Separatrix (dynamical systems)

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

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

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