以一类具有Farey Tree特性的概周期驱动非光滑系统为研究对象,证实了奇异非混沌吸引子的存在性,并进一步分析了它的统计学特性。首先,利用相图和功率谱定性方法分析奇异非混沌吸引子的分形特性。再利用最大李雅普诺夫指数、相敏感指数、功率谱、递归图和有限时间李雅普诺夫指数分布定量方法进一步描述奇异非混沌吸引子的性质。结果表明,该系统存在奇异非混沌吸引子,该奇异非混沌吸引子表现出多种不同于其他类型吸引子的统计学特性。
Taking a quasiperiodic driven nonsmooth system with Farey tree as the research object, the existence of the strange nonchaotic attractor is confirmed and its statistical properties are further analyzed. Firstly, the fractal properties of strange nonchaotic attractor are analyzed by using phase diagram and power spectrum qualitative methods. The properties of strange nonchaotic attractor are described by means of maximum Lyapunov exponent, phase sensitive function, recursive analysis, spectral distribution function and finite-time Lyapunov exponent distribution. The results show that there are strange nonchaotic attractors in the system, and strange nonchaotic attractors exhibit a variety of statistical properties different from other types of attractors.
[1] GREBOGI C, OTT E, PELIKAN S, et al. Strange attractors that are not chaotic[J]. Physica D, 1984, 13(1/2): 261-268.
[2] LIU Z, ZHU Z. Strange nonchaotic attractors from quasiperiodically forced Ueda’s circuit[J]. International Journal of Bifurcation and Chaos, 1996, 6(7): 1383-1388.
[3] THAMILMARAN K, SENTHILKUMAR D V, VENKATESAN A, et al. Experimental realization of strange nonchaotic attractors in a quasiperiodically forced electronic circuit[J]. Physical Review E, 2006, 74(3): 036205.
[4] VENKATESAN A, LAKSHMANAN M, PRASAD A, et al. Intermittency transitions to strange nonchaotic attractors in a quasiperiodically driven Duffing oscillator[J]. Physical Review E, 2000, 61(4): 3641.
[5] HEAGY J, DITTO W L. Dynamics of a two-frequency parametrically driven Duffing oscillator[J]. Journal of Nonlinear Science, 1991, 1: 423-455.
[6] DING M, KELSO J A S. Phase-resetting map and the dynamics of quasi-periodically forced biological oscillators[J]. International Journal of Bifurcation and Chaos, 1994, 4(3): 553-567.
[7] ZHOU C, CHEN T. Robust communication via synchronization between nonchaotic strange attractors[J]. Europhysics Letters, 1997, 38(4): 261.
[8] CHACON R, GARCIA-HOZ A M. Route to chaos via strange non-chaotic attractors by reshaping periodic excitations[J]. Europhysics Letters, 2002, 57(1): 7.
[9] RAMASWAMY R. Synchronization of strange nonchaotic attractors[J]. Physical Review E, 1997, 56(6): 7294.
[10] LAROZE D, BECERRA-ALONSO D, GALLAS J A C, et al. Magnetization dynamics under a quasiperiodic magnetic field[J]. IEEE Transactions on Magnetics, 2012, 48(11): 3567-3570.
[11] MITSUI T, AIHARA K. Dynamics between order and chaos in conceptual models of glacial cycles[J]. Climate Dynamics, 2014, 42: 3087-3099.
[12] HEAGY J F, HAMMEL S M. The birth of strange nonchaotic attractors[J]. Physica D: Nonlinear Phenomena, 1994, 70(1/2): 140-153.
[13] YALCINKAYA T, LAI Y C. Blowout bifurcation route to strange nonchaotic attractors[J]. Physical Review Letters, 1996, 77(25): 5039.
[14] KIM S Y, LIM W, OTT E. Mechanism for the intermittent route to strange nonchaotic attractors[J]. Physical Review E, 2003, 67(5): 056203.
[15] SENTHILKUMAR D V, SRINIVASAN K, THAMILMARAN K, et al. Bubbling route to strange nonchaotic attractor in a nonlinear series LCR circuit with a nonsinusoidal force[J]. Physical Review E, 2008, 78(6): 066211.
[16] ZHANG Y, LUO G. Torus-doubling bifurcations and strange nonchaotic attractors in a vibro-impact system[J]. Journal of Sound and Vibration, 2013, 332(21): 5462-5475.
[17] 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[J]. Nonlinear Dynamics, 2017, 87: 1187-1207.
[18] 沈云柱, 张凡辉, 东广霞. 概周期驱动分段Logistic系统的奇异非混沌吸引子[J]. 河北师范大学学报:自然科学版, 2019 (3): 207-212.
SHEN Y Z, ZHANG F H, DONG G X. Strange Non-chaotic attractors in a piecewise logistic system driven by almost periodic function[J]. Journal of Hebei Normal University: Natural Science Edition, 2019 (3): 207-212.
[19] LI G, YUE Y, LI D, et al. The existence of strange nonchaotic attractors in the quasiperiodically forced Ricker family[J]. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020, 30(5): 053124.
[20] LI G, YUE Y, GREBOGI C, et al. Strange nonchaotic attractors in a periodically forced piecewise linear system with noise[J]. Fractals, 2022, 30(1): 2250003.
[21] PRING S R, BUDD C J. The dynamics of regularized discontinuous maps with applications to impacting systems[J]. SIAM Journal on Applied Dynamical Systems, 2010, 9(1): 188-219.
