Bifurcation diagram

In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.

Bifurcations in 1D discrete dynamical systems

Logistic map

Bifurcation diagram of the logistic map. The attractor for any value of the parameter r is shown on the vertical line at that r.

Symmetry breaking in pitchfork bifurcation as the parameter epsilon is varied. epsilon = 0 is the case of symmetric pitchfork bifurcation.

Animation showing the formation of bifurcation diagram

Bifurcation diagram of the circle map. Black regions correspond to Arnold tongues.

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 set of values of the logistic function visited asymptotically from almost all initial conditions.

The bifurcation diagram shows the forking of the 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.

The diagram also shows period doublings from 3 to 6 to 12 etc., from 5 to 10 to 20 etc., and so forth.

Symmetry breaking in bifurcation sets

In a dynamical system such as

{\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.

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 Chaos

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