参外联合激励下的簇发振荡及其机理

当周期激励频率远小于系统固有频率时，会存在快慢耦合效应，与单项激励不同，参外联合激励不仅会导致快子系统平衡曲线和分岔行为的复杂化，也会产生一些特殊的非线性现象，为此，本文以两耦合Hodgkin-Huxley细胞模型为例，引入周期参外联合激励，探讨在频域不同尺度耦合时该系统的簇发振荡的特点及其分岔机制．通过建立相应的快慢子系统，得到慢变参数变化下的快子系统的各种分岔模式以及相应的分岔行为，结合转换相图，揭示耦合系统随激励幅值变化时的动力学行为及其机理．研究表明，在激励幅值较小时，系统表现为概周期振荡，两频率分别近似于快子系统平衡曲线由Hopf分岔引起的两稳定极限环的振荡频率．概周期解随激励幅值的增加进入簇发振荡，导致这些簇发振荡的主要原因是在慢变参数变化的部分区间内，存在唯一稳定的平衡曲线，使得系统的轨迹逐渐趋向该平衡曲线，产生沉寂态，并随着慢变参数的变化，由分岔进入激发态．同时，快子系统中参与簇发振荡的稳定吸引子随激励幅值的变化也会不同，导致不同形式的簇发振荡．另外，与单项激励下的情形不同，联合激励时快子系统的部分稳定吸引子掩埋在其它稳定吸引子内，从而失去对簇发振荡的影响．

Bursting Oscillations and the Mechanism of a Dynamic System with Parametric and External Excitation

When the exciting frequency is far less than the natural frequency, the effect of slow-fast coupling can be observed. Unlike the situation with sole excitation, the parametric and external excitation not only leads to the complexity of the equilibrium branches as well as the bifurcations, but also results in certain special nonlinear phenomena. To explore the evolution of the dynamic system with parametric and external excitation, based on the coupling of two cells of Hodgkin-Huxley type, by introducing parametric and external excitation, the bursting oscillations as well as the bifurcation mechanism of the system with two scales in frequency domain are investigated. By establishing the slow and fast subsystems, the bifurcations of the fast subsystem are derived with the variation of the slow-varying parameter, which can be used to reveal the dynamics and the mechanism of the coupling system upon the combination of the transformed phase portrait. It is found that when the exciting amplitude is relatively small, the system behaves in quasi-periodic oscillations, which can evolve to the bursting oscillations. The related two frequencies can be approximated by the frequencies of the two stable limit cycles via Hopf bifurcations in the fast subsystem. With the increase of the exciting amplitude, bursting oscillations can be observed, the trajectory of which may oscillate down to the sole stable equilibrium branch, appearing in quiescent state, which may bifurcate to repetitive spiking oscillations. Meanwhile, different stable attractors in the fast subsystem involved in the dynamics may result in different forms of bursting oscillations. Furthermore, unlike the situation with a sole excitation, when there coexist the parametric and external excitations, some of the stable attractors may have no influence on the oscillations since they are buried in other stable attractors.