Period-doubling bifurcationIn 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 (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves. A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system.
Bifurcation diagramIn mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.
Bifurcation theoryBifurcation 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 (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.
Drosophila embryogenesisDrosophila embryogenesis, the process by which Drosophila (fruit fly) embryos form, is a favorite model system for genetics and developmental biology. The study of its embryogenesis unlocked the century-long puzzle of how development was controlled, creating the field of evolutionary developmental biology. The small size, short generation time, and large brood size make it ideal for genetic studies. Transparent embryos facilitate developmental studies. Drosophila melanogaster was introduced into the field of genetic experiments by Thomas Hunt Morgan in 1909.
Pitchfork bifurcationIn bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation where the system transitions from one fixed point to three fixed points. Pitchfork bifurcations, like Hopf bifurcations, have two types – supercritical and subcritical. In continuous dynamical systems described by ODEs—i.e. flows—pitchfork bifurcations occur generically in systems with symmetry. The normal form of the supercritical pitchfork bifurcation is For , there is one stable equilibrium at .
EmbryoAn embryo is an initial stage of development of a multicellular organism. In organisms that reproduce sexually, embryonic development is the part of the life cycle that begins just after fertilization of the female egg cell by the male sperm cell. The resulting fusion of these two cells produces a single-celled zygote that undergoes many cell divisions that produce cells known as blastomeres. The blastomeres are arranged as a solid ball that when reaching a certain size, called a morula, takes in fluid to create a cavity called a blastocoel.
Feigenbaum constantsIn mathematics, specifically bifurcation theory, the Feigenbaum constants ˈfaɪɡənˌbaʊm are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate.
Rössler attractorThe Rössler attractor ˈrɒslər is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Rössler interpreted it as a formalization of a taffy-pulling machine. Some properties of the Rössler system can be deduced via linear methods such as eigenvectors, but the main features of the system require non-linear methods such as Poincaré maps and bifurcation diagrams.
Gap geneA gap gene is a type of gene involved in the development of the segmented embryos of some arthropods. Gap genes are defined by the effect of a mutation in that gene, which causes the loss of contiguous body segments, resembling a gap in the normal body plan. Each gap gene, therefore, is necessary for the development of a section of the organism. Gap genes were first described by Christiane Nüsslein-Volhard and Eric Wieschaus in 1980. They used a genetic screen to identify genes required for embryonic development in the fruit fly Drosophila melanogaster.
MorphogenA morphogen is a substance whose non-uniform distribution governs the pattern of tissue development in the process of morphogenesis or pattern formation, one of the core processes of developmental biology, establishing positions of the various specialized cell types within a tissue. More specifically, a morphogen is a signaling molecule that acts directly on cells to produce specific cellular responses depending on its local concentration.
DiffusionDiffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios.
Pair-rule geneA pair-rule gene is a type of gene involved in the development of the segmented embryos of insects. Pair-rule genes are expressed as a result of differing concentrations of gap gene proteins, which encode transcription factors controlling pair-rule gene expression. Pair-rule genes are defined by the effect of a mutation in that gene, which causes the loss of the normal developmental pattern in alternating segments. Pair-rule genes were first described by Christiane Nüsslein-Volhard and Eric Wieschaus in 1980.
Convection–diffusion equationThe convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation, or (generic) scalar transport equation.
Cross-coupling reactionIn organic chemistry, a cross-coupling reaction is a reaction where two different fragments are joined. Cross-couplings are a subset of the more general coupling reactions. Often cross-coupling reactions require metal catalysts. One important reaction type is this: (R, R' = organic fragments, usually aryle; M = main group center such as Li or MgX; X = halide) These reactions are used to form carbon–carbon bonds but also carbon-heteroatom bonds. Cross-coupling reaction are a subset of coupling reactions.
Kumada couplingIn organic chemistry, the Kumada coupling is a type of cross coupling reaction, useful for generating carbon–carbon bonds by the reaction of a Grignard reagent and an organic halide. The procedure uses transition metal catalysts, typically nickel or palladium, to couple a combination of two alkyl, aryl or vinyl groups. The groups of Robert Corriu and Makoto Kumada reported the reaction independently in 1972. The reaction is notable for being among the first reported catalytic cross-coupling methods.
Hiyama couplingThe Hiyama coupling is a palladium-catalyzed cross-coupling reaction of organosilanes with organic halides used in organic chemistry to form carbon–carbon bonds (C-C bonds). This reaction was discovered in 1988 by Tamejiro Hiyama and Yasuo Hatanaka as a method to form carbon-carbon bonds synthetically with chemo- and regioselectivity. The Hiyama coupling has been applied to the synthesis of various natural products.
Coupling reactionIn organic chemistry, a coupling reaction is a type of reaction in which two reactant molecules are bonded together. Such reactions often require the aid of a metal catalyst. In one important reaction type, a main group organometallic compound of the type R-M (where R = organic group, M = main group centre metal atom) reacts with an organic halide of the type R'-X with formation of a new carbon-carbon bond in the product R-R'. The most common type of coupling reaction is the cross coupling reaction. Richard F.
Molecular diffusionMolecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration. Once the concentrations are equal the molecules continue to move, but since there is no concentration gradient the process of molecular diffusion has ceased and is instead governed by the process of self-diffusion, originating from the random motion of the molecules.
Fick's laws of diffusionFick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, D. Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation. A diffusion process that obeys Fick's laws is called normal or Fickian diffusion; otherwise, it is called anomalous diffusion or non-Fickian diffusion.
Diffusion equationThe diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation, when bulk velocity is zero.
