Technical Physics, Vol. 49, No. 5, 2004, pp. 521–525. Translated from Zhurnal TekhnicheskoÏ Fiziki, Vol. 74, No. 5, 2004, pp. 1–5. Original Russian Text Copyright © 2004 by Vecheslavov. THEORETICAL AND MATHEMATICAL PHYSICS Chaotic Layer of a Pendulum under Low- and Medium-Frequency Perturbations V. V. Vecheslavov Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 11, Novosibirsk, 630090 Russia e-mail:
[email protected] Received April 22, 2003 Abstract—The amplitude of the separatrix map and the size of a pendulum chaotic layer are studied numeri- cally and analytically as functions of the adiabaticity parameter at low and medium perturbation frequencies. Good agreement between the theory and numerical experiment is found at low frequencies. In the medium-fre- quency range, the efficiency of using resonance invariants of separatrix mapping is high. Taken together with the known high-frequency asymptotics, the results obtained in this work reconstruct the chaotic layer pattern throughout the perturbation frequency range. © 2004 MAIK “Nauka/Interperiodica”. INTRODUCTION important that actually both unperturbed separatrices Interaction between nonlinear resonances with the consist of two spatially coincident trajectories for for- formation of dynamic chaos in Hamiltonian systems is ward and backward time, respectively. a complex problem, which is far from being solved. In In the case of an analytical potential (as in (1)), the a number of cases, this problem may be reduced to presence of at least one (!) perturbing resonance always studying a pendulum (a fundamental resonance near results in splitting either of the separatrices into two which initial conditions are chosen) subjected to a branches (“whiskers” after Arnold).