Asymptotic Methods in Resonance Analytical Dynamics

Conditionally periodic solutions near the libration points of the restricted circular three-body problem

“…Ryabov (1952Ryabov ( , 1954 investigated some cases that were not considered by Lyapunov and formulated fairly general conditions for the existence of periodic solutions. Many results on periodic solutions are presented in the monographs by Szebehely (1967) and Grebenikov and Ryabov (1971). In his monograph, apart from periodic solutions, Charlier (1902) also considered some general properties of conditionally periodic solutions for Hamiltonian systems.…”

Asymptotic solution of the second order for corrugated rectangular waveguide

“…In this situation, the laboriousness of asymptotic methods [5][6][7][8][9] is not growing, and the error is reduced.…”

“…Their application is difficult for a combination of smooth and small oscillatory perturbation. For the analysis of the resonant interaction of waves in a weakly irregular waveguides with a small oscillating and smooth perturbation parameters the usage of the asymptotic method of Krylov, Bogolyubov, and Mitropolsky (KBM) [7,8] in the complex form [9][10][11][12][13][14][15][16][17][18][19][20][21] is appropriate.…”

Piecewise-linearized methods for oscillators with limit cycles

“…In the past, nonlinear single degree-of-freedom oscillators with limit cycles have been analyzed asymptotically by means of perturbation methods such as the harmonic balance procedure [6][7][8], the Krylov-Bogoliubov-Mitropolski (KBM) or slowly varying amplitude and phase technique [6,7,[9][10][11], etc., which do require the presence of a small parameter and are only valid for small values of the perturbation parameter. The application of either of these techniques has allowed to determine the first-order approximation to a variety of nonlinear oscillators with limit cycles such as the Duffing and van der Pol oscillators.…”