The blue sky catastrophe is a form of orbital indeterminacy, and an element of bifurcation theory.
Orbital dynamics
Blue sky catastrophe is a type of
bifurcation
Bifurcation or bifurcated may refer to:
Science and technology
* Bifurcation theory, the study of sudden changes in dynamical systems
** Bifurcation, of an incompressible flow, modeled by squeeze mapping the fluid flow
* River bifurcation, the ...
of a
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given a ...
. 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
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
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
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
before the bifurcation point. In other words, the orbit ''vanishes into the blue sky''.
Applications of blue sky catastrophe in other fields
The bifurcation has found application in, amongst other places,
slow-fast models of
computational neuroscience
Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to u ...
. The possibility of the phenomenon was raised by
David Ruelle
David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term '' strange attractor'', and developed a ...
and
Floris Takens
Floris Takens (12 November 1940 – 20 June 2010) was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems.
Together with David Ruelle, he predicted that fluid turbulence could develop through a strange attr ...
in 1971, and explored by
R.L. Devaney and others in the following decade. More compelling analysis was not performed until the 1990s.
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 is born in a global saddle-loop bifurcation, becomes chaotic in a period doubling cascade, and disappears in the blue sky catastrophe.
References
Blue Sky Catastrophearticle in
Scholarpedia
''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content in science and medicine.
''Scholarpedia'' articles are written ...
Andrey Shilnikov- studies the blue sky catastrophe and other topics in dynamical neuroscience.
E. Meca et al. Phys. Rev. Lett. 92, 234501 (2004)- Blue Sky Catastrophe in fluid dynamics.
Further reading
*
* {{cite journal , title= Publisher's Note: Blue Sky Catastrophe in Double-Diffusive Convection , authors= Esteban Meca, Isabel Mercader, Oriol Batiste, Laureano Ramirez-Piscina , volume= 92 , number= 25 , publication-date= 22 June 2004 , page=259901 , doi= 10.1103/PhysRevLett.92.259901 , journal= Physical Review Letters , publisher= American Physical Society
External links
Andrey Shilnikov- studies the blue sky catastrophe and other topics in dynamical neuroscience.
Bifurcation theory
Celestial mechanics