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Poincaré–Lindstedt Method
In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms—terms growing without bound—arising in the straightforward application of perturbation theory to weakly nonlinear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ... problems with finite oscillatory solutions. The method is named after Henri Poincaré, and Anders Lindstedt. The article gives several examples. The theory can be found in Chapter 10 of Nonlinear Differential Equations and Dynamical Systems by Verhulst. Example: the Duffing equation The undamped, unforced Duffing equation is given by :\ddot + x + \varepsi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In regular perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Function
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called ''aperiodic''. Definition A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period. A funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secular Variation
The secular variation of a time series is its long-term, non-periodic variation (see '' Decomposition of time series''). Whether a variation is perceived as secular or not depends on the available timescale: a variation that is secular over a timescale of centuries may be a segment of what is, over a timescale of millions of years, a periodic variation. Natural quantities often have both periodic and secular variations. Secular variation is sometimes called secular trend or secular drift when the emphasis is on a linear long-term trend. The term is used wherever time series are applicable in history, economics, operations research, biological anthropology, and astronomy (particularly celestial mechanics) such as VSOP (planets). Etymology The word ''secular'', from the Latin root ''saecularis'' ("of an age, occurring once in an age"), has two basic meanings: I. Of or pertaining to the world (from which secularity is derived), and II. Of or belonging to an age or long period. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. He has further been called "the Carl Friedrich Gauss, Gauss of History of mathematics, modern mathematics". Due to his success in science, along with his influence and philosophy, he has been called "the philosopher par excellence of modern science". As a mathematician and physicist, he made many original fundamental contributions to Pure mathematics, pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. Poincaré is regarded as the cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anders Lindstedt
Anders Lindstedt (27 June 1854 – 16 May 1939) was a Swedish mathematician, astronomer, and actuarial scientist, known for the Lindstedt-Poincaré method. Life and work Lindstedt was born in a small village in the district of Sundborns, Dalecarlia a province in central Sweden.Hvar 8 dag, 10:de Årg, No 11, 13 december 1908, sid. 162'.Memoir Anders Lindstedt 27 June 1854-16 May 1939, Journal of the Institute of Actuaries, 70 (1939) p. 269/ref> He obtained a PhD from the University of Lund aged 32 and was subsequently appointed as a lecturer in astronomy. He later went on to a position at the University of Dorpat (then belonging to Russia, now University of Tartu in Estonia) where he worked for around seven years on theoretical astronomy. He combined practical astronomy with an interest in theory, developing especially an interest in the three-body problem This work was to influence Poincaré whose work on the three-body problem led to the discovery that there can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duffing Equation
The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain Harmonic oscillator, damped and driven oscillators. The equation is given by \ddot + \delta \dot + \alpha x + \beta x^3 = \gamma \cos (\omega t), where the (unknown) function x = x(t) is the displacement at time , \dot is the first derivative of x with respect to time, i.e. velocity, and \ddot is the second time-derivative of x, i.e. acceleration. The numbers \delta, \alpha, \beta, \gamma and \omega are given constants. The equation describes the motion of a damped oscillator with a more complex potential than in simple harmonic motion (which corresponds to the case \beta=\delta=0); in physical terms, it models, for example, an elastic pendulum whose spring's stiffness does not exactly obey Hooke's law. The Duffing equation is an example of a dynamical system that exhibits chaos theory, chaotic behavior. Moreover, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secular Term
Secularity, also the secular or secularness (from Latin , or or ), is the state of being unrelated or neutral in regards to religion. The origins of secularity can be traced to the Bible itself. The concept was fleshed out through Christian history into the modern era. Since the Middle Ages, there have been clergy not pertaining to a religious order called "secular clergy". Furthermore, secular and religious entities were not separated in the medieval period, but coexisted and interacted naturally. The word ''secular'' has a meaning very similar to profane as used in a religious context. Today, anything that is not directly connected with religion may be considered secular, in other words, neutral to religion. Secularity does not mean , but . Many activities in religious bodies are secular, and though there are multiple types of secularity or secularization, most do not lead to irreligiosity. Linguistically, a process by which anything becomes secular is named ''secularization' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. The theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886. The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a '' convergent'' Taylor seri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leading-order
The leading-order terms (or leading-order corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.math.ucdavis.edu/~hunter/notes/asy.pdf The sizes of the different terms in the equation(s) will change as the variables change, and hence, which terms are leading-order may also change. A common and powerful way of simplifying and understanding a wide variety of complicated mathematical models is to investigate which terms are the largest (and therefore most important), for particular sizes of the variables and parameters, and analyse the behaviour produced by just these terms (regarding the other smaller terms as negligible). This gives the main behaviour – the true behaviour is only small deviations away from this. The main behaviour may be captured sufficiently well by just the strictly leading-order terms, or it may be decided that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Der Pol Oscillator
In the study of dynamical systems, the van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a non-Conservative force, conservative, oscillating system with non-linear damping. It evolves in time according to the second-order differential equation - \mu(1-x^2) + x = 0, where is the position coordinate system, coordinate—which is a function (mathematics), function of the time —and is a scalar (mathematics), scalar parameter indicating the nonlinearity and the strength of the damping. History The Van der Pol oscillator was originally proposed by the Dutch electrical engineering, electrical engineer and physicist Balthasar van der Pol while he was working at Philips. Van der Pol found stable oscillations, which he subsequently called relaxation oscillator, relaxation-oscillations and are now known as a type of limit cycle, in electrical circuits employing vacuum tubes. When these circuits are driven near the limit cycle, they become entrainment (phys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
