# Bifurcation diagram

Template:No footnotes 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

### Logistic map

An example is the bifurcation diagram of the logistic map:

$x_{n+1}=rx_{n}(1-x_{n}).\,$ 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. ${\ddot {x}}+f(x;\mu )+\epsilon g(x)=0$ ,
which is structurally stable when $\mu \neq 0$ , if a bifurcation diagram is plotted, treating $\mu$ as the bifurcation parameter, but for different values of $\epsilon$ , the case $\epsilon =0$ is the symmetric pitchfork bifurcation. When $\epsilon \neq 0$ , we say we have a pitchfork with broken symmetry. This is illustrated in the animation on the right.