关键词:多频激励;频域两尺度;簇发振荡;分岔机理 Abstract The complicated behaviors as well as the bifurcation mechanism of the dynamical systems with different time scales have become one of the hot subjects at home and abroad, since they often behave in bursting attractors characterized by the combinations between large-amplitude oscillations and small-amplitude oscillations. Since the slow-fast analysis was employed to investigate the mechanism of the special forms of movements, a lot of results related to the bursting oscillations in autonomous systems with two scales in time domain have been obtained. Recently, based on the transformed phase portraits, different types of bursting oscillations as well as the mechanism in the vector fields with single periodic excitation have been presented. However, few works has been published related to the systems with multiple periodic excitations, the dynamics of which still remains an open problem. The main purpose of the manuscript is to explore effect of the multiple scales in such systems. As a example, based on a relatively simple four-dimensional Chua’s circuit, by introducing two periodically changed electric sources, when the two exciting frequencies are strictly resonant, both of which are far less than the natural frequency of the system, a dynamical model with scales under two periodic excitations is established. Note that the combination of the two exciting terms can be transformed as a function of a periodic term with single frequency, which can be regarded as a slow-varying parameter. The equilibrium branches as well as the associated bifurcations with the variation of the slow-varying parameter can be derived by employing the characteristics analysis of the equilibrium points. It is found that the distribution of the equilibrium branches as well as the bifurcation details may changed with the variation the amplitudes of the excitations, which may influence the attractors of the whole dynamical system. Three typical cases corresponding to the different situations of the equilibrium branches are considered, in which different forms of bursting oscillations are observed. Based on the transformed phase portraits, the bifurcation mechanism of the bursting oscillations has been presented. It is found that the trajectory may move almost strictly along one of the stable equilibrium branches, while jumping to another stable equilibrium branch may occur at the fold bifurcation points, the transient process of which leads to the large-amplitude oscillations corresponding to spiking states. Furthermore, it is pointed out that when more fold bifurcation points involve the behaviors of the system, more complicated bursting oscillations may appear.
Keywords:multi-frequency excitation;two scales in frequency domain;bursting oscillations;bifurcation mechanism -->0 PDF (2881KB)元数据多维度评价相关文章收藏文章 本文引用格式导出EndNoteRisBibtex收藏本文--> 夏雨, 毕勤胜, 罗超, 张晓芳. 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 https://doi.org/10.6052/0459-1879-18-017 XiaYu, BiQinsheng, LuoChao, ZhangXiaofang. BEHAVIORS OF MODIFIED CHUA’S OSCILLATOR TWO TIME SCALES UNDER TWO EXCITATOINS WITH FREQUENCY RATIO AT 1:2[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(2): 362-372 https://doi.org/10.6052/0459-1879-18-017
四维修正蔡氏电路中存在着丰富的动力学行 为[27],在该电路的基础上,引入两频率不同的周期电流源 ,如图1所示,其中非线性阻尼的伏安特性方程为 , 为无量纲参数, 和 量纲分别为(V<sup>-2</sup>) 和(V<sup>-4</sup>). 其相应的动力学方程可以表示为 $ \left.\!\! \begin{array}{c} d V_1 / d t = \dfrac{1}{C_1 }[G_1 (V_2 - V_1 ) - G_1 g(V_1 ) + i_G ] \\ d V_2 / d t = \dfrac{1}{C_2 }[G_1 (V_1 - V_2 ) + i_{L_1 } + i_{L_2 } ] \\ d i_{L_1 } / d t = -\dfrac{1}{L_1 }(V_2 - R_2 i_{L_2 } ) \\ d i_{L_2 } / d t = - \dfrac{1}{L_2 }V_2 \end{array}\!\!\right\} (1)$ 显示原图|下载原图ZIP|生成PPT 图 1双频外激励电路模型 -->Fig. 1Circuit model with two periodic excitations related to different frequencies -->
其中 , . 引入变换 , , , , ,取定 ,式(1)可以表示为如下无量纲化形式 $ \left.\!\!\begin{array}{c} d x / d \tau = \alpha [y - \delta (x + bx^3 + cx^5)] + w \\ d y / d \tau = \kappa x - y + u + v \\ d u / d \tau = - \gamma y + v \\ d v / d \tau = - \beta y \end{array}\!\!\right\} (2)$ 其中, , , , , , , . 和 为参考电位. 激励项 ,其中 , , . 当两外激励项不存在时,系统会由平衡点失稳进入周期振荡,进而进入混沌,而对于某些参数条件,存在着对应于两个正Lyapunov指数[28]. 周期外激励的存在,同样会导致系统由周期振荡进入混沌. 假设外激励为零,可以计算系统的固有频率,而非线性系统的固有频率与系统的运动状态有关,如当系统逐渐稳定于某一焦点时,其固有频率为该焦点雅可比矩阵的一对共轭复根的虚部值,而当系统表现为周期振荡时,其固有频率为该振荡的频率. 在此取 , ,其他参数取常规量时,外激励频率与系统的固有频率之间会存在量级差距. 显然,状态变量主要按照系统的固有频率振荡,而两激励项则按照其激励频率振荡,导致不同尺度之间的耦合,从而产生诸如簇发等类型的特殊振荡模式.
2 分岔分析
由于两外激励频率处于严格的共振关系,假设 ,则两慢变项可以表示为 . 同时,由于激励频率远小于系统的固有频率,因此,可以将 视为慢变参数,即慢子系统,从而得到相应的广义自治系统,即快子系统,进而将整个系统看作快慢两子系统的耦合. 也即快子系统为 $ \left.\!\!\begin{array}{c} d x / d \tau = \alpha [y - \delta (x + bx^3 + cx^5)] + A_1 (2W^2 - 1) + A_2 W \\ d y / d \tau = \kappa x - y + u + v \\ d u / d \tau = - \gamma y + v \\ d v / d \tau = - \beta y \end{array}\!\!\right\} (3) $ 而慢子系统为 . 为揭示快慢耦合复杂行为的产生机制,首先分析快子系统的分岔行为. 其平衡点可以表示为 , 其中 满足 (4) 其相应的特征方程为 (5) 式中, , , , ,其中 . 当参数满足 , , , 时, 为稳定的. 当参数满足 ,也即FB $ \left.\!\!\begin{array}{l} 1 + 3bX_0^2 + 5cX_0^4 = 0 \\ - \alpha \delta (X_0 + bX_0^3 + cX_0^5 ) + A_1 (2W^2 - 1) + A_2 W = 0 \end{array}\!\! \right \} (6) $ 可能会产生fold分岔,导致不同平衡点之间的跳跃. 而当参数满足HB $ h_1 h_2 h_3 - h_1^2 h_4 - h_3^2 = 0 (7)$ 则 可能会由Hopf分岔失稳,导致频率为 的周期振荡,在这两分岔集的交点处,也即FHB $ \left.\begin{array}{l} 1 + 3bX_0^2 + 5cX_0^4 = 0 \\ - \alpha \delta (X_0 + bX_0^3 + cX_0^5 ) + A_1 (2W^2 - 1) + A_2 W = 0 \\ \alpha \kappa - \gamma = 0 \end{array}\!\!\right\} (8)$ 可能会产生余维二fold-Hopf分岔,导致平衡点向远离该平衡点所在平衡曲线的极限环的跳跃. 固定参数 , , , , , , , . 图2分别给出了 , 和 时3种典型的不同外激励幅值下的平衡曲线 及其分岔图. 从图2(d)中可以看出,整个外激励项 的变化并不关于 对称,从而导致图2(a) 图2(c)中的平衡曲线也不关于 对称. 同时,从图2(d)中发现,3种不同情形下的整个外激励项变化曲线S , S 和S 均通过 和 点,说明3种情形下在该两点存在相同的平衡点分布. 由于3条变化曲线所对应的值在右端变化范围大,表明在平衡曲线图中右端存在着更多的平衡点,这也可以由图2(a) 图2(c)中的平衡曲线得到证实. 由于 ,因此 ,也即 的变化存在两极值边界 ,分别用 表示. 在 时,由图2(a)可知,平衡曲线由3段组成,其中EB 和EB 是稳定的,EB 是不稳定的, 分别由fold分岔点FB 和FB 连接. 显示原图|下载原图ZIP|生成PPT 图 2平衡曲线及其分岔图 -->Fig. 2The equilibrium branches as well as bifurcation points -->
为揭示其相应的簇发振荡机制,图6给出了在 平面上的转换相图及其与相应快子系统平衡曲线的叠加图. 与图4(a)相比,图6(a)在 平面上形成了3个滞后环结构,多条平衡曲线以及相应的分岔点参与了其簇发振荡的结构. 显示原图|下载原图ZIP|生成PPT 图 6时 (a) 平面上的转换相图; (b) 转换相图与平衡曲线的叠加图 -->Fig. 6(a) Transformed phase portrait on the plane; (b) The overlap of equilibrium branches and transformed phase portrait on the plane for -->
对于严格共振的多频激励系统,可以通过适当变换,将所有激励项表示为某一激励项的代数表达. 当激励频率远小于系统的固有频率时,可以将该激励项作为慢变参数,从而建立相应的快子系统和单一慢变量的慢子系统. 通过对快子系统的平衡点及分岔分析,结合转换相图,可以揭示不同簇发振荡的产生机制. 基于相对简单的模型,本文应用上述方法得到了3种典型的簇发振荡,并给出了其相应的产生机制. 由于快子系统的平衡曲线上仅存在fold分岔,发现整个系统的轨线会随参数的不同,沿不同稳定平衡曲线运动,而fold分岔会导致轨迹在不同平衡曲线上的跳跃,产生相应的激发态. 激发态可以用从分岔点向相应稳定平衡曲线的暂态过程来近似,其振荡幅值的变化和振荡频率也可用相应平衡点特征值的实部和虚部来描述. 参与簇发振荡的平衡曲线分岔点越多,其相应簇发振荡吸引子的结构也越复杂. 必须指出的是,当系统中的各激励频率不存在严格共振关系,同时激励频率远小于系统的固有频率时,可以将激励频率与时间之积视为慢变量,同样可以进行相关分析. 在这种情况下,需要一系列的放大图才能将不同簇发振荡的机制描述清楚,我们将另外讨论这种情形. The authors have declared that no competing interests exist.
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