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Chaotic motions and fractal basin boundaries in spring-pendulum system. (English) Zbl 1168.70319
Summary: This study investigates the chaotic response of the spring-pendulum system. In this system besides of strange attractors, multiple regular attractors may co-exist for some values of system parameters, and it is important to study the global behavior of the system using the basin boundaries of the attractors. Here multiple scales method is used to distinguish the regions of stable and unstable attractors. Early studies show that there are unstable regions for the spring-pendulum system. In this study using bifurcation diagrams and Poincaré maps, it is shown that in some cases the response becomes quasi-periodic or chaotic for some deviations from external and internal resonance frequencies. Also it will be shown that the response is sensitive to the value of damping parameters, which may result in chaotic response. Results show that the jumping phenomena may occur when multiple regular attractors exist. Using basin boundaries of attractors it is also shown that in some regions these boundaries are fractal.

MSC:
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Agliari, A.; Gardini, L.; Mira, C., On the fractal structure of basin boundaries in two-dimensional noninvertible maps, Int. J. bifurcat. chaos, 13, 7, 1767-1785, (2003) · Zbl 1056.37058
[2] Bhattacharyya, R., Behavior of a rubber spring pendulum, J. appl. mech., 67, 2, 332-337, (2000) · Zbl 1110.74343
[3] Defreitas, M.T.; Viana, R.L.; Grebogi, C., Multistability, basin boundary structure, and chaotic behavior in a suspension bridge model, Int. J. bifurcat. chaos, 14, 3, 927-950, (2004) · Zbl 1129.74318
[4] El-Bassiouny, A.F., Parametrically excited nonlinear systemsa comparison of two methods, Int. J. math. math. sci., 32, 12, 739-761, (2002) · Zbl 1012.65141
[5] Garira, W.; Bishop, S.R., Oscillatory orbits of the parametrically excited pendulum, Int. J. bifurcat. chaos, 13, 10, 2949-2958, (2003) · Zbl 1061.70013
[6] Georgiou Ioannis, T., On the global geometric structure of the dynamics of the elastic pendulum, Nonlinear dyn., 18, 1, 51-68, (1999) · Zbl 0963.70013
[7] Grebogi, C.; McDonald, S.W.; Ott, E.; York, J.A., Final state sensitivityan obstruction to predictability, Phys. lett. A, 99, 9, 415-418, (1983)
[8] Grebogi, C.; Ott, E.; York, J.A., Fractal basin boundaries, long-lived chaotic transients and unstable – unstable pair bifurcation, Phys. rev. lett., 50, 13, 935-938, (1983)
[9] Lee, W.K.; Hsu, C.S., A global analysis of a harmonically excited spring-pendulum system with internal resonance, J. sound vib., 171, 3, 335-359, (1994) · Zbl 0925.70264
[10] Lee, W.K.; Park, H.D., Chaotic dynamics of a harmonically excited spring-pendulum system with internal resonance, Nonlinear dyn., 14, 211-229, (1997) · Zbl 0907.70015
[11] Li, G.X.; Moon, F.C., Criteria for chaos of a three-well potential oscillator with homoclinic and hetroclinic orbits, J. sound vib., 136, 1, 17-34, (1990) · Zbl 1235.74096
[12] Miles, J., Resonantly forced motion of two quadratically coupled oscillators, Physica D, 13, 247-260, (1984) · Zbl 0578.70023
[13] Moon, F.C.; Li, G.X., Fractal basin boundaries and homoclinic orbits for periodic motions in a two-well potential, Phys. rev. lett., 55, 14, 1439-1442, (1985)
[14] Nayfeh, A.H., Parametric excitation of two internally resonant oscillators, J. sound vib., 119, 1, 95-109, (1987) · Zbl 1235.70128
[15] Nayfeh, T.A.; Asrar, W.; Nayfeh, A.H., Three-mode interactions in harmonically excited systems with quadratic nonlinearities, Nonlinear dyn., 58, 1033-1041, (1991)
[16] Nayfeh, A.H.; Balachandran, B., Experimental investigation of resonantly forced oscillations of a two-degree-of-freedom structure, Int. J. non-linear mech., 25, 2/3, 199-209, (1990)
[17] Nayfeh, A.H.; Balachandran, B.; Colbert, M.A.; Nayfeh, M.A., An experimental investigation of complicated response of a two-degree-of-freedom structure, ASME J. appl. mech., 56, 960-967, (1989)
[18] Nayfeh, A.H.; Mook, D.T.; Marshall, L.R., Nonlinear coupling in pitch and roll modes in ship motions, J. hydronaut., 7, 4, 145-152, (1973)
[19] Nayfeh, A.H.; Zavodney, L.D., Experimental observation of amplitude- and phase-modulated response of two internally coupled oscillators to a harmonic excitation, ASME J. appl. mech., 55, 706-710, (1988)
[20] Sethna, P.R., Vibrations of dynamical systems with quadratic nonlinearities, J. appl. mech., 32, 576-582, (1965)
[21] Szemplinska-Stupnicka, W.; Tyrkiel, E.; Zubrzycki, A., The global bifurcations that lead to transient tumbling chaos in a parametrically driven pendulum, Int. J. bifurcat. chaos, 10, 9, 2161-2175, (2000) · Zbl 0965.70036
[22] Tousi, S.; Bajaj, A.K., Period-doubling bifurcations and modulated motions in force mechanical systems, ASME J. appl. mech., 52, 446-452, (1985)
[23] Zaki, K.; Noah, S.; Rajagopal, K.R.; Srinivasa, A.R., Effect of nonlinear stiffness on the motion of a flexible pendulum, Nonlinear dyn., 27, 1, 1-18, (2002) · Zbl 1050.70014
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