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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] |
Supercritical Hopf Bifurcation
Supercritical may refer to: Physics and technology Condensed matter physics * Critical temperature, TC, a temperature above which distinct liquid and gas phases do not exist for a given material ** Supercritical drying, a process used to remove liquid in a precisely controlled way, similar to freeze drying ** Supercritical fluid, a substance at a temperature and pressure above its thermodynamic critical point: *** Supercritical carbon dioxide: **** Supercritical fluid chromatography, a form of liquid chromatography using supercritical carbon dioxide as the mobile phase ***Supercritical water: **** Supercritical steam generator, a steam generator operating above the critical point of water, hence having no water–steam separation **** Supercritical water oxidation or SCWO, a process that occurs in water at temperatures and pressures above a mixture's thermodynamic critical point **** Supercritical water reactor (SCWR), a Generation IV nuclear reactor concept that uses supercri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm Series
In mathematics, the Sturm series associated with a pair of polynomials is named after Jacques Charles François Sturm. Definition Let p_0 and p_1 two univariate polynomials. Suppose that they do not have a common root and the degree of p_0 is greater than the degree of p_1. The ''Sturm series'' is constructed by: : p_i := p_ q_ - p_ \text i \geq 0. This is almost the same algorithm as Euclid's but the remainder p_ has negative sign. Sturm series associated to a characteristic polynomial Let us see now Sturm series p_0,p_1,\dots,p_k associated to a characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ... P in the variable \lambda: : P(\lambda)= a_0 \lambda^k + a_1 \lambda^ + \cdots + a_ \lambda + a_k where a_i for i in \ are rational functions in \mathbb(Z) wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Routh–Hurwitz Stability Criterion
In control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts. German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. Suppose is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is \mathbf J_ = \frac, or explicitly :\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^ f_1 \\ \vdots \\ \nabla^ f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cdots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical field theory. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics, which is a quantum field theory. Fundamental physical aspects of classical electrodynamics are presented in many texts, such as those by Feynman, Leighton and Sands, Griffiths, Panofsky and Phillips, and Jackson. History The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of optics centuries before light was understood to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brusselator
The Brusselator is a theoretical model for a type of autocatalytic reaction. The Brusselator model was proposed by Ilya Prigogine and his collaborators at the Université Libre de Bruxelles. It is a portmanteau of Brussels and oscillator. It is characterized by the reactions : A \rightarrow X : 2X + Y \rightarrow 3X : B + X \rightarrow Y + D : X \rightarrow E Under conditions where A and B are in vast excess and can thus be modeled at constant concentration, the rate equations become :\left\ = \left\ + \left\^2 \left\ - \left\ \left\ - \left\ \, :\left\ = \left\ \left\ - \left\^2 \left\ \, where, for convenience, the rate constants have been set to 1. The Brusselator has a fixed point at :\left\ = A \, :\left\ = \,. The fixed point becomes unstable when : B>1+A^2 \, leading to an oscillation of the system. Unlike the Lotka–Volterra equation, the oscillations of the Brusselator do not depend on the amount of reactant present initially. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorenz Attractor
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the " butterfly effect" stems from the real-world implications of the Lorenz attractor, namely that in a chaotic physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable. The shape of the Lorenz attractor itself, when plotted in phase space, may also be seen to resemble a butterfly. Overview In 1963, Edward Lorenz, with the help of Ellen Fetter who was responsible for the numerical s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belousov–Zhabotinsky Reaction
A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillators is the inclusion of bromine and an acid. The reactions are important to theoretical chemistry in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior. These reactions are far from equilibrium and remain so for a significant length of time and evolve chaotically. In this sense, they provide an interesting chemical model of nonequilibrium biological phenomena; as such, mathematical models and simulations of the BZ reactions themselves are of theoretical interest, showing phenomenon as noise-induced order. An essential aspect of the BZ reaction is its so called "excitability"; under the influence of stimuli, patterns develop in what would otherwise be a perfectly quies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glycolysis
Glycolysis is the metabolic pathway that converts glucose () into pyruvate (). The free energy released in this process is used to form the high-energy molecules adenosine triphosphate (ATP) and reduced nicotinamide adenine dinucleotide (NADH). Glycolysis is a sequence of ten reactions catalyzed by enzymes. Glycolysis is a metabolic pathway that does not require oxygen (In anaerobic conditions pyruvate is converted to lactic acid). The wide occurrence of glycolysis in other species indicates that it is an ancient metabolic pathway. Indeed, the reactions that make up glycolysis and its parallel pathway, the pentose phosphate pathway, occur in the oxygen-free conditions of the Archean oceans, also in the absence of enzymes, catalyzed by metal. In most organisms, glycolysis occurs in the liquid part of cells, the cytosol. The most common type of glycolysis is the ''Embden–Meyerhof–Parnas (EMP) pathway'', which was discovered by Gustav Embden, Otto Meyerhof, and Jakub Karol ... [...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] |
Paradox Of Enrichment
The paradox of enrichment is a term from population ecology coined by Michael Rosenzweig in 1971. He described an effect in six predator–prey models where increasing the food available to the prey caused the predator's population to destabilize. A common example is that if the food supply of a prey such as a rabbit is overabundant, its population will grow unbounded and cause the predator population (such as a lynx) to grow unsustainably large. That may result in a crash in the population of the predators and possibly lead to local eradication or even species extinction. The term 'paradox' has been used since then to describe this effect in slightly conflicting ways. The original sense was one of irony; by attempting to increase the carrying capacity in an ecosystem, one could fatally imbalance it. Since then, some authors have used the word to describe the difference between modelled and real predator–prey interactions. Rosenzweig used ordinary differential equation models ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
