Separatrix (mathematics)

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

[1] Consider the differential equation describing the motion of a simple pendulum: where

In this system there is a conserved quantity H (the Hamiltonian), which is given by

With this defined, one can plot a curve of constant H in the phase space of system.

The type of resulting curve depends upon the value of H. If

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.

then the curve is open, and this corresponds to the pendulum forever swinging through complete circles.

In this system the separatrix is the curve that corresponds to

The region inside the separatrix has all those phase space curves which correspond to the pendulum oscillating back and forth, whereas the region outside the separatrix has all the phase space curves which correspond to the pendulum continuously turning through vertical planar circles.

In the FitzHugh–Nagumo model, when the linear nullcline pierces the cubic nullcline at the left, middle, and right branch once each, the system has a separatrix.

The separatrix itself is the stable manifold for the saddle point in the middle.

The separatrix is clearly visible by numerically solving for trajectories backwards in time.