mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a separatrix is the boundary separating two modes of behaviour in a differential equation.Blanchard, Paul, ''Differential Equations'', 4th ed., 2012, Brooks/Cole, Boston, MA, pg. 469.

Examples

Simple pendulum

Consider the differential equation describing the motion of a simple

pendulum A pendulum is a device made of 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 i ...

: :

+ \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 (physics), drag). This is the steady gain in speed caused exclusively by gravitational attraction. All bodi ...

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 The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

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 . 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.

FitzHugh–Nagumo model

In the

FitzHugh–Nagumo model The FitzHugh–Nagumo model (FHN) describes a prototype of an excitable system (e.g., a neuron A neuron (American English), neurone (British English), or nerve cell, is an membrane potential#Cell excitability, excitable cell (biology), cell t ...

, when the linear nullcline pierces the cubic nullcline at the left, middle, and right branch once each, the system has a separatrix. Trajectories to the left of the separatrix converge to the left stable equilibrium, and similarly for the right. The separatrix itself is the

stable manifold In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repell ...

for the saddle point in the middle. Details are found in the page. The separatrix is clearly visible by numerically solving for trajectories ''backwards in time''. Since when solving for the trajectories forwards in time, trajectories diverge from the separatrix, when solving backwards in time, trajectories converge to the separatrix.

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 ...

. Dynamical systems