Abstract：Here, the fast-slow effect of two-scale coupling in frequency domain and its mechanism were studied. A typical chaotic system was taken as an example, and the combination of parametric and external excitation was introduced. It was observed that when the two excitation frequencies are far less than the system’s natural frequency, special behavior of bursting oscillation occurs in the system. The system dynamic characteristics were studied in two cases of the two excitation frequencies satisfying conditions of strict resonance and non-strict resonance, respectively. Bifurcation analyses of a generalized autonomous system were performed with variation of slowly varying variable in the two cases to obtain different types bursting oscillations and their bifurcation mechanism. It was found that in the strict resonance case, the variation of the ratio of the two excitation frequencies can cause bursting oscillations with different scroll numbers; in the non-strict resonance case, with the variation of frequency, periodic and almost periodic bursting oscillations alternately appear in the system.
张晓芳，董颖涛，韩修静，毕勤胜. 参外联合激励下一类混沌系统的动力学机理[J]. 振动与冲击, 2021, 40(1): 183-191.
ZHANG Xiaofang, DONG Yingtao, HAN Xiujing, BI Qinsheng. Dynamic mechanism of a class of chaotic systems under combination of parametric and external excitation. JOURNAL OF VIBRATION AND SHOCK, 2021, 40(1): 183-191.
[1] 韩修静，江波, 毕勤胜. 快慢型超混沌Lorenz系统分析[J]. 物理学报，2009， 58(9):6006-6015. HAN Xiujing，JIANG Bo，BI Qinsheng. Analysis of the fast-slow hyperchaotic Lorenz system [J]. Acta Physica Sinica，2009，58(9):6006-6015.
[2] TANTET A，LUCARINI V，Dijkstra H A.
Resonances in a Chaotic Attractor Crisis of the
Lorenz Flow [J]. Journal of Statistical Physics
2018，170(3):584-616.
[3] 阮静雅，孙克辉，牟俊. 基于忆阻器反馈的
Lorenz超混沌系统及其电路实现[J]. 物理学报，
2016，65(19): 19-29.
RUAN Jingya，SUN Kehui，MOU Jun. Memristor-
based Lorenz hyper-chaotic system and its circuit
Implementation [J]. Acta Physica Sinica，2016，
65(19): 19-29.
[4] DANCA M F，FECKAN M，KUZNETSOV N V，et al. Complex dynamics，hidden attractors and contin- uous approximation of a fractional-order hyperchaotic PWC system [J]. Nonlinear Dynamics，2018，91(16):1-18.
[5] DAI H，YUE X，DAN X，et al. Chaos and chaotic transients in an aeroelastic system [J]. Journal of Sound & Vibration，2014，333(26):7267-7285.
[6] 古华光，惠磊，贾冰. 一类位于加周期分岔中
的貌似混沌的随机神经放电节律的识别[J]. 物理学报，2012，61(8):000072-81.
GU Guanghua，HUI Lei，JIA Bing. Identification of a stochastic neural firing rhythm lying in period-adding bifurcation and resembling chaos [J]. Acta Physica Sinica，2012，61(8):000072-81.
[7] 孙伟鹏，王贺元，阚猛. Willamowski-Rossler系统混沌行为的数值仿真及控制与同步研究[J]. 动力学与控制学报，2018，16(1): 35-40.
SUN Weipeng，WANG Heyuan，KAN Meng.
Numerical simulation,control,and synchronization
research on chaotic behavior of Willamowski-
Rossler system [J]. Journal of Dynamics and
Control，2018，16(1): 35-40.
[8] 高秉建，陆君安，陈爱敏. 一个unified系统与Rossler系统的组合研究[J]. 物理学报，2006， 55(9):4450-4454.
GAO Bingjian，LU Junan，CHEN Aimin. A novel chaotic system via combining a unified system with Rossler system[J]. Acta Physica Sinica，2006，55(9):4450-4454.
[9] OZKAYNAK F，ÇELIK V，OZER A B. A new S-box construction method based on the fractional-order chaotic Chen system[J]. Signal，Image and Video Processing，2017，11(4):659-664.
[10] ELABBASY E，AGIZA H，EL-DESSOKY M. Synchronization of Modified Chen System[J]. International Journal of Bifurcation & Chaos，2004，14(11):3969-3979.
[11] LU J G，CHEN G. A note on the fractional-order Chen system[J]. Chaos Solitons & Fractals，2006，27(3):685-688.
[12] EFFATI S， NIK H S，JAJARMI A. Hyperchaos control of the hyperchaotic Chen system by optimal control design[J]. Nonlinear Dynamics，2013，73(1-2):499-508.
[13] 冯朝文，蔡理，张立森，等. SETMOS在蔡氏
电路中的特性研究[J]. 物理学报，2010，59(12):
8420-8425.
FENG Chaowen，CAI Li，ZHANG Lisen，et al.
Characteristic of hybrid single electron transistor and metal oxide semiconductor structure in Chua's circuit [J]. 2010，59(12):8420-8425.
[14] ALKAHTANI B S T. Chua's circuit model with
Atangana-Baleanu derivative with fractional order[J]. Chaos，Solitons & Fractals，2016，89:
547-551.
[15] LI J，LIU Y，WEI Z. Zero-Hopf bifurcation and Hopf bifurcation for smooth Chua's system[J]. Advances in Difference Equations，2018，2018
(1):141.
[16] DADRAS S, MOMENI H R. A novel three-
dimensional autonomous chaotic system generat-
ing two，three and four-scroll attractors[J]. Physics Letters A，2009，373(40):3637-3642.
[17] YENICERI R，YALCIN M E. Multi-scroll
chaotic attractors from a generalized time-delay
sampled-data system[J]. International Journal of
Circuit Theory & Applications，2016，44(6):1263
-1276.
[18] WANG Z，QI G，SUN Y，et al. A new type of
four-wing chaotic attractors in 3-D quadratic autonomous systems[J]. Nonlinear Dynamics，
2010，60(3):443-457.
[19] 夏雨，毕勤胜，罗超，等. 双频1:2激励下修
正蔡氏振子两尺度耦合行为[J]. 力学学报，
2018，50(2):362-372.
XIA Yu，BI Qinsheng，LUO Chao，et al. Behaviors of modified chua's oscillator two time scales under two excitatoins with frequency ratio at 1:2. Chinese Journal of Theoretical and Applied Mechanics，2018，50(2):362-372.
[20] 张晓芳，陈章耀，毕勤胜. 周期激励下Chen
系统的簇发现象分析[J]. 物理学报，2010，
59(6):3802-3809.
ZHANG Xiaofang，CHEN Zhangyao，BI Qinsheng. Analysis of bursting phenomena in Chen's system with periodic excitation[J]. Acta Physica Sinica，2010， 59(6):3802-3809.
[21] 张正娣，刘亚楠，李静，等. 分段Filippov系
统的簇发振荡及擦边运动机理[J]. 物理学报，2018，67(11):40-49.
ZHANG Zhengdi，LIU Yanan，LI Jing，et al. Bursting oscillations and mechanism of sliding movement in piecewise Filippov system [J]. Acta Physica Sinica，2018，67(11):40-49.
[22] LEI H，LIU L，LIU C，et al. The nonlinear
bifurcation and chaos of coupled heave and pitch motions of a truss spar platform[J]. Journal of Ocean University of China，2015，14(5):795-802.
[23] BAO B C，HU A H，BAO H，et al. Three-Dimensional Memristive Hindmarsh-Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors. Complexity，2018， 3872573，1-11.
[24] BAO B C，HU A H，XU Q，et al. AC-induced coexisting asymmetric bursters in the improved Hindmarsh-Rose model[J]. Nonlinear Dynamics，2018，92:1695-1706.
[25] BAO H，LIU W B，HU A H. Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction[J]. Nonlinear Dynamics，2019，95:43-56.
[26] BAO B C，WU P Y，BAO H，et al. Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator[J]. Chaos，Solitons & Fractals，2018，109:146-153.
[27] BI Q S，LI S L，KURTHS J，et al. The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures[J]. Nonlinear Dynamics，2016，85(2): 1-13.
[28] 邢雅清，陈小可，张正娣，等. 多平衡态下簇发振荡产生机理及吸引子结构分析[J]. 物理学报，2016，65(9):1-9.
XING Yaqing，CHEN Xiaoke，ZHANG Zhengdi
et al. Mechanism of bursting oscillations with
multiple equilibrium states and the analysis of
the structures of the attractors[J]. Acta Physica
Sinica，2016，65(9):1-9.
[29] FALLAH H. Symmetric Fold/Super-Hopf Burst- ing，Chaos and Mixed-Mode Oscillations in Pernarowski Model of Pancreatic Beta-Cells.[J]. International Journal of Bifurcation & Chaos，2016，26(09):1630022.
[30] IZHIKEVICH E M. Neural excitability，spiking and bursting[J]. International Journal of Bifurcation & Chaos，2000，10(06):1171-1266.
[31] 陈章耀,王亚茗,张春,等.双状态切换下BVP振子的复杂行为分析[J]. 力学学报，2016，48(4):953-962.
CHEN Zhangyao，WANG Yaming，ZHANG Chun，et al. Complicated behaviors as well as the mechanism in BVP oscillator with switches related to two states[J]. Chinese Journal of Theoretical and Applied Mechanics，2016，48(4):953-962.
[32] BI Qinsheng. The mechanism of bursting phenomena in Belousov-Zhabotinsky (BZ) chemical reaction with multiple time scales[J]. Science China Technological Sciences ，2010，
53(2):748-760.
[33] 张晓芳，吴磊，毕勤胜. 不同激励频率比下复合模态振荡的结构特性分析[J]. 中国科学：技术科学，2017，47(06):106-114.
ZHANG Xiaofang，WU Lei，BI Qinsheng. Science China Technological Sciences，2017，47(06):106-114.
