Structural Instability of a Supercritical Pitchfork Bifurcation

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.
Depending on the values for the parameters and , there are one or three real roots for the above cubic equation. The stability of the steady states is given by the sign of the function: . This Demonstration gives the bifurcation diagram (- plot) and the loci of the steady states. The stable steady states are indicated with blue dots while the unstable steady state is show with a red dot for user-set values of and . The blue region shown in the bifurcation diagram bounds the loci of unstable steady states for all values of , . The - diagram features two regions: in the red region there are three steady states; in the green region there is only one steady state. The blue curve in the same plot gives the loci for two distinct steady states (note that one of the steady states has multiplicity two). Finally, the Demonstration plots the amplitude trajectories for two different initial conditions, and , which lead to the two stable steady states when and are selected appropriately.

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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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