×

Strange nonchaotic attractors in a nonsmooth dynamical system. (English) Zbl 1476.37058

Summary: Strange nonchaotic attractors (SNAs) have fractal geometric structure, but are nonchaotic in the dynamical sense. Since C. Grebogi et al. [Physica D 13, 261–268 (1984; Zbl 0588.58036)] discovered SNA in 1984, it has become one of the important topics in nonlinear dynamics. Now, the study of SNAs has been mainly confined to smooth dynamics with quasiperiodic excitation or random excitation. In this paper, we consider a class of single-degree-of-freedom gear dynamical system with quasiperiodic forcing. We show that the gear transmission system can be modeled as a three-dimensional piecewise linear system, which belongs to a typical class of nonsmooth system. We then show that SNAs do exist in such nonsmooth dynamical system with quasiperiodic force. The dynamical behavior of the nonsmooth system is analyzed as a parameter is varied. The dynamics is analyzed through phase diagrams and bifurcation diagrams, Lyapunov exponents, singular continuous power spectrum, phase sensitivity of time series and rational approximations.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37C75 Stability theory for smooth dynamical systems
37G35 Dynamical aspects of attractors and their bifurcations

Citations:

Zbl 0588.58036
PDFBibTeX XMLCite
Full Text: DOI

References:

[2] Alsedà, L.; Misiurewicz, M., Attractors for unimodal quasiperiodically forced maps, J Diff Equations Appl, 14, 10-11, 1175-1196 (2008) · Zbl 1153.37020
[3] Jäger, T., Strange non-chaotic attractors in quasiperiodically forced circle maps, Commun Math Phys, 289, 1, 253-289 (2009) · Zbl 1171.37035
[4] Zhang, Y., Wada basins of strange nonchaotic attractors in a quasiperiodically forced system, Phys Lett A, 377, 18, 1269-1273 (2013) · Zbl 1290.37016
[5] Keller, G., A note on strange nonchaotic attractors, Fundamenta Mathematicae, 151, 2 (1996) · Zbl 0899.58033
[6] Suresh, K.; Prasad, A.; Thamilmaran, K., Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chua’s oscillator, Phys Lett A, 377, 8, 612-621 (2013) · Zbl 1428.37023
[7] Mitsui, T.; Crucifix, M.; Aihara, K., Bifurcations and strange nonchaotic attractors in a phase oscillator model of glacial-interglacial cycles, Physica D, 306, 25-33 (2015) · Zbl 1364.86022
[8] Jalnine, A. Y.; Kuznetsov, S. P., Autonomous strange nonchaotic oscillations in a system of mechanical rotators, Regular Chaotic Dyn, 22, 3, 210-225 (2017) · Zbl 1387.34054
[9] Wang, X.; Zhan, M.; Lai, C. H., Strange nonchaotic attractors in random dynamical systems, Phys Rev Lett, 92, 7, 074-102 (2004)
[10] Ding, M.; Grebogi, C.; Ott, E., Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic, Phys Rev A, 39, 5, 2593-2598 (1989)
[11] Romeiras, F. J.; Bondeson, A.; Ott, E., Quasiperiodic forcing and the observability of strange nonchaotic attractors, Physica Scripta, 40, 3, 442-444 (2006) · Zbl 1063.37520
[12] Heagy, J.; Ditto W, L., Dynamics of a two-frequency parametrically driven duffing oscillator, J Nonlinear Sci, 1, 4, 423-455 (1991) · Zbl 0794.34028
[14] Anishchenko, V. S.; Vadivasova, T. E.; Sosnovtseva, O., Strange nonchaotic attractors in autonomous and periodically driven systems, Phys Rev E, 54, 3231-3234 (1996)
[16] Lindner, J. F.; Kohar, V.; Kia, B., Strange nonchaotic stars, Phys Rev Lett, 114, 5, 054-101 (2015)
[18] Zhang, Y.; Luo, G., Torus-doubling bifurcations and strange nonchaotic attractors in a vibro-impact system, J Sound Vib, 332, 21, 5462-5475 (2013)
[19] Chithra, A.; Mohamed, I. R.; Rizwana, R., Observation of chaotic and strange nonchaotic attractors in a simple multi-scroll system, J Comput Electron, 17, 13, 1-10 (2017)
[20] Sathish M, A.; Venkatesan, A.; Lakshmanan, M., Strange nonchaotic attractors for computation, Phys Rev E, 97, 5, Article 052212 pp. (2018)
[21] Ding, M.; Grebogi, C.; Ott, E., Dimensions of strange nonchaotic attractors, Phys Lett A, 137, 4, 167-172 (1989)
[22] Glendinning, P.; Jaeger, T.; Keller, G., How chaotic are strange nonchaotic attractors, Nonlinearity, 19, 9, 2005-2022 (2006) · Zbl 1109.37013
[23] Shen, Y.; Zhang, Y., Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations, Nonlinear Dyn, 96, 1405-1428 (2019) · Zbl 1437.37048
[24] Senthilkumar, D. V.; Srinivasan, K.; Thamilmaran, K.; Lakshmanan, M., Bubbling route to strange nonchaotic attractor in a nonlinear series LCR circuit with a nonsinusoidal force, Phys Rev E, 78, Article 066211 pp. (2018)
[25] Suresh, K.; Prasad, A.; Thamilmaran, K., Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chua’s oscillator, Phys Lett A, 377, 8, 612-621 (2013) · Zbl 1428.37023
[26] Mohammed, O. D.; Rantatalo, M., Dynamic response and time-frequency analysis for gear tooth crack detection, Mech Syst Signal Process, 66, 612-624 (2015)
[27] Jiang, Y.; Zhu, H.; Li, Z., The nonlinear dynamics response of cracked gear system in a coal cutter taking environmental multi-frequency excitation forces into consideration, Nonlinear Dyn, 84, 1, 203-222 (2016)
[28] Raffaella, A.; Petrescu, R. V.V; Apicella, A., A dynamic model for gears, Am J Eng Appl Sci, 10, 2, 484-490 (2017)
[29] Halse, C. K.; Wilson, R. E.; Bernardo, M. D., Coexisting solutions and bifurcations in mechanical oscillators with backlash, J Sound Vib, 305, 4, 854-885 (2007) · Zbl 1242.70045
[31] Nayfeh, A. H., Balachandran B, applied nonlinear dynamics (1995), John Wiley & Sons, Inc: John Wiley & Sons, Inc New York, NY · Zbl 0848.34001
[32] Pikovsky, A. S.; Zaks, M. A.; Feudel, U.; Kurths, J., Singular continuous spectra in dissipative dynamics, Phys Rev E, 52, 1, 285-296 (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.