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  "body": "\n## Example\n\nLet $a, b, c$ be real parameters.\nThe Rössler system is the [[autonomous system|0673D85QKZ]] in $\\R^3$\n$$x' = -y - z, \\qquad y' = x + a y, \\qquad z' = b + z (x - c).$$\nOtto Rössler proposed it in 1976 with $a = b = 0.2$ and $c = 5.7$.\nFor these values, numerical solutions approach a bounded set with a folded band shape, the Rössler attractor.\n\nThe [[equilibrium points|0G4X36AE0X]] are found as follows.\nThe equation $y' = 0$ gives $x = -a y$, and $x' = 0$ gives $z = -y$.\nThen $z' = 0$ becomes $a y^2 + c y + b = 0$.\nSo for $a \\ne 0$ and $c^2 \u003e 4ab$ there are exactly two equilibria, with $y = \\left(-c \\pm \\sqrt{c^2 - 4ab}\\right)/(2a)$.\n\n## Remarks\n\nThe only nonlinear term is $z x$, so the system is simpler than the Lorenz system, which has two.\nIn the $(x, y)$-plane, the first two equations give a spiral that moves outward when $0 \u003c a \u003c 2$.\nWhen $x$ exceeds $c$, the variable $z$ grows fast and folds the solution back toward the centre.\nThis stretching and folding is the mechanism of the chaotic behaviour seen in simulations.\n\n## Explanations\n\n### L1 human\n\nThe Rössler system is a three-dimensional flow built to show chaos with as few terms as possible.\nA solution spirals outward in a plane, then is lifted up and thrown back to the middle.\nRepeating this stretch and fold mixes nearby solutions, like kneading dough.\n",
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