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  "body": "\n## Example\n\nLet $\\delta, \\alpha, \\beta, \\gamma, \\omega$ be real constants.\nThe Duffing equation is the second-order [[ordinary differential equation|0NT64JV96A]]\n$$x'' + \\delta x' + \\alpha x + \\beta x^3 = \\gamma \\cos(\\omega t).$$\nHere $\\delta$ is a damping coefficient, $\\alpha x + \\beta x^3$ is a restoring force with a cubic term, and $\\gamma \\cos(\\omega t)$ is a periodic forcing.\n\nTake $\\delta = \\gamma = 0$, $\\alpha = -1$ and $\\beta = 1$, so $x'' - x + x^3 = 0$.\nThis is a [[Hamiltonian system|0EDKGA2FAK]] with $H(x, y) = \\frac{1}{2} y^2 - \\frac{1}{2} x^2 + \\frac{1}{4} x^4$ and $y = x'$.\nIts equilibria are $(0, 0)$, a [[hyperbolic|0C5PREPZWG]] saddle, and $(\\pm 1, 0)$, two centres.\nThe function $x(t) = \\sqrt{2}\\, \\operatorname{sech} t$ is a solution.\nCheck: with $u = \\operatorname{sech} t$ one has $u'' = u - 2u^3$, so $x'' - x + x^3 = \\sqrt{2}(u - 2u^3) - \\sqrt{2} u + 2\\sqrt{2} u^3 = 0$.\nSince $x(t) \\to 0$ and $x'(t) \\to 0$ as $t \\to \\pm\\infty$, this solution is a [[homoclinic orbit|6DBSQKZK4W]] of the saddle.\nBy the symmetry $x \\mapsto -x$, there is a second one in the half-plane $x \u003c 0$.\n\n## Remarks\n\nSolutions of the unforced, undamped equation stay on level sets of $H$.\nSo each solution is an equilibrium, a periodic solution, or a homoclinic solution.\nWith small damping and small forcing, the two homoclinic orbits can break into a tangle of stable and unstable manifolds.\nThe [[Melnikov method|A755BDRP1A]] decides when this happens in terms of $\\gamma$, $\\delta$ and $\\omega$.\nIn that regime, numerical solutions of the Duffing equation look chaotic.\nFor $\\alpha \u003e 0$ and small $\\beta$, the equation is a perturbed harmonic oscillator whose frequency depends on the amplitude.\n\n## Explanations\n\n### L1 human\n\nThe Duffing equation is a spring whose force is not exactly proportional to the stretch, pushed back and forth periodically.\nWith $\\alpha \u003c 0 \u003c \\beta$ the spring has two stable rest positions, like a ball in a double well.\nA forced ball can jump between the two wells in an irregular way, which is a classical example of chaos.\n",
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