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Feigenbaum Function
In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum: * the solution to the Feigenbaum-Cvitanović functional equation; and * the scaling function that described the covers of the attractor of the logistic map Feigenbaum-Cvitanović functional equation This functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. Discovered by Mitchell Feigenbaum and Predrag Cvitanović,Footnote on p. 46 of Feigenbaum (1978) states "This exact equation was discovered by P. Cvitanović during discussion and in collaboration with the author." the equation is the mathematical expression of the universality of period doubling. It specifies a function ''g'' and a parameter by the relation : g(x) = - \alpha g( g(\frac x ) ) with the initial conditions * ''g''(0) = 1, * ''g''′(0) = 0, and * ''g''′′ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
