## Frequent Links

# Bifurcation diagram

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (March 2013) |

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.

## Contents

## Bifurcations in 1D discrete dynamical systems

### Logistic map

An example is the bifurcation diagram of the logistic map:

- <math> x_{n+1}=rx_n(1-x_n). \,</math>

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

The map is <math>x_{n+1}=x_n^2-c</math>.

## Symmetry breaking in bifurcation sets

In a dynamical system such as

- <math> \ddot {x} + f(x;\mu) + \epsilon g(x) = 0</math>,

which is structurally stable when <math> \mu \neq 0 </math>, if a bifurcation diagram is plotted, treating <math> \mu </math> as the bifurcation parameter, but for different values of <math> \epsilon </math>, the case <math> \epsilon = 0</math> is the symmetric pitchfork bifurcation. When <math> \epsilon \neq 0 </math>, we say we have a pitchfork with *broken symmetry.* This is illustrated in the animation on the right.

## See also

## References

- Paul Glendinning, "Stability, Instability and Chaos", Cambridge University Press, 1994.
- Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000.

## External links

- Logistic Map Simulation. A Java applet simulating the Logistic Map by Yuval Baror.
- The Logistic Map and Chaosde:Bifurkationsdiagramm