Bifurcation theory provides models of transitions and instabilities as some control parameter is varied. One of the most important types of bifurcation is the pitchfork bifurcation observed in problems that have a symmetry. For example, a beam deflection under a load that exceeds the buckling threshold can me modeled by a supercritical pitchfork bifurcation. Does anyone know in which phenomenon one can observe the subcritical pitchfork bifurcation?
Dear Abdallah,
please have a look at the generation of APD (action potential duration) alternans in cardiac electrophysiology, which have been studied and demonstrated both theoretically and experimentally.
Here you have a couple of links, involving an intuitive Java applet and a reference to the subcritical bifurcation case:
http://thevirtualheart.org/visualization/java/Hopf2a.html
http://www.maths.usyd.edu.au/u/gottwald/preprints/alternans.pdf
Abdallah,
You can stick with buckling. Torsional buckling of a rod
under controlled tension and a fixed angle of twist is
described by a subcritical pitchfork bifurcation if the
twist angle is sufficiently large. See
http://www.sciencedirect.com/science/article/pii/S0020740302001832
Regards,
Gert van der Heijden
The Rayleigh Taylor instability of a circular interface under control of a rotating magnetic field.
http://dx.doi.org/10.1017/jfm.2013.424
Best,
Andreas
A cylindrical liquid bridge in absence of gravity forces (or compensated by an outer bath of the same density) experiences a breakup before reaching the critical slenderness (length/diameter=pi) due to a subcritical pitchfork bifurcation. Axial microgravity can be viewed as an imperfection leading to amphoric stable shapes.
See J. M. Vega and J. M. Perales, "Almost cylindrical isorotating liquid
bridges for small Bond numbers," in ESA SP-191 (ESA/ESTEC, Noordwijk, 1983), p. 247.
Regards,
Helio González
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