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  "body": "\n## Definition\n\nLet $H \\colon \\R^2 \\to \\R$ be smooth and let $f = (\\partial H / \\partial y, -\\partial H / \\partial x)$, so that $u' = f(u)$ is a planar [[Hamiltonian system|0EDKGA2FAK]].\nLet $p$ be a [[hyperbolic|0C5PREPZWG]] saddle of this system, and let $q_0(t)$ be a solution whose orbit is a [[homoclinic orbit|6DBSQKZK4W]] of $p$.\nLet $g \\colon \\R^2 \\times \\R \\to \\R^2$ be smooth and [[periodic|0ATF6Q0633]] of period $T \u003e 0$ in its last variable.\nConsider the perturbed system\n$$u' = f(u) + \\varepsilon\\, g(u, t), \\qquad \\varepsilon \\text{ small}.$$\nFor $a, b \\in \\R^2$ put $a \\wedge b = a_1 b_2 - a_2 b_1$.\nThe *Melnikov function* of the perturbation along $q_0$ is\n$$M(t_0) = \\int_{-\\infty}^{\\infty} f(q_0(t)) \\wedge g\\big(q_0(t), t + t_0\\big)\\, dt, \\qquad t_0 \\in \\R.$$\nThe *Melnikov method* is the study of the perturbed system through the zeros of $M$.\n\n## Remarks\n\nThe integral converges because $f(q_0(t)) \\to f(p) = 0$ exponentially fast as $t \\to \\pm\\infty$.\nThe function $M$ is $T$-periodic in $t_0$.\nFor small $\\varepsilon$, the perturbed system has a hyperbolic periodic orbit near $p$.\nThe distance between its stable and unstable manifolds, measured along the normal to the unperturbed loop at $q_0(0)$, is $\\varepsilon M(t_0) / \\norm{f(q_0(0))} + O(\\varepsilon^2)$, with a suitable sign convention.\nThe basic theorem says: if $M$ has a simple zero, then for small $\\varepsilon \\ne 0$ these manifolds intersect transversally; if $M$ has no zero, they do not intersect.\n\nExample: for $H = \\frac{1}{2} y^2 - \\frac{1}{2} x^2 + \\frac{1}{4} x^4$ and $g = (0, -\\delta y + \\gamma \\cos(\\omega t))$, the system is the damped, forced Duffing equation.\nHere $q_0(t) = (\\sqrt{2} \\operatorname{sech} t, -\\sqrt{2} \\operatorname{sech} t \\tanh t)$, and a direct computation gives $M(t_0) = -\\frac{4}{3} \\delta + c\\, \\gamma \\sin(\\omega t_0)$ with a constant $c$ that depends only on $\\omega$.\n\nNon-example: if $g(u, t) = (0, t)$, then $g$ is not periodic in $t$.\nThe perturbed system is then not periodic in time, there is no Poincaré map of period $T$, and the theorem above does not apply.\n",
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