Abstract
We investigate the system\(\ddot x - x\cos \varepsilon 1 + x^3 = 0\) in which ε≪1 by using averaging and elliptie functions. It is shown that this system is applicable to the dynamics of the familiar rotating-plane pendulum. The slow foreing permits us to envision an ‘instantancous phase portrait’ in the\(x - \dot x\) phase plane which exhibits a center at the origin when cos ε1≤0 and a saddle and associated double homoclinic loop separatrix when cos ɛ 1 > 0. The chaos in this problem is related to the question of on which side (left (=L) or right (=R)) of the reappearing double homoclinic loop separatrix a motion finds itself. We show that the sequence of L's and R's exhibits sensitive dependence on initial conditions by using a simplified model which assumes that motions cross the instantancous separatrix instantancously. We also present an improved model which ‘patches’ a separatrix boundary layer onto the averaging model. The predictions of both models are compared with the results of numerical integration.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V. I.,Mathematical Methods of Classical Mechanics, Springer, New York, 1984.
Bender, C. M. and Orzag, S. A.:Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
Byrd, P. and Friedman, M.,Handbook of Elliptic Integrals for Engineers and Physicists, Springer, Berlin. 1954.
Cary, J. R., Escande, D. F., and Tennyson, J. L.: ‘Adiabatic-invariant change due to separatrix crossing’,Phys. Rev. A 34, 1986, 4256–4275.
Cary, J. R. and Skodje, R. T., ‘Phase change between separatrix crossings’,Physica D 36, 1989, 287–316.
Coppola, V. T.,Averaging of Strongly Nonlinear Oscillators Using Elliptic Functions, Ph. D. Dissertation. Cornell University. 1989.
Coppola, V. T. and Rand, R. H., ‘Averaging using elliptic functions: Approximation of limit cycles’,Acta Mechanica 81, 1990, 125–142.
Coppola, V. T. and Rand, R. H., ‘Computer algebra, elliptic functions and chaos’, inProceedings of 1990 ASME International Computers in Engineering Conference. Vol. 1. Boston. Aug. 6–10, 1990, 193–200.
Devaney, R.,An Introduction to Chaotic Dynamics, Addison-Wesley, New York, 1987.
Escande, D. F. ‘Hamiltonian Chaos and Adiabaticity’, preprint.
Goldstein, H.,Classical Mechanics, end ed., Addison-Wesley, Reading. Mass. 1980.
Greenwood, D.,Principles of Dynamics, Prentice Hall, Englewood Cliffs, N.J., 1965.
Guckenheimer, J. and Holmes, P.:Nonlinear Oscillations. Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York. 1983.
Henrard, J. ‘The adiabatic invariant in classical mechanics’, preprint.
Kath, W. K., ‘Slowly varying phase plane and boundary-layer theory’Studies in Applied Math. 72, 1985, 221–239.
Keane, M., ‘Interval exchange transformations’,Math. Z. 141, 1975, 25–31.
Kevorkian, J. and Cole, J. D.,Perturbation Methods in Applied Mathematics, Springer, New York, 1981.
Menyuk, C. R., ‘Particle motion in the field of a modulated wave’,Phys. Rev. A. 31, 1985, 3282–3290.
Rand, R. H.,Computer Algebra in Applied Mathematics: An lutroduction to MACSYMA, Pitman, Boston, 1984.
Rand, R. H. and Armbruster, D.,Perturbation Methods. Bifurcation Theory and Computer Algebra, Springer, New York, 1987.
Sanders, J. A. and Verhulst, F.,Averaging Methods in Nonlinear Dynamical Systems, Springer, New York, 1985.
Shaw, S. W. and Wiggins, S., ‘Chaotic dynamics of a whirling pendulum’,Physica D 31, 1988, 190–211.
Simmonds, J. and Mann, J.,A First Look at Perturbation Theory, Kreiger Publishing, Malabar. FI. 1986.
Timofeev, A. V.. ‘On the constancy of an adiabatic invariant when the nature of the motion changes’.Sov. Phys. JETP 48, 1978, 656–659.
Wiggins, S.. ‘On the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations’SIAM J. Appl. Math. 48, 1988, 262–285.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Coppola, V.T., Rand, R.H. Chaos in a system with a periodically disappearing separatrix. Nonlinear Dyn 1, 401–420 (1990). https://doi.org/10.1007/BF01893171
Issue Date:
DOI: https://doi.org/10.1007/BF01893171