Bifurcation diagram

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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.

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[edit] Bifurcations in the Logistic Map

Bifurcation diagram of the logistic map
Bifurcation diagram of the logistic map

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 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.

As an example: For x(n+1) = x(n).^2 -c; the code in Matlab can be written as:

close all;
clear all;
c=0; 
y=0.0;  
 
hold on 
while c < 4
        for i=1:100; 
            y = y.^2 -c; %converge the iteration
        end 
        for i=1:20 
 
            y = y.^2 - c;
            plot(c,y,'.'); % plot the converged points
        end 
        c=c+0.01;
end


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

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

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

which is structurally stable when \mu \neq 0 , if a bifurcation diagram is plotted, treating μ as the bifurcation parameter, but for different values of ε, the case ε = 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.

[edit] See also

[edit] 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.

[edit] External links