Abstract
It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.
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References
Feudel, U., Kuznetsov, S., Pikovsky, A.: Strange Nonchaotic Attractors: Dynamics Between Order and Chaos in Quasiperiodically Forced Systems. World Scientific, Singapore (2006)
Prasad, A., Negi, S.S., Ramaswamy, R.: Strange nonchaotic attractors. Int. J. Bifurc. Chaos 11, 291–309 (2001)
Prasad, A., Nandi, A., Ramaswamy, R.: Aperiodic nonchaotic attractors, strange and otherwise. Int. J. Bifurc. Chaos 17, 3397–3407 (2007)
Grebogi, C., Ott, E., Pelikan, S., Yorke, J.A.: Strange attractors that are not chaotic. Physica D 13, 261–268 (1984)
Jäger, T.H.: The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. Mem. Am. Math. Soc. 945, 1–106 (2009)
Bjerklov, K.: SNA’s in the quasi-periodic quadratic family. Commun. Math. Phys. 286, 137–161 (2009)
Groger, M., Jäger, T.H.: Dimensions of attractors in pinched skew products. Commun. Math. Phys. 320, 101–119 (2013)
Ditto, W.L., Spano, M.L., Savage, H.T., Rauseo, S.N., Heagy, J.F., Ott, E.: Experimental observation of a strange nonchaotic attractor. Phys. Rev. Lett. 65, 533–536 (1990)
Zhou, T., Moss, F., Bulsara, A.: Observation of a strange nonchaotic attractor in a multistable potential. Phys. Rev. A 45, 5394–5400 (1992)
Thamilmaran, K., Senthilkumar, D.V., Venkatesan, A., Lakshmanan, M.: Experimental realization of strange nonchaotic attractors in a quasiperiodically forced electronic circuit. Phys. Rev. E 74, 036205 (2006)
Heagy, J.F., Hammel, S.M.: The birth of strange nonchaotic attractors. Physica D 70, 140–153 (1994)
Nishikawa, T., Kaneko, K.: Fractalization of a torus as a strange nonchaotic attractor. Phys. Rev. E 54, 6114–6124 (1996)
Kim, J.W., Kim, S.Y., Hunt, B., Ott, E.: Fractal properties of robust strange nonchaotic attractors in maps of two or more dimensions. Phys. Rev. E 67, 036211 (2003)
Hunt, B.R., Ott, E.: Fractal properties of robust strange nonchaotic attractors. Phys. Rev. Lett. 87, 254101 (2001)
Prasad, A., Ramaswamy, R., Satija, I., Shah, N.: Collision and symmetry breaking in the transition to strange nonchaotic attractors. Phys. Rev. Lett. 83, 4530–4533 (1999)
Prasad, A., Mehra, V., Ramaswamy, R.: Intermittency route to strange nonchaotic attractors. Phys. Rev. Lett. 79, 4127–4130 (1997)
Verkatesan, A., Murali, K., Lakshmanan, M.: Birth of strange nonchaotic attractors through type III intermittency. Phys. Lett. A 259, 246–253 (1999)
Kim, S.Y., Lim, W., Ott, E.: Mechanism for the intermittent route to strange nonchaotic attractors. Phys. Rev. E 67, 056203 (2003)
Osinga, H.M., Feudel, U.: Boundary crisis in quasiperiodically forced systems. Physica D 141, 54–64 (2000)
Witt, A., Feudel, U., Pikovsky, A.S.: Birth of strange nonchaotic attractors due to interior crisis. Physica D 109, 180–190 (1997)
Kim, S.Y., Lim, W.: Mechanism for boundary crises in quasiperiodically forced period-doubling systems. Phys. Lett. A 334, 160–168 (2005)
Lim, W., Kim, S.Y.: Interior crises in quasiperiodically forced period-doubling systems. Phys. Lett. A 355, 331–336 (2006)
Yalcinkaya, T., Lai, Y.C.: Blowout bifurcation route to strange nonchaotic attractors. Phys. Rev. Lett. 77, 5039–5042 (1996)
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, 066211 (2008)
Lindner, J.F., Kohar, V., Kia, B., Hippke, M., Learned, J.G., Ditto, W.L.: Strange nonchaotic stars. Phys. Rev. Lett. 114, 054101 (2015)
Zhou, C.S., Chen, T.L.: Robust communication via synchronization between nonchaotic strange attractors. Europhys. Lett. 38, 261–265 (1997)
Chacon, R., Gracia-Hoz, A.M.: Route to chaos via strange non-chaotic attractors by reshaping periodic excitations. Europhys. Lett. 57, 7–13 (2002)
Ramaswamy, R.: Synchronization of strange nonchaotic attractors. Phys. Rev. E 56, 7294–7296 (1997)
Laroze, D., Becerra-Alonso, D., Gallas, J.A.C., Pleiner, H.: Magnetization dynamics under a quasiperiodic magnetic field. IEEE Trans. Magn. 48, 3567–3570 (2012)
Mitsui, T., Aihara, K.: Dynamics between order and chaos in conceptual models of glacial cycles. Clim. Dyn. 42, 3087–3099 (2013)
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)
Premraj, D., Suresh, K., Palanivel, J., Thamilmaran, K.: Dynamic bifurcation and strange nonchaos in a two-frequency parametrically driven nonlinear oscillator. Commun. Nonlinear Sci. Numer. Simul. 50, 103–114 (2017)
Venkatesan, A., Lakshmanan, M.: Interruption of torus bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: mechanisms and their characterizations. Phys. Rev. E 63, 026219 (2001)
Venkatesan, A., Lakshmanan, M., Prasad, A., Ramaswamy, R.: Intermittency transitions to strange nonchaotic attractors in a quasiperiodically driven Duffing oscillator. Phys. Rev. E 61, 3641–3651 (2000)
Chen, H., Llibre, J., Tang, Y.: Global dynamics of a SD oscillator. Nonlinear Dyn. 91, 1755–1777 (2018)
Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Physica D 241, 1826–1844 (2012)
Zhao, X., Schaeffer, D.G.: Alternate pacing of border-collision period-doubling bifurcations. Nonlinear Dyn. 50, 733–742 (2007)
Lin, D.C., Oguamanam, D.C.D.: A numerical study of the dynamics of three-mass system on frictional tracks. Nonlinear Dyn. 94, 2047–2058 (2018)
Luo, G.W., Lv, X.H., Zhu, X.F., Shi, Y.Q., Du, S.S.: Diversity and transition characteristics of sticking and non-sticking periodic impact motions of periodically forced impact systems with large dissipation. Nonlinear Dyn. 94, 1047–1079 (2018)
Simpson, D.J.W., Meiss, J.D.: Aspects of bifurcation theory for piecewise-smooth, continuous systems. Physica D 241, 1861–1868 (2012)
Long, X.H., Lin, G., Balachandran, B.: Grazing bifurcations in an elastic structure excited by harmonic impactor motions. Physica D 237, 1129–1138 (2008)
Arulgnanam, A., Prasad, A., Thamilmaran, K., Daniel, M.: Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with experimental and analytical confirmation. Chaos Soliton Fractals 75, 96–110 (2015)
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, 612–621 (2013)
Yue, Y., Miao, P., Xie, J.: Coexistence of strange nonchaotic attractors and a special mixed attractor caused by a new intermittency in a periodically driven vibro-impact system. Nonlinear Dyn. 87, 1–21 (2016)
Zhang, Y., Luo, G.: Torus-doubling bifurcations and strange nonchaotic attractors in a vibro-impact system. J. Sound Vib. 332, 5462–5475 (2013)
Avrutin, V., Schanz, M.: Border-collision period-doubling scenario. Phys. Rev. E 70, 026222 (2004)
Pikovsky, A.S., Feudel, U.: Characterizing strange nonchaotic attractors. Chaos 5, 253–260 (1995)
Prasad, A., Mehra, V., Ramaswamy, R.: Strange nonchaotic attractors in the quasiperiodically forced logistic map. Phys. Rev. E 57, 1576–1584 (1998)
Ngamga, E.J., Nandi, A., Ramaswamy, R., Romano, M.C., Thiel, M., Kurths, J.: Recurrence analysis of strange nonchaotic dynamics. Phys. Rev. E 75, 036222 (2007)
Ngamga, E.J., Buscarino, A., Frasca, M., Fortuna, L., Prasad, A., Kurths, J.: Recurrence analysis of strange nonchaotic dynamics in driven excitable systems. Chaos 18, 013128 (2008)
Marwan, N., Romano, M.C., Thiel, M., Kurths, J.: Recurrence plots for the analysis of complex systems. Phys. Rep. 438, 237–329 (2007)
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The authors are deeply indebted to all anonymous reviewers and the editor for their careful reading of the manuscript, as well as for their fruitful comments and advice which led to an improvement of this paper. This work was supported by the National Natural Science Foundation of China (Nos. 11732014, 11572205 and 11702111).
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Shen, Y., Zhang, Y. Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations. Nonlinear Dyn 96, 1405–1428 (2019). https://doi.org/10.1007/s11071-019-04862-5
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DOI: https://doi.org/10.1007/s11071-019-04862-5