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The main purpose of this study is to investigate the characteristics as well as the bifurcation mechanisms of the bursting oscillations in the asymmetrical dynamical system with two scales in the frequency domain. Since the slow-fast Hodgkin-Huxley model was established to successfully reproduce the activities of neuron, the complicated dynamics of the system with multiple time scales has become a hot research topic due to the wide engineering background. The dynamical system with multiple scales often presents periodic oscillations coupled by large-amplitude oscillations at spiking states and small-amplitude oscillations at quiescent states, which are connected by bifurcations. Up to now, most of the reports concentrate on bursting oscillations in the symmetric systems, in which there exists only one form of spiking oscillations and quiescence, respectively. Here we explore some typical forms of bursting behavior in an asymmetrical dynamical system with periodic excitation, in which there exists an order gap between the exciting frequency and the natural frequency. As an example, based on the typical Chua's oscillator, by introducing an asymmetrical controller and a periodically changed current source, and choosing suitable parameter values, we establish an asymmetrical dynamical system with two scales in the frequency domain. Since the exciting frequency is much smaller than the natural frequency, the whole periodic exciting term can be regarded as a slowly-varying parameter, leading to the fast subsystem in autonomous form. Since all the equilibrium curves and relevant bifurcations are presented in the form related to the slowly-varying parameter, the transformed phase portraits describing the evolution relationship between the state variables and the slowly-varying parameter are employed to account for the mechanism of the bursting oscillations. With the variation of the slowly-varying parameter, different equilibrium states and relevant bifurcations in the fast subsystem are presented. It is found that for different parameter values, multiple balance curves of the fast subsystem may coexist, which affect the structure of the bursting attractor. For the other parameters fixed to certain values, the balance curve with the variation of the slowly-varying parameter is presented. Three typical cases with different exciting amplitudes are considered, corresponding to different situations of coexistence of equilibrium states in the fast subsystem. In the first case, there exist at most three stable equilibrium points in the fast subsystem. Bursting attractor that oscillates around the three points can be observed, in which fold and Hopf bifurcations lead to the alternations between spiking states and quiescent states, while in the second case, saddle on the limit cycle bifurcation may cause the repetitive spiking oscillations to jump to the equilibrium curve. In the third case with relatively large exciting amplitude, only two equilibrium curves may involve the bursting oscillations, in which fold bifurcations lead to the alternation between the quiescent states and spiking states. Unlike the structures of bursting oscillations in the symmetric system, different forms of asymmetrical bursting oscillations with different periodic exciting amplitudes can be observed, the mechanisms of which are presented. It is pointed out that the change of the external exciting amplitude, does not only cause the variation of the attracting basins corresponding to different stable equilibrium branches, but also leads to the change of the temporal intervals when the trajectory passes different bifurcation points, respectively, which results in different patterns of bursting oscillations. Furthermore, since the slowly-varying parameter determined by the whole exciting term changes between two extreme values determined by the amplitude, the trajectory of the bursting oscillations of the transformed phase portrait returns at the two extreme values. The properties of equilibrium branches between the two extreme values determine the forms of the moving attractors.
[1] Cardin P T, de Moraes J R, da Silva P R 2015 J. Math. Anal. Appl. 423 1166
[2] Hodgkin A L, Huxley A F 1990 Bull. Math. Biol. 52 25
[3] Sánchez A D, G.Izús G, Erba M G, Deza R R 2014 Phys. Lett. A 378 1579
[4] Rakaric Z, Kovacic I 2016 Mech. Syst. Signal Process. 81 35
[5] Chumakov G A, Chumakova N A 2003 Chem. Eng. J. 91 151
[6] Jia F L, Xu W, Li H N, Hou L Q 2013 Acta Phys. Sin. 62 100503 (in Chinese) [贾飞蕾, 徐伟, 李恒年, 侯黎强 2013 62 100503]
[7] Yang S C, Hong H P 2016 Eng. Struct. 123 490
[8] Rakaric Z, Kovacic I 2016 Mech. Syst. Signal Process. 81 35
[9] Li X H, Hou J Y 2016 Int. J. Non-Linear Mech. 81 165
[10] Yu B S, Jin D P, Pang Z J 2014 Sci. Sin. Phys. Mech. Astron. 44 858 (in Chinese) [余本嵩, 金栋平, 庞兆君 2014 中国科学: 物理学 力学 天文学 44 858]
[11] Bi Q S 2012 Sci. China Ser. E10 2820
[12] Kim S, Lim W 2016 Neural Networks 79 53
[13] Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171
[14] Izhikevich E M 2003 Trends Neurosci. 26 161
[15] Shimizu K, Saito Y, Sekikawa M, Inaba N 2012 Physica D 241 1518
[16] Shilnikov A, Kolomiets M 2008 Int. J. Bifurcation Chaos 18 2141
[17] Han X J, Xia F B, Ji P, Bi Q S, Kurths J 2016 Commun. Nonlinear Sci. Numer. Si. 36 517
[18] Bi Q S, Zhang R, Zhang Z D 2014 Appl. Math. Comput. 243 482
[19] Yu Y, Zhang Z D, Bi Q S, Gao Y 2016 Appl. Math. Model 40 1816
[20] Zhang X F, Wu L, Bi Q S 2016 Chin. Phys. B 25 070501
[21] Xing Y Q, Chen X K, Zhang Z D, Bi Q S 2016 Acta Phys. Sin. 65 090501 (in Chinese) [邢雅清, 陈小可, 张正娣, 毕勤胜 2016 65 090501]
[22] Zheng S, Han X J, Bi Q S 2011 Commun. Nonlinear Sci. Numer. Si. 16 1998
[23] Vo T, Kramer M A, Kaper T J 2016 Phys. Rev. Lett. 117 268101
[24] Zvonko R, Ivana K 2016 Mech. Syst. Signal Process. 81 35
[25] Milicevic K, Nyarko E K, Biondic I 2016 Nonlinear Dyn. 81 51
[26] Srinivasan K, Chandrasekar V K, Pradeep R G 2016 Commun. Nonlinear Sci. Numer. Si. 39 156
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[1] Cardin P T, de Moraes J R, da Silva P R 2015 J. Math. Anal. Appl. 423 1166
[2] Hodgkin A L, Huxley A F 1990 Bull. Math. Biol. 52 25
[3] Sánchez A D, G.Izús G, Erba M G, Deza R R 2014 Phys. Lett. A 378 1579
[4] Rakaric Z, Kovacic I 2016 Mech. Syst. Signal Process. 81 35
[5] Chumakov G A, Chumakova N A 2003 Chem. Eng. J. 91 151
[6] Jia F L, Xu W, Li H N, Hou L Q 2013 Acta Phys. Sin. 62 100503 (in Chinese) [贾飞蕾, 徐伟, 李恒年, 侯黎强 2013 62 100503]
[7] Yang S C, Hong H P 2016 Eng. Struct. 123 490
[8] Rakaric Z, Kovacic I 2016 Mech. Syst. Signal Process. 81 35
[9] Li X H, Hou J Y 2016 Int. J. Non-Linear Mech. 81 165
[10] Yu B S, Jin D P, Pang Z J 2014 Sci. Sin. Phys. Mech. Astron. 44 858 (in Chinese) [余本嵩, 金栋平, 庞兆君 2014 中国科学: 物理学 力学 天文学 44 858]
[11] Bi Q S 2012 Sci. China Ser. E10 2820
[12] Kim S, Lim W 2016 Neural Networks 79 53
[13] Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171
[14] Izhikevich E M 2003 Trends Neurosci. 26 161
[15] Shimizu K, Saito Y, Sekikawa M, Inaba N 2012 Physica D 241 1518
[16] Shilnikov A, Kolomiets M 2008 Int. J. Bifurcation Chaos 18 2141
[17] Han X J, Xia F B, Ji P, Bi Q S, Kurths J 2016 Commun. Nonlinear Sci. Numer. Si. 36 517
[18] Bi Q S, Zhang R, Zhang Z D 2014 Appl. Math. Comput. 243 482
[19] Yu Y, Zhang Z D, Bi Q S, Gao Y 2016 Appl. Math. Model 40 1816
[20] Zhang X F, Wu L, Bi Q S 2016 Chin. Phys. B 25 070501
[21] Xing Y Q, Chen X K, Zhang Z D, Bi Q S 2016 Acta Phys. Sin. 65 090501 (in Chinese) [邢雅清, 陈小可, 张正娣, 毕勤胜 2016 65 090501]
[22] Zheng S, Han X J, Bi Q S 2011 Commun. Nonlinear Sci. Numer. Si. 16 1998
[23] Vo T, Kramer M A, Kaper T J 2016 Phys. Rev. Lett. 117 268101
[24] Zvonko R, Ivana K 2016 Mech. Syst. Signal Process. 81 35
[25] Milicevic K, Nyarko E K, Biondic I 2016 Nonlinear Dyn. 81 51
[26] Srinivasan K, Chandrasekar V K, Pradeep R G 2016 Commun. Nonlinear Sci. Numer. Si. 39 156
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