Everything about The Blue Sky Catastrophe totally explained
The
blue sky catastrophe is a type of
bifurcation of a
periodic orbit. In other words, it describes a sort of behaviour stable solutions of a set of differential equations can undergo as the equations are gradually changed. This type of bifurcation is characterised by both the
period and
length of the orbit approaching infinity as the control parameter approaches a finite bifurcation value, but with the orbit still remaining within a
bounded part of the
phase space, and without loss of
stability before the bifurcation point. In other words, the orbit
vanishes into the blue sky.
The bifurcation has found application in, amongst other places,
slow-fast models of
computational neuroscience. The possibility of the phenomenon was raised by
David Ruelle and
Floris Takens in 1971, and explored by
R.L. Devaney and others in the following decade. More compelling analysis wasn't performed until the 1990's.
This bifurcation has also been found in the context of
fluid dynamics, namely in double-diffusive convection of a small
Prandtl number fluid. Double diffusive convection occurs when convection of the fluid is driven by both thermal and concentration gradients, and the temperature and concentration diffusivities take different values. The bifurcation is found in an orbit that's born in a global saddle-loop bifurcation, becomes chaotic in a period doubling cascade, and disappears in the blue sky catastrophe.
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