Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
Perhaps his best-known work is his introduction of cycle expansions—that is, expansions based on using periodic orbit theory—to approximate chaotic dynamics in a controlled perturbative way.
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This resembles the Born approximation, in that the details of the problem are treated as a perturbation.
The equilibrium here is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state.
The perturbation theory of toroidal invariant manifolds of dynamical systems was developed here by academician M. M. Bogolyubov, Yu. O. Mitropolsky, academician of the NAS of Ukraine and the Russian Academy of Sciences, and A. M. Samoilenko, academician of the NAS of Ukraine.
This selection rule arises from a first-order perturbation theory approximation of the time-dependent Schrödinger equation.