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  "body": "\n## Definition\n\nLet $(x_n)_{n \\ge n_0}$ be a [[real sequence|04CC6K5VDT]] that is a solution of a difference equation and is not identically zero.\nThe solution is *oscillatory* if for every integer $N \\ge n_0$ there is an integer $n \\ge N$ with\n$$x_n \\, x_{n+1} \\le 0.$$\nOtherwise it is *nonoscillatory*, that is, it is eventually positive or eventually negative.\n\n## Remarks\n\nExample: $x_n = (-1)^n$ solves $x_{n+1} = -x_n$.\nFor every $n$ one has $x_n x_{n+1} = -1 \\le 0$, so the solution is oscillatory.\n\nNon-example: $x_n = 2^{-n}$ solves $x_{n+1} = x_n / 2$.\nHere $x_n x_{n+1} \u003e 0$ for all $n$, so the condition fails already for $N = n_0$, and the solution is nonoscillatory.\n\nOne also speaks of oscillation about a value $x^*$, often an equilibrium, meaning that $(x_n - x^*)$ is oscillatory.\nFor the linear equation $x_{n+1} = a x_n$ with $a \\ne 0$, every nonzero solution is oscillatory exactly when $a \u003c 0$.\n",
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