Basins of Attraction and Transient Chaos in a Gear-Rattling Model

Abstract

The authors numerically investigate basins of attraction of coexisting periodic and chaotic attrac tors in a gear-rattling impact model. These attractors are strongly dependent on small changes of the initial conditions. Gradually varying a control parameter, the size of these basins of attraction is modified by global bifurcations of their boundaries. Moreover, the topology of these basins is also modified by appearance or disappearance of coexisting attractors. Furthermore, for the considered control parameter range, the frac tal basin boundaries are so interleaved that trajectories are practically unpredictable in some regions of phase space. The authors also examine an example of a crisis on which a chaotic attractor is converted into a chaotic transient that goes to a periodic attractor. For this crisis, the authors show the evolution of transient lifetime dependence of the initial conditions as the control parameter is varied.

Keywords

Gear-rattling impact basins of attraction transient chaos

Get full access to this article

View all access options for this article.

References

Casas, F. , Chin, W. , Grebogi, C. , and Ott, E. , 1996, "Universal grazing bifurcations in impact oscillators ," Physical Review E 53, 134-139.

Grebogi, C. , Ott, E. , and Yorke, J.A. , 1983, "Crises, sudden changes in chaotic attractors and transient chaos," Physica D 7, 181-200.

Grebogi, C. , Ott, E. , and Yorke, J.A. , 1986, "Critical exponent of chaotic transients in nonlinear dynamical systems," Physical Review Letters 57, 1284-1287.

Grebogi, C. , Ott, E. , and Yorke, J.A. , 1987, "Chaos, strange attractors, a fractal basin boundaries in nonlinear dynamics," Science 238, 632-638.

Hinrichs, N. , Oestreich, M. , and Popp, K. , 1997, "Dynamics of oscillators with impact and friction ," Chaos, Solitons, and Fractals 8(4), 535-558.

Kaharaman, A. and Singh, R. , 1990, "Nonlinear dynamics of a spur gear pair," Journal of Sound and Vibration 142, 49-75.

Karagiannis, K. and Pfeiffer, F. , 1991, "Theoretical and experimental investigations of gear-rattling ," Nonlinear Dynamics 2, 367-387.

Li, G.X. , Rand, R.H. , and Moon, F.C. , 1990, "Bifurcations and chaos in forced zero-stiffness impact oscillator," International Journal Non-Linear Mechanics 25(4), 417-432.

Moon, F.C. , 1998, Applied Dynamics With Applications to Multibody and Mechatronic Systems, John Wiley, New York.

10.

Nordmark, A.B. , 1991, "Non-periodic motion caused by grazing incidence in impact oscillator," Journal of Sound and Vibration 145(2), 279-297.

11.

Nusse, H. and Yorke, J.A. , 2000, "Fractal basin boundaries generated by basin cells and the geometry of mixing chaotic flows," Physical Review Letters 84, 626-629.

12.

Okolewska, B.B. and Kapitaniak, T. , 1998, "Co-existing attractors of impact oscillator," Chaos, Solitons, and Fractals 9(8), 1439-1443.

13.

Okolewska, B.B. and Peterka, F. , 1998, "An investigation of the dynamic sys with impacts ," Chaos, Solitons, and Fractals 9(8), 1321-1338.

14.

Peterka, F. and Vacik, J. , 1992, "Transition to chaotic motion in mechanical systems with impacts," Journal of Sound and Vibration 154, 95-115.

15.

Pfeiffer, F. , 1988, "Seltsame Attraktoren in Zahnradgetrieben," Ingenieur Archiv 58, 113-125.

16.

Pfeiffer, F. and Kunert, A. , 1990, "Rattling models from deterministic to stochastic processes ," Nonlinear Dynamics 1, 63-74.

17.

Thompson, J.M.T. , 1996, "Global dynamics oscillators: Fractal basins and intermediate bifurcations," in Nonlinear Mathematics and Its Applications , P. J. Aston , ed., pp. 1-47, Cambridge University Press, Cambridge, UK.

18.

Win, W. , Ott, E. , Nusse, H.E. , and Grebogi, C. , 1995, "Universal behavior of impact oscillators near grazing incidence," Physics Letters A 201, 197-204.