-example: for the same equation, the solution with $x(0) = 2$ is $x(t) = \\coth(t + c)$, where $c \u003e 0$ and $\\coth c = 2$.\nIt tends to $1$ as $t \\to +\\infty$, but it blows up at $t = -c$, so clause 1 fails.\n\nA heteroclinic orbit from $p$ to $q$ lies in the unstable set of $p$ and in the stable set of $q$.\nIf $p = q$ in the same construction, the orbit is called a homoclinic orbit.\nSome authors also allow $p$ and $q$ to be periodic orbits instead of equilibria.\n",
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