Abstract
Melnikov's method is applied to the planar double pendulum proving it to be a chaotic system. The parameter space of the double pendulum is discussed, and the integrable cases are identified. In the neighborhood of the integrable case of two uncoupled pendulums Melnikov's integral is evaluated using residue calculus. In the two limiting cases of one pendulum becoming a rotator or an oscillator, the parameter dependence of chaos, i.e. the width of the separatrix layer is analytically discussed. The results are compared with numerically computed Poincaré surfaces of section, and good agreement is found.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Holmes, P., Marsden, J.E.: Commun. Math. Phys.82, 523 (1982)
Poincaré, H.: Les méthodes nouvelles de la méchanique céleste. New York: Dover 1957
Landau, L.D., Lifschitz, E.M.: Theoretische Physik I. Berlin: Akademie Verlag 1964
Richter, P.H., Scholz, H.-J.: Das ebene Doppelpendel — The planar double pendulum, Publikationen zu wissenschaftlichen Filmen, Sektion Technische Wissenschaften/Naturwissenschaften,9 IWF, Göttingen, 1986
Richter, P.H., Scholz, H.-J.: Chaos in classical mechanics: the double pendulum: In: Stochastic Phenomena and Chaotic Behavior in Complex Systems. Schuster, P. (ed.) Berlin, Heidelberg, New York: Springer 1984
Guggenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Berlin, Heidelberg, New York: Springer 1983
Koch, B.P., Bruhn, B.: J. Phys. A25, 3945 (1992)
Gonzalez, M.O.: Classical complex analysis. New York: Marcel Dekker 1992
Abramowitz, M., Stegun, I.A., (eds.): Handbook of mathematical functions. New York: Dover 1972
Gradshtevn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. New York: Academic Press 1980
Byrd, P.F., Friedmann, M.D.: Handbook of elliptic integrals for engineers and scientists. Berlin, Heidelberg, New York: Springer 1971
Lichtenberg, A.J., Lieberman, M.A.: Regular and stochastic motion. Berlin, Heidelberg, New York: Springer 1983
Richter, P.H., Scholz, H.-J., Wittek, A.: A Breathing Chaos, Nonlinearity3, 45 (1990)
Wolfram, S.: Mathematica. Redwood City, Calif. Addison-Wesley 1991