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Blue Sky Catastrophe
The blue sky catastrophe is a form of orbital indeterminacy, and an element of bifurcation theory. Orbital dynamics Blue sky catastrophe is a type of bifurcation of a periodic orbit. In other words, it describes a sort of behaviour stable solutions of a set of differential equations can undergo as the equations are gradually changed. This type of bifurcation is characterised by both the period and length of the orbit approaching infinity as the control parameter approaches a finite bifurcation value, but with the orbit still remaining within a bounded part of the phase space, and without loss of stability before the bifurcation point. In other words, the orbit ''vanishes into the blue sky''. Applications of blue sky catastrophe in other fields The bifurcation has found application in, amongst other places, slow-fast models of computational neuroscience. The possibility of the phenomenon was raised by David Ruelle and Floris Takens in 1971, and explored by R.L. Devaney and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bifurcation Theory
Bifurcation 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. Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them . Bifurcation types It is useful to divide bifurcations into two principal classes: * Local bifurcations, which can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Orbit
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a mapping ''f'' from a set ''X'' into itself, :f: X \to X, a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that :\ f_n(x) = x where f_n is the ''n''th iterate of ''f''. The smallest positive integer ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a periodic function). If there exist distinct ''n'' and ''m'' such that :f_n(x) = f_m(x) then ''x'' is called a preperiodic point. All periodic points are preperiodic. If ''f'' is a diffeomorphism of a differentiable manifold, so that the derivative f_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbital Period
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars. For celestial objects in general, the sidereal period ( sidereal year) is referred to by the orbital period, determined by a 360° revolution of one body around its primary, e.g. Earth around the Sun, relative to the fixed stars projected in the sky. Orbital periods can be defined in several ways. The tropical period is more particularly about the position of the parent star. It is the basis for the solar year, and respectively the calendar year. The synodic period incorporates not only the orbital relation to the parent star, but also to other celestial objects, making it not a mere different approach to the orbit of an object around its parent, but a period of orbital relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in. Various terms for the length of a fixed object are used, and these include height, which is vertical length or vertical extent, and width, breadth or depth. Height is used when there is a base from which vertical measurements can be taken. Width or breadth usually refer to a shorter dimension when length is the longest one. Depth is used for the third dimension of a three dimensional object. Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded
Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded emotionality, a concept within communication theory that stems from emotional labor and bounded rationality Linguistics * Boundedness (linguistics), whether a situation has a clearly defined beginning or end Mathematics * Boundedness axiom, the axiom schema of replacement * Bounded deformation, a function whose distributional derivatives are not quite well-behaved-enough to qualify as functions of bounded variation, although the symmetric part of the derivative matrix does meet that condition * Bounded growth, occurs when the growth rate of a mathematical function is constantly increasing at a decreasing rate * Bounded operator, a linear transformation ''L'' between normed vector spaces for which the ratio of the norm of ''L''(''v'') to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the outer product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. Introduction In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase-space trajectory for the system) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit (dynamics)
In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As a phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems. For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces. Definition Given a dynamical system (''T'', ''M'', Φ) with ''T'' a group, ''M'' a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Neuroscience
Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to understand the principles that govern the development, structure, physiology and cognitive abilities of the nervous system. Computational neuroscience employs computational simulations to validate and solve mathematical models, and so can be seen as a sub-field of theoretical neuroscience; however, the two fields are often synonymous. The term mathematical neuroscience is also used sometimes, to stress the quantitative nature of the field. Computational neuroscience focuses on the description of biologically plausible neurons (and neural systems) and their physiology and dynamics, and it is therefore not directly concerned with biologically unrealistic models used in connectionism, control theory, cybernetics, quantitative psychology, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Ruelle
David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term ''strange attractor'', and developed a new theory of turbulence. Biography Ruelle studied physics at the Université Libre de Bruxelles, obtaining a PhD degree in 1959 under the supervision of Res Jost. He spent two years (1960–1962) at the ETH Zurich, and another two years (1962–1964) at the Institute for Advanced Study in Princeton, New Jersey. In 1964, he became professor at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France. Since 2000, he has been an emeritus professor at IHES and distinguished visiting professor at Rutgers University. David Ruelle made fundamental contributions in various aspects of mathematical physics. In quantum field theory, the most important contribution is the rigorous formulation of scattering processes based on Wigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floris Takens
Floris Takens (12 November 1940 – 20 June 2010) was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems. Together with David Ruelle, he predicted that fluid turbulence could develop through a strange attractor, a term they coined, as opposed to the then-prevailing theory of accretion of modes. The prediction was later confirmed by experiment. Takens also established the result now known as the Takens's theorem, which shows how to reconstruct a dynamical system from an observed time-series. He was the first to show how chaotic attractors could be learned by neural networks. Takens was born in Zaandam in the Netherlands. He attended schools in The Hague and in Zaandam before serving in the Dutch army for one year (1960–1961). At the University of Amsterdam he concluded his undergraduate and graduate studies. He was granted a doctorate in mathematics in 1969 under the supervision of Nicolaas Kuiper for a thesis entitled ''The minimal number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
