An asymptotic symmetry of the rapidly forced pendulum

Technical Report · Sat Sep 01 00:00:00 EDT 1990

DOI:https://doi.org/10.2172/6504138· OSTI ID:6504138

Chang, Yi-Hua ^[1]; Segur, H ^[2]

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1. Colorado Univ., Boulder, CO (USA). Applied Mathematics Program State Univ. of New York, Buffalo, NY (USA). Dept. of Mathematics
2. Colorado Univ., Boulder, CO (USA). Applied Mathematics Program

The inhomogeneous differential equation (x{double prime} + sin x = {delta} sin (t + t{sub 0}){var epsilon}) describes the motion of a sinusoidally forced pendulum. The orbits that connect the two saddle points of the unforced ({delta} = 0) pendulum, (x = {pi}) and (x = {minus}{pi}), are called separatrices. If {var epsilon} {Omicron}(1), then one can use Melnikov's method to show that these separatrices can split for weak forcing ({delta} {much lt} 1), and that the perturbed motion is chaotic. If {var epsilon} 1, Melnikov's method fails because the perturbation term is not analytic in {var epsilon} at {var epsilon} = 0. In this paper we show that for {delta} {much lt} 1 and {var epsilon} {much lt} 1, the solution of the perturbed problem exhibits a symmetry to all orders in an asymptotic expansion. From the asymptotic expansion it follows that the separatrices split by an amount that is at most transcendentally small. This proof differs from that of Holmes, Marsden and Scheurle. 16 refs.

• Research Organization: Colorado Univ., Boulder, CO (USA). Applied Mathematics Program; State Univ. of New York, Buffalo, NY (USA). Dept. of Mathematics
• Sponsoring Organization: DOE/ER
• DOE Contract Number: FG02-86ER25021
• OSTI ID: 6504138
• Report Number(s): DOE/ER/25021-T1; ON: DE91001868
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS
990200 -- Mathematics & Computers
ASYMPTOTIC SOLUTIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
HAMILTONIANS
ITERATIVE METHODS
MATHEMATICAL OPERATORS
PENDULUMS
QUANTUM OPERATORS
SYMMETRY