Blue sky catastrophe
Encyclopedia
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 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.
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
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
Orbital period
The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...
and length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
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
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
, and without loss of stability
Stability
-Mathematics:*Stability theory, the study of the stability of solutions to differential equations and dynamical systems**Lyapunov stability**Structural stability*Stability of a point in geometric invariant theory....
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
Computational neuroscience
Computational neuroscience is the study of brain function in terms of the information processing properties of the structures that make up the nervous system...
. The possibility of the phenomenon was raised by David Ruelle
David Ruelle
David Pierre Ruelle is a Belgian-French mathematical physicist. He has worked on statistical physics and dynamical systems. With Floris Takens he coined the term strange attractor, and founded a new theory of turbulence...
and Floris Takens
Floris Takens
Floris Takens 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 attractor, a term they coined, as opposed to the then-prevailing theory of accretion of modes....
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
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, 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.