mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a separatrix is the boundary separating two modes of behaviour in a

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

.Blanchard, Paul, ''Differential Equations'', 4th ed., 2012, Brooks/Cole, Boston, MA, pg. 469.

Example

Consider the differential equation describing the motion of a simple

pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...

: :

+ \sin\theta=0.

where

\ell

denotes the length of the pendulum,

g

the

gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...

and

\theta

the angle between the pendulum and vertically downwards. In this system there is a conserved quantity H (the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

), which is given by

H = \frac - \frac\cos\theta.

With this defined, one can plot a curve of constant ''H'' in the

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...

of system. The phase space is a graph with

\theta

along the horizontal axis and

\dot

on the vertical axis – see the thumbnail to the right. The type of resulting curve depends upon the value of ''H''. If

H<-\frac

then no curve exists (because

\dot

must be imaginary). If

-\frac then the curve will be a simple closed

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

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

\frac 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 H=\frac{\ell} . It separates — hence the name — the phase space into two distinct areas, each with a distinct type of motion. 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.

References

* Logan, J. David, ''Applied Mathematics'', 3rd Ed., 2006, John Wiley and Sons, Hoboken, NJ, pg. 65.

External links

Separatrix
from

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...

. Dynamical systems