# Structural Instability of a Supercritical Pitchfork Bifurcation

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A dynamical system is structurally unstable when small perturbations alter the qualitative behavior of trajectories. An example of structural instability is the flow pattern that can occur in a two-dimensional flow field subjected to a sudden expansion [1]. In that case the stability analysis can be reduced to studying the following generic amplitude equation: . The steady-state amplitudes of the perturbation are the real solutions of the following nonlinear equation: . Here, is called the imperfection parameter. When , the dynamical system exhibits the classical supercritical pitchfork bifurcation.

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Contributed by: Brian G. Higgins and Housam Binous (June 2011)

Open content licensed under CC BY-NC-SA

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References

[1] J. Mizushima and Y. Shiotani, "Structural Instability of the Bifurcation Diagram for Two-Dimensional Flow in a Channel with a Sudden Expansion," *Journal of Fluid Mechanics*, 420, 2000 pp. 131–145.

[2] P. G. Drazin and W. H. Reid, *Hydrodynamic Stability*, Cambridge: Cambridge University Press, 1981.

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