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  "body": "\n## Definition\n\nLet $X \\subseteq \\R^k$, let $I \\subseteq \\R$ be an open interval, and let $f_\\mu \\colon X \\to X$, $\\mu \\in I$, be a family of maps such that $(\\mu, x) \\mapsto f_\\mu(x)$ is [[continuous|0X3WGX95FE]].\nA parameter $\\mu_0 \\in I$ is a *bifurcation value* of the family if the following holds.\nEvery open interval $J \\ni \\mu_0$ contains some $\\mu$ such that $f_\\mu$ is not [[topologically conjugate|HNXXAPQKG3]] to $f_{\\mu_0}$.\nOne then says that the family, or the difference equation $x_{n+1} = f_\\mu(x_n)$, undergoes a *bifurcation* at $\\mu_0$.\n\n## Remarks\n\nExample: for $f_\\mu(x) = \\mu x$ on $\\R$ with $\\mu \\in (0, \\infty)$, the value $\\mu_0 = 1$ is a bifurcation value.\nThe map $f_1$ is the identity, so every point is fixed, while for $\\mu \\ne 1$ the only fixed point is $0$.\nA conjugacy maps fixed points bijectively to fixed points, so $f_\\mu$ is not conjugate to $f_1$.\n\nNon-example: in the same family, $\\mu_0 = 1/2$ is not a bifurcation value.\nFor $0 \u003c a, b \u003c 1$ the homeomorphism $h(x) = \\sgn(x) \\abs{x}^{\\ln b / \\ln a}$ satisfies $h(a x) = b\\, h(x)$, so all $f_\\mu$ with $\\mu \\in (0, 1)$ are conjugate to each other.\n\nMany texts use a local version, in which only the behaviour near a fixed point or a periodic orbit is compared.\nA local bifurcation of a fixed point $x^*$ of a smooth family can only occur when an eigenvalue of $D f_{\\mu_0}(x^*)$ lies on the unit circle.\nFor one-dimensional maps this means $f_{\\mu_0}'(x^*) = 1$ or $f_{\\mu_0}'(x^*) = -1$.\n",
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