Abstract
Resonant capture describes the behavior of a weakly coupled multi-degree-of-freedom system when two or more of its uncoupled frequencies become locked in resonance. Flow on the region of phase space near the resonance (the resonance manifold) involves a region bounded by a separatrix in the uncoupled (ε = 0) system. Capture corresponds to motions which appear to cross into the interior of the separated region for ε > 0.
We offer two approximate methods for estimating which initial conditions lead to capture: an energy method and a perturbation method based on invariant manifold theory. These methods are applied to a model problem involving the spinup of an unbalanced rotor attached to an elastic support.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beigie, D., Leonard, A., and Wiggins, S., ‘Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems’, Nonlinearity 4, 1991, 775–819.
Bosley, D. L. and Kevorkian, J., ‘Sustained resonance in very slowly varying oscillatory Hamiltonian systems’, SIAM Journal of Applied Mathematics 51, 1991, 439–471.
Bourland, F. J. and Haberman, R., ‘Capture and the connection formulas for the transition across a separatrix’, in Asymptotic Analysis and the Numerical Solution of P.D.E.’s, Kaper, H. G. and Garbey, M. (eds.), Dekker, New York, 1991, pp. 17–30.
Bourland, F. J. and Haberman, R., ‘Separatrix crossing: Time-invariant potentials with dissipation’, SIAM Journal of Applied Mathematics 50, 1990, 1716–1744.
Bridge, J., ‘Chaos in dynamical systems with periodically disappearing separatrices’, Ph.D. Thesis, Cornell University, 1992.
Bridge, J. and Rand, R., ‘Chaos and symbol sequences in systems with a periodically-disappearing figure-eight separatrix’, in Bifurcation Phenomena and Chaos in Thermal Convection, Bau, H. H., Bertram, L.A., and Korpela, S. A. (eds.), American Society of Mechanical Engineering, HTD, Vol. 124, 1992, pp. 47–55.
Cary, J. R. and Skodje, R. T., ‘Phase change between separatrix crossings’, Physica D 36, 1989, 287–316.
Cary, J. R., Escande, D. F., and Tennyson, J. L., ‘Adiabatic-invariant change due to separatrix crossing’, Physics Reviews A 34, 1986, 4256–4275.
Coppola, V. T. and Rand, R. H., ‘Chaos in a system with a periodically disappearing separatrix’, Nonlinear Dynamics 1, 1990, 401–420.
Hall, C. D. and Rand, R. H., ‘Spinup dynamics of axial dual-spin spacecraft’, in Astronomies 1991, Advances in the Astronautical Sciences, Kaufman, B., Alfriend, K. T., Roehrich, R. L., and Dasenbrock, R. R. (eds.) Vol. 76, American Astronautical Society, 1992.
Henrard, J., ‘Capture into resonance: An extension of the use of adiabatic invariants’, Celestial Mechanics 27, 1982, 3–22.
Henrard, J., ‘A second fundamental model for resonance’, Celestial Mechanics 30, 1983, 197–218.
Kaper, T. J., Kovacic, G., and Wiggins, S., ‘Melnikov functions, action, and lobe area in Hamiltonian systems’, Los Alamos Report LA-UR 90–2455, 1990.
Lochak, P. and Meunier, C., Multiphase Averaging for Classical Systems, Springer-Verlag, Berlin-New York, 1988.
Neishtadt, A. I., ‘Passage through a separatrix in a resonance problem with a slowly-varying parameter’, Journal of Applied Mathematics and Physics (PMM) 39, 1975, 594–605.
Neishtadt, A. I., ‘Averaging and passage through resonances’, in Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990.
Rand, R. H., Kinsey, R. J., and Mingori, D. L., ‘Dynamics of spinup through resonance’, International Journal of Nonlinear Mechanics 27, 1992, 489–502.
Robinson, C., ‘Sustained resonance for a nonlinear system with slowly varying coefficients’, SIAM Journal of Mathematical Analysis 14, 1983, 847–860.
Sanders, J. A. and Verhulst, F., Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, Berlin-New York, 1985.
Vakakis, A. F., ‘Exponentially small splittings of manifolds in a rapidly forced Duffing system’, Journal of Sound and Vibration 170, 1994, 119–129.
Yee, R. K., ‘Spinup dynamics of a rotating system with limited torque’, M.S. Thesis, UCLA, 1981.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Quinn, D., Rand, R., Bridge, J. (1995). The Dynamics of Resonant Capture. In: Bajaj, A.K., Shaw, S.W. (eds) Advances in Nonlinear Dynamics: Methods and Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0367-1_1
Download citation
DOI: https://doi.org/10.1007/978-94-011-0367-1_1
Received:
Accepted:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4164-5
Online ISBN: 978-94-011-0367-1
eBook Packages: Springer Book Archive