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The Poincaré recurrence time is the length of time elapsed until the recurrence. This time may vary greatly depending on the exact initial state and required degree of closeness. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume.
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.
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
In ergodic theory, Kac's lemma, demonstrated by mathematician Mark Kac in 1947, [1] is a lemma stating that in a measure space the orbit of almost all the points contained in a set of such space, whose measure is (), return to within an average time inversely proportional to ().
In quantum mechanics, the quantum revival [1] is a periodic recurrence of the quantum wave function from its original form during the time evolution either many times in space as the multiple scaled fractions in the form of the initial wave function (fractional revival) or approximately or exactly to its original form from the beginning (full ...
A Poincaré plot, named after Henri Poincaré, is a graphical representation used to visualize the relationship between consecutive data points in time series to detect patterns and irregularities in the time series, revealing information about the stability of dynamical systems, providing insights into periodic orbits, chaotic motions, and bifurcations.
Poincaré's recurrence theorem: certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. Poincaré–Bendixson theorem : a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.
In 1896, the mathematician Ernst Zermelo advanced a theory that the second law of thermodynamics was absolute rather than statistical. [7] Zermelo bolstered his theory by pointing out that the Poincaré recurrence theorem shows statistical entropy in a closed system must eventually be a periodic function; therefore, the Second Law, which is always observed to increase entropy, is unlikely to ...