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The blue sky catastrophe is a form of orbital indeterminacy, and an element of bifurcation theory.
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.
Granular convection is a phenomenon where granular material subjected to shaking or vibration will exhibit circulation patterns similar to types of fluid convection. It is sometimes called the Brazil nut effect, when the largest of irregularly shaped particles end up on the surface of a granular material containing a mixture of variously sized objects. This name derives from the example of a typical container of mixed nuts, in which the largest will be Brazil nuts. The phenomenon is also known as the muesli effect since it is seen in packets of breakfast cereal containing particles of different sizes but similar density, such as muesli mix.
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and many other phenomena governed by Einstein's theory of general relativity. A currently active field of research in numerical relativity is the simulation of relativistic binaries and their associated gravitational waves.
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems and discrete systems.
In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. With the doubled period, it takes twice as long for the numerical values visited by the system to repeat themselves.
Predrag Cvitanović is a theoretical physicist regarded for his work in nonlinear dynamics, particularly his contributions to periodic orbit theory.
The double-exchange mechanism is a type of a magnetic exchange that may arise between ions in different oxidation states. First proposed by Clarence Zener, this theory predicts the relative ease with which an electron may be exchanged between two species and has important implications for whether materials are ferromagnetic, antiferromagnetic, or exhibit spiral magnetism. For example, consider the 180 degree interaction of Mn-O-Mn in which the Mn "eg" orbitals are directly interacting with the O "2p" orbitals, and one of the Mn ions has more electrons than the other. In the ground state, electrons on each Mn ion are aligned according to the Hund's rule:
In fluid mechanics, external flow is a flow that boundary layers develop freely, without constraints imposed by adjacent surfaces. It can be defined as the flow of a fluid around a body that is completely submerged in it. Examples include fluid motion over a flat plate and flow over curved surfaces such as a sphere, cylinder, airfoil, or turbine blade, water flowing around submarines, and air flowing around a truck; a 2000 paper analyzing the latter used computational fluid dynamics to model the three-dimensional flow structure and pressure distribution on the external surface of the truck. In a 2008 paper, external flow was said to be "arguably is the most common and best studied case in soft matter systems.
In statistical mechanics the Percus–Yevick approximation is a closure relation to solve the Ornstein–Zernike equation. It is also referred to as the Percus–Yevick equation. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function. The approximation is named after Jerome K. Percus and George J. Yevick.
The Magnetic Prandtl number (Prm) is a dimensionless quantity occurring in magnetohydrodynamics which approximates the ratio of momentum diffusivity (viscosity) and magnetic diffusivity. It is defined as:
A composite fermion is the topological bound state of an electron and an even number of quantized vortices, sometimes visually pictured as the bound state of an electron and, attached, an even number of magnetic flux quanta. Composite fermions were originally envisioned in the context of the fractional quantum Hall effect, but subsequently took on a life of their own, exhibiting many other consequences and phenomena.
A trojan wave packet is a wave packet that is nonstationary and nonspreading. It is part of an artificially created system that consists of a nucleus and one or more electron wave packets, and that is highly excited under a continuous electromagnetic field.
Stellar pulsations are caused by expansions and contractions in the outer layers as a star seeks to maintain equilibrium. These fluctuations in stellar radius cause corresponding changes in the luminosity of the star. Astronomers are able to deduce this mechanism by measuring the spectrum and observing the Doppler effect. Many intrinsic variable stars that pulsate with large amplitudes, such as the classical Cepheids, RR Lyrae stars and large-amplitude Delta Scuti stars show regular light curves.
Double ionization is a process of formation of doubly charged ions when laser radiation is exerted on neutral atoms or molecules. Double ionization is usually less probable than single-electron ionization. Two types of double ionization are distinguished: sequential and non-sequential.
Quantum scarring refers to a phenomenon where the eigenstates of a classically chaotic quantum system have enhanced probability density around the paths of unstable classical periodic orbits. The instability of the periodic orbit is a decisive point that differentiates quantum scars from the more trivial observation that the probability density is enhanced in the neighborhood of stable periodic orbits. The latter can be understood as a purely classical phenomenon, a manifestation of the Bohr correspondence principle, whereas in the former, quantum interference is essential. As such, scarring is both a visual example of quantum-classical correspondence, and simultaneously an example of a (local) quantum suppression of chaos.
In mathematics, a periodic travelling wave is a periodic function of one-dimensional space that moves with constant speed. Consequently, it is a special type of spatiotemporal oscillation that is a periodic function of both space and time.
In physics, the hydrodynamic quantum analogs refer to experimentally-observed phenomena involving bouncing fluid droplets over a vibrating fluid bath that behave analogously to several quantum-mechanical systems.
Dwight Barkley is a professor of mathematics at the University of Warwick.
Alexander Avraamovitch Golubov is a doctor of physical and mathematical sciences, associate professor at the University of Twente (Netherlands). He specializes in condensed matter physics with the focus on theory of electronic transport in superconducting devices. He made key contributions to theory of Josephson effect in novel superconducting materials and hybrid structures, and to theory of multiband superconductivity.
Tin-Lun "Jason" Ho is a Chinese-American theoretical physicist, specializing in condensed matter theory, quantum gases, and Bose-Einstein condensates. He is known for the Mermin-Ho relation.
Robert Everett Ecke is an American experimental physicist who is a Laboratory Fellow and Director Emeritus of the Center for Nonlinear Studies (CNLS) at Los Alamos National Laboratory and Affiliate Professor of Physics at the University of Washington. His research has included chaotic nonlinear dynamics, pattern formation, rotating Rayleigh-Bénard convection, two-dimensional turbulence, granular materials, and stratified flows. He is a Fellow of the American Physical Society (APS) and of the American Association for the Advancement of Science (AAAS), was Chair of the APS Topical Group on Statistical and Nonlinear Physics, served in numerous roles in the APS Division of Fluid Dynamics, and was the Secretary of the Physics Section of the AAAS.