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The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems. The theorem is named after Henri Poincaré, who discussed it in 1890.
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior.Maps may be parameterized by a discrete-time or a continuous-time parameter.
Poincaré recurrence theorem, on sufficient conditions for recurrence to take place in dynamical systems; Poincaré-Bendixson theorem, on the existence of attractors for two-dimensional dynamical systems; Poincaré–Birkhoff–Witt theorem, concerning lie algebras and their universal envelopes; Poincaré lemma
A two-dimensional Poincaré section of the forced Duffing equation. In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system.
In the field of differential equations Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the Poincaré sphere and the Poincaré map. Poincaré on "everybody's belief" in the Normal Law of Errors (see normal distribution for an account of that "law")
This is effectively the modern statement of the Poincaré recurrence theorem. A sketch of a proof of the equivalence of these four properties is given in the article on the Hopf decomposition . Suppose that μ ( X ) < ∞ {\displaystyle \mu (X)<\infty } and τ {\displaystyle \tau } is measure-preserving.
This map describes the Poincaré's surface of section of the motion of a simple mechanical system known as the kicked rotator.The kicked rotator consists of a stick that is free of the gravitational force, which can rotate frictionlessly in a plane around an axis located in one of its tips, and which is periodically kicked on the other tip.
Graph of tent map function Example of iterating the initial condition x 0 = 0.4 over the tent map with μ = 1.9. In mathematics, the tent map with parameter μ is the real-valued function f μ defined by ():= {,}, the name being due to the tent-like shape of the graph of f μ.