Bifurcation diagram
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In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria or fixed points) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable fixed points with a solid line and unstable fixed points with a dotted line.
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[edit] Bifurcations in the 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. Only the stable solutions are shown here, there are many other unstable solutions which are not shown in this diagram.
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.
[edit] 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.
[edit] See also
[edit] External links
- The Logistic Map and Chaos by Elmer G. Wiensde:Bifurkationsdiagramm
