# 3.17] What is a Bifurcation? (nonlinear science)

## Description

This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by
others.

# 3.17] What is a Bifurcation? (nonlinear science)

(Thanks to Zhen Mei for Contributions to this answer)

A bifurcation is a qualitative change in dynamics upon a small variation in

the parameters of a system.

Many dynamical systems depend on parameters, e.g. Reynolds number, catalyst

density, temperature, etc. Normally a gradually variation of a parameter in

the system corresponds to the gradual variation of the solutions of the

problem. However, there exists a large number of problems for which the number

of solutions changes abruptly and the structure of solution manifolds varies

dramatically when a parameter passes through some critical values. For

example, the abrupt buckling of a stab when the stress is increased beyond a

critical value, the onset of convection and turbulence when the flow

parameters are changed, the formation of patterns in certain PDE's, etc. This

kind of phenomena is called bifurcation, i.e. a qualitative change in the

behavior of solutions of a dynamics system, a partial differential equation or

a delay differential equation.

Bifurcation theory is a method for studying how solutions of a nonlinear

problem and their stability change as the parameters varies. The onset of

chaos is often studied by bifurcation theory. For example, in certain

parameterized families of one dimensional maps, chaos occurs by infinitely

many period doubling bifurcations

(See http://www.stud.ntnu.no/~berland/math/feigenbaum/)

There are a number of well constructed computer tools for studying

bifurcations. In particular see [5.2] for AUTO and DStool.

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