Separatrix (dynamical systems)

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

Example

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

{d^2\theta\over dt^2}%2B{g\over l} \sin\theta=0.

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

H = \frac{\dot{\theta}^2}{2} - \frac{g}{l}\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 \theta along the horizontal axis and \dot{\theta} on the vertical axis - see the thumbnail to the right. The type of resulting curve depends upon the value of H.

If H<-\frac{g}{l} then no curve exist (\dot{\theta} must be imaginary).

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

If \frac{g}{l}<H then the curve is open, and this corresponds to pendulum forever swinging through complete circles.

In this system the separatix is the curve that corresponds to H=\frac{g}{l}. It separates (hence the name) the phase space into two distinct areas. Within the separatrix corresponds to oscillating motion back and forth, whereas the outside of the separatrix corresponds to motion with the pendulum continuously turning through circles.

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