Pitchfork Bifurcation in Dynamical Systems

A pitchfork bifurcation occurs when increasing the parameter causes the graph of the function to intersect the line . See Example 2.31 in [1]. Intersections with the line correspond to fixed points for the map, which are plotted in the figure at the top right; solid lines represent stable fixed points and dashed lines represent unstable fixed points. Eigenvalues inside the unit circle correspond to stable fixed points; eigenvalues outside correspond to unstable fixed points. The eigenvalues for the fixed points at particular values of are shown at the bottom.

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Reference
[1] A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, New York: Wiley, 1995.
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