KrylovBogoliubov
KrylovBogoliubov refers to the mathematical framework developed jointly by Russian mathematicians Alexander Alexandrovich Krylov and Nikolay Nikolaevich Bogoliubov, which provides foundational results in the theory of nonlinear dynamical systems. The most widely cited contribution is the Krylov–Bogolyubov theorem, which establishes the existence of invariant measures for continuous dynamical systems. By constructing a sequence of probability measures derived from averaging the iterates of a point under a continuous map or flow, the theorem shows that under mild compactness and continuity assumptions a limit measure exists and is invariant under the dynamics. This result has become a cornerstone in the study of ergodic theory and statistical mechanics, allowing researchers to rigorously define and analyze equilibrium states for systems that lack explicit solutions.
Beyond invariant measures, Krylov and Bogoliubov introduced methods for analyzing the long-term behaviour of nonlinear waves
The duo’s work also contributed to the rigorous formulation of the Boltzmann equation and the foundations of
Today, KrylovBogoliubov is recognized both as a specific theorem and as a broader methodological legacy that