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Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum. (English) Zbl 0607.70027
Experimental evidence is presented which supports the theory that homoclinic orbits in a Poincaré map associated with a phase space flow are precursors of chaotic motion. A permanent magnet rotor in crossed steady and time-varying magnetic fields is shown to satisfy a set of third order differential equations analogous to the forced pendulum or to a particle in a combined periodic and traveling wave force field. Critical values of magnetic torque and forcing frequency are measured for chaotic oscillations of the rotor and are found to be consistent with a lower bound for the existence of homoclinic orbits derived by the method of Melnikov. The fractal nature of the strange attractor is revealed by a Poincaré map triggered by the angular position of the rotor. Numerical simulations using the model also agree well with both theoretical and experimental criteria for chaos.

70K50Transition to stochasticity (general mechanics)
70K05Phase plane analysis, limit cycles (general mechanics)
37-99Dynamic systems and ergodic theory (MSC2000)
37D45Strange attractors, chaotic dynamics
Full Text: DOI
[1] Holmes, P. J.: Phil trans. Roy. soc.. 292, 419 (1979)
[2] Greenspan, B. D.; Holmes, P. J.: G.barenblattg.ioossd.d.josephnonlinear dynamics and turbulence. Nonlinear dynamics and turbulence (1983) · Zbl 0532.58019
[3] Guckenheimer, J.; Holmes, P. J.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields. (1983) · Zbl 0515.34001
[4] Moon, F. C.; Li, G-X: Phys. rev. Lett.. 55, 1439 (1985)
[5] Moon, F. C.: J. appl. Mech.. 47, 638 (1980)
[6] Mcdonald, S. W.; Grebogi, C.; Ott, E.; Yorke, J. A.: Physica. 17D, 125 (1985)
[7] Zaslavskii, G. M.; Chirikov, B. V.: Sov. phys. Uspekhi. 14, 549 (1972)
[8] Croquette, V.; Poitou, C.: J. phys. Lett.. 42, L-537 (1981)
[9] Odyniec, M.; Chua, L. O.: IEEE trans. Circuits and systems. 32, 34 (1985)
[10] F.M.A. Salam and S.S. Sastry, IEEE Trans. on Circuits and Systems, to appear.
[11] Hockett, K. G.; Holmes, P. J.: Ergod. th. Dynam. sys.. 6, 205 (1986)
[12] Gwinn, E. G.; Westervelt, R. M.: Phys. rev. Lett.. 54, 1613 (1985)
[13] Koch, B. P.; Leven, R. W.: Physica. 16D, 1 (1985)
[14] Mclaughlin, J. B.: J. stat. Phys.. 24, 375 (1981)
[15] Moon, F. C.: Phys. rev. Lett.. 53, 962 (1984)