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Bifurcation diagram

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

File:LogisticMap BifurcationDiagram.png
Bifurcation diagram of the logistic map. The attractor for any value of the parameter r is shown on the vertical line at that r.
File:Diagram bifurkacji anim small.gif
Animation showing the formation of bifurcation diagram
File:Circle map bifurcation.jpeg
Bifurcation diagram of the circle map. Black regions correspond to Arnold tongues.

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

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

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