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Kuramoto Model
The Kuramoto model (or Kuramoto–Daido model), first proposed by , is a mathematical model used to 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 oscillators, and it has found widespread applications in areas such as 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 \omega_i, and each is coupled equally to all other oscillators ... [...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 (also known as nonlinear science) and 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 entrainment phenomenon in coupled oscillator systems. Biography He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamilton Function
Hamilton may refer to: People * Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname ** The Duke of Hamilton, the premier peer of Scotland ** Lord Hamilton (other), several Scottish, Irish and British peers, and some members of the judiciary, who may be referred to simply as ''Hamilton'' ** Clan Hamilton, an ancient Scottish kindred * Alexander Hamilton (1755–1804), first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States * Lewis Hamilton, a British Formula One driver *William Rowan Hamilton (1805–1865), Irish physicist, astronomer, and mathematician for whom ''Hamiltonian mechanics'' is named *Hamílton (footballer) (born 1980), Togolese footballer Places Australia * Hamilton, New South Wales, suburb of Newcastle * Hamilton Hill, Western Australia, suburb of Perth * Hamilton, South Australia * Hamilton, Tasmania * Hamilton, Victoria Queensl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heteroclinic Cycle
In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits. If a heteroclinic cycle is asymptotically stable, approaching trajectories spend longer and longer periods of time in a neighbourhood of successive equilibria. In generic dynamical systems heteroclinic connections are of high co-dimension, that is, they will not persist if parameters are varied. Robust heteroclinic cycles A robust heteroclinic cycle is one which persists under small changes in the underlying dynamical system. Robust cycles often arise in the presence of symmetry or other constraints which force the existence of invariant hyperplanes. A prototypical example of a robust heteroclinic cycle is the Guckenheimer–Holmes cycle. This cycle has also been studied in the context of rotating convection, and as three competing species in population dynamics. See also * Heteroclinic b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodgkin–Huxley Model
The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical characteristics of excitable cells such as neurons and muscle cells. It is a continuous-time dynamical system. Alan Hodgkin and Andrew Huxley described the model in 1952 to explain the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. They received the 1963 Nobel Prize in Physiology or Medicine for this work. Basic components The typical Hodgkin–Huxley model treats each component of an excitable cell as an electrical element (as shown in the figure). The lipid bilayer is represented as a capacitance (Cm). Voltage-gated ion channels are represented by electrical conductances (''g''''n'', where ''n'' is the specific ion channel) that depend on both voltage and time. Leak channels are rep ... [...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 whom 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 pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michelle Girvan
Michelle Girvan (born 1977) is an American physicist and 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 The Santa Fe Institute (SFI) is an independent, nonprofit theoretical research institute located in Santa Fe, New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point. A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf. Overview Supercritical and subcritical Hopf bifurcations The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical. The normal form of a Hopf bifurcation is: ::\frac ... [...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 perturbative analysis of quantum field theories with an internal symmetry group such as SO(N) or 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 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 O(N) φ4 — the scalar field φ takes on values in the real vector representation of O(N). Using the index notation for the N " flavors" with the Einstein summation convention and because O(N) is orthogonal, no distinction will be made between covariant and contravariant ... [...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] |
