Mathematical constants by continued fraction representation
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
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An example is the bifurcation diagram of the logistic map:
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
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The map is .
Symmetry breaking in bifurcation sets
In a dynamical system such as
which is structurally stable when , if a bifurcation diagram is plotted, treating as the bifurcation parameter, but for different values of , the case is the symmetric pitchfork bifurcation. When , 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
- Logistic Map Simulation. A Java applet simulating the Logistic Map by Yuval Baror.
- The Logistic Map and Chaos