Perturbation vs Bifurcation Theory

[neurocomp2003]

I know what Bifurcation Theory is.
but What is Perturbation theory? is it similar to bifurcation theory?

 

[PerennialII]

Might say a "steady" bifurcation, or the non-existence of a bifurcation as a result of a perturbation, is actually a perturbation (of sorts ... if this wording makes any sense ) (both have a perturbation, but the response of 'some system' to a perturbation is 'fundamentally' different). Suppose in general could talk about a method of solution (approximate) where a small perturbation is added to an exact solution to result in a complementary solution - with the purpose of it being an improvement over the original one in one way or the other (perturbation theory itself is used "all over" physics and engineering).

...and a better explanation: http://en.wikipedia.org/wiki/Perturbation_theory

 

[HallsofIvy]

While they can both be used to analyze a problem, as PerennialII indicated, "perturbation" and "bifurcation" are very different techniques. In fact, it is nt quite correct to talk about "bifurcation" as a technique- it is something that happens to a system, not something we do to the problem as is "perturbation".

The "Perturbation technique" is a method of (approximately) solving a non-linear problem by writing the "non-linear" part as a power series in some parameter. Taking only the linear terms of the power series gives a solution to the linearized version of the problem. Taking higher powers sucessively allows us to write the problem in terms of the previous solutions.

A "bifurcation", on the other hand, involves a problem that already depends on some parameter. If, for some value of that parameter, a single equilibrium solution splits into two (or more), that is a "bifurcation value" of the parameter.

Here's an interesting "bifurcation" that has nothing to do with differential equations or "perturbation": set up to powerful, equal strength, light bulbs, separated horizontally by distance L. If a screen is "sufficiently far away", the light will be most intense at the point on the screen directly away from the point halfway between the two bulbs. As you move the screen close to the two bulbs, that will remain true- until the screen is $\sqrt{3}L$. Then the "center point" will become a local minimum, separating two local maxima of brightness which move toward the two bulbs as the screen is brought still closer.

 

[J77]

Here's an interesting "bifurcation" that has nothing to do with differential equations or "perturbation": set up to powerful, equal strength, light bulbs, separated horizontally by distance L. If a screen is "sufficiently far away", the light will be most intense at the point on the screen directly away from the point halfway between the two bulbs. As you move the screen close to the two bulbs, that will remain true- until the screen is $\sqrt{3}L$. Then the "center point" will become a local minimum, separating two local maxima of brightness which move toward the two bulbs as the screen is brought still closer.

As another example, if you have two folds in a parameter, with eigenvalues at zero, you can vary another parameter to form a cusp point. However, if the folds are in the direction of, say, the amplitude of the solution you can also vary another parameter to get these folds to come together in amplitude - like the light in Halls' example - but this time, it's not a bifurcation.

Arnold deals with stuff like this in his Castastrophe book (Springer) - but he calls most things perestroika not bifurcation

 

[arildno]

It should, however, be emphasized that there is a difference between "regular" and "singular" perturbations.
In a "regular" (or proper) perturbation, the exact "perturbed" solution looks very much alike the unperturbed approximation.
In the case of a singular perturbation, for example in boundary layer theory, this is by no means the case (for the "outer solution").
A tiny, non-zero parameter may remain unneglectable (in contrast to the situation in a regular perturbation case).