In mathematics, a separatrix is the boundary separating two modes of behaviour in a differential equation.

Consider the differential equation describing the motion of a simple pendulum:

where ${\displaystyle \ell }$ denotes the length of the pendulum, ${\displaystyle g}$ the gravitational acceleration and ${\displaystyle \theta }$ the angle between the pendulum and vertically downwards. In this system there is a conserved quantity H (the Hamiltonian), which is given by

${\displaystyle H={\frac {{\dot {\theta }}^{2}}{2}}-{\frac {g}{\ell }}\cos \theta .}$

With this defined, one can plot a curve of constant H in the phase space of system. The phase space is a graph with ${\displaystyle \theta }$ along the horizontal axis and ${\displaystyle {\dot {\theta }}}$ on the vertical axis – see the thumbnail to the right. The type of resulting curve depends upon the value of H.

If ${\displaystyle H<-{\frac {g}{\ell }}}$ then no curve exists (because ${\displaystyle {\dot {\theta }}}$ would then be imaginary).

If ${\displaystyle -{\frac {g}{\ell }}<H<{\frac {g}{\ell }}}$ then the curve will be a simple closed curve which is nearly circular for small H and becomes "eye" shaped when H approaches the upper bound. These curves correspond to the pendulum swinging periodically from side to side.

If ${\displaystyle {\frac {g}{\ell }}<H}$ then the curve is open, and this corresponds to the pendulum forever swinging through complete circles. Note that ${\displaystyle \theta =\pm 180}$ deg. corresponds to the pendulum reaching the top, after which it leaves the phase-space plot on one side and reappears again on the other.