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Kuramoto Model
The Kuramoto model (or Kuramoto–Daido model), first proposed by , is a mathematical model used in describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated by the behavior of systems of chemical and biological process, biological oscillators, and it has found widespread applications in areas such as Neural oscillation#Mathematical description, neuroscience and oscillating flame dynamics. Kuramoto was quite surprised when the behavior of some physical systems, namely coupled arrays of Josephson junctions, followed his model. The model makes several assumptions, including that there is weak coupling, that the oscillators are identical or nearly identical, and that interactions depend sinusoidally on the phase difference between each pair of objects. Definition In the most popular version of the Kuramoto model, each of the oscillators is considered to have its own intrinsic natural frequency ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoshiki Kuramoto
(born 1940) is a Japanese physicist in the Nonlinear Dynamics group at Kyoto University who formulated the Kuramoto model and is also known for the Kuramoto–Sivashinsky equation. He is also the discoverer of so-called chimera states in networks of coupled oscillators. Kuramoto specializes in Nonlinear dynamics, nonlinear dynamics (also known as Nonlinear system, nonlinear science) and Nonequilibrium thermodynamics, non-equilibrium statistical mechanics. Notably, he has worked on the network dynamics created by limit cycle oscillators. Among his accomplishments is the derivation of the Kuramoto–Sivashinsky equation, which describes the phase instability of oscillating fields. This is regarded as the first example of spatiotemporal chaos. Another achievement is his proposal of a solvable model for oscillator populations, now known as the Kuramoto model. Other achievements include deriving the complex Ginzburg–Landau equation in reaction-diffusion systems and studying the entra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Map
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant (mathematics), invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form geometric shapes that resemble tongues, in which case they are called Arnold tongues. Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and Electrocardiography, cardiac electric waves. Sometimes the frequency of oscillation depends on, or is constrained (i.e., ''phase-locked'' or ''mode-locked'', in some contexts) based on some quantity, and it is often of interest to study this relation. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Rényi Model
In the mathematical field of graph theory, the Erdős–Rényi model refers to one of two closely related models for generating random graphs or the evolution of a random network. These models are named after Hungarians, Hungarian mathematicians Paul Erdős and Alfréd Rényi, who introduced one of the models in 1959. Edgar Gilbert introduced the other model contemporaneously with and independently of Erdős and Rényi. In the model of Erdős and Rényi, all graphs on a fixed vertex set with a fixed number of edges are equally likely. In the model introduced by Gilbert, also called the Erdős–Rényi–Gilbert model, each edge has a fixed probability of being present or absent, statistical independence, independently of the other edges. These models can be used in the probabilistic method to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs. Definition There are two clo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neuropercolation
Percolation (from the Latin word ''percolatio'', meaning filtration) is a theoretical model used to understand the way activation and diffusion of neural activity occurs within neural networks.Friedenbreg, J., Silverman, G. (2012). ''Cognitive Science: An Introduction to the Study of Mind'' (2nd Ed.). Thousand Oaks, CA: SAGE Publications. Percolation is a model used to explain how neural activity is transmitted across the various connections within the brain. Percolation theory can be easily understood by explaining its use in epidemiology. Individuals who are infected with a disease can spread the disease through contact with others in their social network. Those who are more social and come into contact with more people will help to propagate the disease quicker than those who are less social. Factors such as occupation and sociability influence the rate of infection. Now, if one were to think of ''neurons'' as ''individuals'' and ''synaptic connections'' as the ''social bonds' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fitness Model (network Theory)
In complex network theory, the fitness model is a model of the evolution of a network: how the links between nodes change over time depends on the fitness of nodes. Fitter nodes attract more links at the expense of less fit nodes. It has been used to model the network structure of the World Wide Web. Description of the model The model is based on the idea of fitness, an inherent competitive factor that nodes may have, capable of affecting the network's evolution. According to this idea, the nodes' intrinsic ability to attract links in the network varies from node to node, the most efficient (or "fit") being able to gather more edges in the expense of others. In that sense, not all nodes are identical to each other, and they claim their degree increase according to the fitness they possess every time. The fitness factors of all the nodes composing the network may form a distribution ρ(η) characteristic of the system been studied. Ginestra Bianconi and Albert-László Barabási p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michelle Girvan
Michelle Girvan (born 1977) is an American physicist and network science, network scientist whose research combines methods from dynamical systems, graph theory, and statistical mechanics and applies them to problems including epidemiology, gene regulation, and the study of Information cascades. She is one of the namesakes of the Girvan–Newman algorithm, used to detect community structure in complex systems. Girvan is a professor of physics at the University of Maryland, College Park. Education and career Girvan graduated from the Massachusetts Institute of Technology in 1999, with a double major in mathematics and physics and a minor in political science. She completed a Ph.D. in physics at Cornell University in 2004. Her dissertation, ''The Structure and Dynamics of Complex Networks'', was supervised by Steven Strogatz. After postdoctoral research at the Santa Fe Institute, she joined the University of Maryland faculty in 2007. Recognition In 2017 Girvan was named a Fellow o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Bifurcation
In the mathematics of dynamical systems and differential equations, a Hopf bifurcation is said to occur when varying a parameter of the system causes the set of solutions (trajectories) to change from being attracted to (or repelled by) a fixed point, and instead become attracted to (or repelled by) an oscillatory, periodic solution. The Hopf bifurcation is a two-dimensional analog of the pitchfork bifurcation. Many different kinds of systems exhibit Hopf bifurcations, from radio oscillators to railroad bogies. Trailers towed behind automobiles become infamously unstable if loaded incorrectly, or if designed with the wrong geometry. This offers a gut-sense intuitive example of a Hopf bifurcation in the ordinary world, where stable motion becomes unstable and oscillatory as a parameter is varied. The general theory of how the solution sets of dynamical systems change in response to changes of parameters is called bifurcation theory; the term ''bifurcation'' arises, as the set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large N Limit
In quantum field theory and statistical mechanics, the 1/''N'' expansion (also known as the "large ''N''" expansion) is a particular perturbation theory, perturbative analysis of quantum field theories with an internal symmetry group theory, group such as special orthogonal group, SO(N) or special unitary group, SU(N). It consists in deriving an expansion for the properties of the theory in powers of 1/N, which is treated as a small parameter. This technique is used in Quantum chromodynamics, QCD (even though N is only 3 there) with the gauge group SU(3). Another application in particle physics is to the study of AdS/CFT dualities. It is also extensively used in condensed matter physics where it can be used to provide a rigorous basis for mean-field theory. Example Starting with a simple example — the orthogonal group, O(N) Quartic interaction, φ4 — the scalar field φ takes on values in the real number, real vector representation of O(N). Using the index notation for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transformation
Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Transformation (genetics), genetic alteration of a cell by DNA uptake In mathematics * Transformation (function), concerning functions from sets to themselves. For functions in the broader sense, see function (mathematics). **Affine transformation, in geometry **Linear transformation between modules in linear algebra. Also called a linear map. ***Transformation matrix which represent linear maps in linear algebra. *Integral transform, between a function in one domain to a function in another * Natural transformation between functors in category theory. * Unitary transformation, between two Hilbert spaces * Geometric transformation, between sets of points in geometry **Infinitesimal transformation, a limiting case of a geometrical transformatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spatial Synchronization Patterns In Variants Of The Kuramoto Model
Spatial may refer to: *Dimension *Space *Three-dimensional space *Spatial (platform) Spatial is a Unity-powered UGC gaming platform that enables developers to publish and monetize multiplayer games across web, mobile, and VR. Spatial focuses on games developed using the Unity game engine and the C# programming language. The com ... See also * * {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bose–Einstein Condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low Density, densities is cooled to temperatures very close to absolute zero#Relation with Bose–Einstein condensate, absolute zero, i.e. . Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which microscopic Quantum mechanics, quantum-mechanical phenomena, particularly wave interference#Quantum interference, wavefunction interference, become apparent Macroscopic quantum phenomena, macroscopically. More generally, condensation refers to the appearance of macroscopic occupation of one or several states: for example, in BCS theory, a superconductor is a condensate of Cooper pairs. As such, condensation can be associated with phase transition, and the macroscopic occupation of the state is the order parameter. Bose–Einstein condensate was first predicted, generally, in 1924–1925 by Albert Einstein, credit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
