In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values a variable of a system can obtain in function of a parameter of the system.

An example is the bifurcation diagram of the logistic map. In this case, the parameter r is shown on the horizontal axis of the plot and the vertical axis shows the density of 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. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.

An interesting feature of this diagram is that as the periods go to infinity, r remains finite. When r is greater than (approximately) 3.57, the orbits become chaotic. Hence this bifurcation diagram demonstrates a nice example of the importance of chaos theory in even very simple non-linear systems.

Symmetry breaking in bifurcation sets

$\backslash ddot\; \{x\}\; +\; f(x;\backslash mu)\; +\; \backslash epsilon\; g(x)\; =\; 0$,

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

See also

• Sarkovskii's theorem

External Links

• The Logistic Map and Chaos by Elmer G. Wiens

Chaos theory | Bifurcation theory

Каскад бифуркаций | Bifurkationsdiagramm