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  "body": "\n## Definition\n\nLet $\\Lambda \\subseteq \\R^k$ be an open set of parameters.\nFor each $\\mu \\in \\Lambda$, let $\\dot x = F(x, \\mu)$ be an [[ordinary differential equation|0NT64JV96A]] on an open set of $\\R^n$, with $F$ smooth in $(x, \\mu)$.\nTwo such systems are *topologically equivalent* if a homeomorphism maps the orbits of one onto the orbits of the other.\nThe homeomorphism must preserve the direction of time.\nA parameter value $\\mu_0 \\in \\Lambda$ is a *bifurcation value* if the following holds.\nEvery neighbourhood of $\\mu_0$ contains some $\\mu$ whose system is not topologically equivalent to the system at $\\mu_0$.\n*Bifurcation theory* studies the bifurcation values of such families and the changes of the [[phase portrait|0T3GK0XCBC]] that occur at them.\n\n## Remarks\n\nThe same definitions apply to families of maps $x \\mapsto f_\\mu(x)$, with topological conjugacy in place of equivalence.\nExample: for $\\dot x = \\mu - x^2$ on $\\R$, the value $\\mu_0 = 0$ is a bifurcation value.\nFor $\\mu \u003c 0$ there is no [[equilibrium|0N8NGKDZDZ]], for $\\mu = 0$ there is one, and for $\\mu \u003e 0$ there are two, at $\\pm\\sqrt{\\mu}$.\nThis is a [[saddle-node bifurcation|MAP5J6HK0T]].\nNon-example: $\\mu_0 = 1$ is not a bifurcation value of this family.\nFor all $\\mu$ near $1$ there are two equilibria, $\\sqrt{\\mu}$ attracting and $-\\sqrt{\\mu}$ repelling, and the phase portraits are equivalent.\nA system whose topological type is unchanged by every small perturbation of the vector field, not only by changes of $\\mu$, is [[structurally stable|D4MGFAHVMT]].\nLocal bifurcations concern a neighbourhood of an equilibrium or periodic orbit; global ones, such as homoclinic bifurcations, involve whole orbits.\n",
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