In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable solutions with a solid line and unstable solutions with a dotted line.

Bifurcations in 1D discrete dynamical systems

Bifurcation diagram of the circle map.

An example is the bifurcation diagram of the logistic map:

The bifurcation parameter r is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function.

The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.

Real quadratic map

See also: Complex quadratic polynomial

The map is .

Symmetry breaking in bifurcation sets

Symmetry breaking in pitchfork bifurcation as the parameter epsilon is varied.

epsilon = 0 is the case of symmetric pitchfork bifurcation

In a dynamical system such as

which is structurally stable when ,

if a bifurcation diagram is plotted, treating as the bifurcation parameter, but for different values of ,

the case is the symmetric pitchfork bifurcation.

When , we say we have a pitchfork with broken symmetry. These is easily representedd in in 2, 3, and 4 dimimensional graphing approaches.

In lieu of a figure and legend, a text desription of a a representative bifurcation diagram is given below.

Fixed points:

Stabel Steady State

Saddle Steady State

Stable vs. Unstabler limit cycle max / min

Singularities

SN1 Saddle Node

Hopf Bifurcation

Sniper Bifurcation

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