多平衡态下簇发振荡产生机理及吸引子结构分析

以周期激励下受控Lorenz模型为例, 考察了多平衡态共存下激励频率与系统固有频率之间存在量级差距也即存在频域上的不同尺度时的耦合效应. 由于激励频率远小于系统的固有频率, 因此将整个激励项视为慢变参数, 分析随慢变参数变化下的各种分岔模式及其相应的分岔行为, 指出在一定条件下, 不同平衡点会产生Hopf分岔和fold分岔. 根据分岔条件的不同, 给出了两种典型情况下的簇发振荡, 并通过引入转换相图, 揭示了不同簇发的产生机理, 指出多平衡态和多种分岔共存不仅会导致沉寂态和激发态的多样性, 而且会使得不同沉寂态和激发态之间存在着不同的转换形式.     

Bursting phenomena as well as the bifurcation mechanism in a coupled BVP oscillator with periodic excitation

We explore the complicated bursting oscillations as well as the mechanism in a high-dimensional dynamical system.By introducing a periodically changed electrical power source in a coupled BVP oscillator, a fifth-order vector field with two scales in frequency domain is established when an order gap exists between the natural frequency and the exciting frequency.Upon the analysis of the generalized autonomous system, bifurcation sets are derived, which divide the parameter space into several regions associated with different types of dynamical behaviors. Two typical cases are focused on as examples,in which different types of bursting oscillations such as sub Hopf/sub Hopf burster, sub Hopf/fold-cycle burster, and doublefold/fold burster can be observed. By employing the transformed phase portraits, the bifurcation mechanism of the bursting oscillations is presented, which reveals that different bifurcations occurring at the transition between the quiescent states(QSs) and the repetitive spiking states(SPs) may result in different forms of bursting oscillations. Furthermore, because of the inertia of the movement, delay may exist between the locations of the bifurcation points on the trajectory and the bifurcation points obtained theoretically.     

Non-smooth bursting analysis of a Filippov-type system with multiple-frequency excitations

The main purpose of this paper is to explore the patterns of the bursting oscillations and the non-smooth dynamical behaviours in a Filippov-type system which possesses parametric and external periodic excitations. We take a coupled system consisting of Duffing and Van der Pol oscillators as an example. Owing to the existence of an order gap between the exciting frequency and the natural one, we can regard a single periodic excitation as a slow-varying parameter, and the other periodic excitations can be transformed as functions of the slow-varying parameter when the exciting frequency is far less than the natural one. By analysing the subsystems, we derive equilibrium branches and related bifurcations with the variation of the slow-varying parameter. Even though the equilibrium branches with two different frequencies of the parametric excitation have a similar structure, the tortuousness of the equilibrium branches is diverse, and the number of extreme points is changed from 6 to 10. Overlying the equilibrium branches with the transformed phase portrait and employing the evolutionary process of the limit cycle induced by the Hopf bifurcation, the critical conditions of the homoclinic bifurcation and multisliding bifurcation are derived. Numerical simulation verifies the results well.     

慢变控制下Chen系统的复杂行为及其机理

由于Chen系统的控制分析大都是基于同一时间尺度,而两时间尺度耦合问题的相关研究基本上局限于单维慢变量情形.本文探讨了基于慢时间尺度上的Duffing振子,即含有两维慢子系统控制下Chen系统的动力学演化过程.给出了诸如对称式fold/fold、对称式fold/Hopf、对称式homoclinic/homoclinic等不同形式的簇发振荡行为,并揭示了其相应的产生机制,指出慢子系统中两维慢变量的相互影响导致系统产生了类似于周期激励下的簇发行为.     

Bursting oscillations as well as the bifurcation mechanism in a non-smooth chaotic geomagnetic field model

Based on the chaotic geomagnetic field model, a non-smooth factor is introduced to explore complex dynamical behaviors of a system with multiple time scales. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamic behaviors. Due to the existence of non-smooth factors, different types of bifurcations are presented in spiking states, such as grazing-sliding bifurcation and across-sliding bifurcation. In addition, the non-smooth fold bifurcation may lead to the appearance of a special quiescent state in the interface as well as a non-smooth homoclinic bifurcation phenomenon. Due to these bifurcation behaviors, a special transition between spiking and quiescent state can also occur.     

Bursting phenomena as well as the bifurcation mechanism in controlled Lorenz oscillator with two time scales

A controlled Lorenz model with fast-slow effect has been established, in which there exist order gap between the variables associated with the controller and the original Lorenz oscillator, respectively. The conditions of fold bifurcation as well as Hopf bifurcation for the fast subsystem are derived to investigate the mechanism of the behaviors of the whole system. Two cases in which the equilibrium points of the fast subsystem behave in different characteristics have been considered, leading to different dynamical evolutions with the change of coupling strength. Several types of bursting phenomena, such as fold/fold burster, fold/Hopf burster, near-fold/Hopf burster, fold/near-Hopf buster have been observed. Theoretical analysis shows that the bifurcations points which connect the quiescent state and the repetitive spiking state agree well with the turning points of the trajectories of the bursters. Furthermore, the mechanism of the period-adding bifurcations, resulting in the rapid change of the period of the movements, is presented.     

Homoclinic bifurcation in Chua’s circuit

We report our experimental observations of the Shil’nikov-type homoclinic chaos in asymmetry-induced Chua’s oscillator. The asymmetry plays a crucial role in the related homoclinic bifurcations. The asymmetry is introduced in the circuit by forcing a DC voltage. For a selected asymmetry, when a system parameter is controlled, we observed transition from large amplitude limit cycle to homoclinic chaos via a sequence of periodic mixed-mode oscillations interspersed by chaotic states. Moreover, we observed two intermediate bursting regimes. Experimental evidences of homoclinic chaos are verified with PSPICE simulations.     

Pre-B?tzinger复合体的从簇到峰放电的同步转迁及分岔机制

Pre-Bötzinger复合体是兴奋性耦合的神经元网络,通过产生复杂的放电节律和节律模式的同步转迁参与调控呼吸节律.本文选用复杂簇和峰放电节律的单神经元数学模型构建复合体模型,仿真了与生物学实验相关的多类同步节律模式及其复杂转迁历程,并利用快慢变量分离揭示了相应的分岔机制.当初值相同时,随着兴奋性耦合强度的增加,复合体模型依次表现出完全同步的“fold/homoclinic”,“subHopf/subHopf”簇放电和周期1峰放电.当初值不同时,随耦合强度增加,表现为由“fold/homoclinic”,到“fold/fold limit cycle”、到“subHopf/subHopf”与“fold/fold limit cycle”的混合簇放电、再到“subHopf/subHopf”簇放电的相位同步转迁,最后到反相同步周期1峰放电.完全(同相)同步和反相同步的周期1节律表现出了不同分岔机制.反相峰同步行为给出了与强兴奋性耦合容易诱发同相同步这一传统观念不同的新示例.研究结果给出了preBötzinger复合体的从簇到峰放电节律的同步转迁规律及复杂分岔机制,反常同步行为丰富了非线性动力学的内涵.     

Analysis of the T-point-Hopf bifurcation

In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-2 heteroclinic cycle occurs. If the parameter space is three-dimensional, such a bifurcation is located generically on a curve. A more degenerate scenario appears when this curve reaches a surface of Hopf bifurcations of one of the equilibria involved in the heteroclinic cycle. We are interested in the analysis of this codimension-3 bifurcation, which we call T-point-Hopf. In this work we propose a model, based on the construction of a Poincaré map, that describes the global behavior close to a T-point-Hopf bifurcation. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved. The predictions deduced from this model strongly agree with the numerical results obtained in a modified van der Pol-Duffing electronic oscillator.     

Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.     

两时间尺度下非光滑广义蔡氏电路系统的簇发振荡机理

通过引入周期变化的电流源并选择适当参数, 使得周期激励频率与系统固有频率之间存在量级差距, 建立了两时间尺度即快慢耦合非光滑广义蔡氏电路模型. 基于相应的广义自治系统, 考察了其不同区域中的平衡态及其稳定性, 得到了不同分岔行为及其相应的临界条件. 同时, 利用广义Clarke导数得到的广义Jacobian矩阵, 探讨了系统轨迹穿越非光滑分界面时的各种非常规分岔模式, 进而结合广义相图, 深入分析了Fold/Fold周期簇发振荡以及Fold/Hopf周期簇 发振荡两种典型的周期簇发行为及其相应的分岔机制. 关键词： 非光滑 广义蔡氏电路 两时间尺度 分岔机制     

Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system

Border-collision bifurcations arise when the periodic trajectory of a piecewise-smooth system under variation of a parameter crosses into a region with different dynamics. Considering a three-dimensional map describing the behavior of a DC/DC power converter, the Letter discusses a new type of border-collision bifurcation that leads to the birth of a “bilayered torus”. This torus consists of the union of two saddle cycles, their unstable manifolds, and a stable focus cycle. When changing the parameters, the bilayered torus transforms through a border-collision bifurcation into a resonance torus containing the stable cycle and a saddle. The Letter also presents scenarios for torus destruction through homoclinic and heteroclinic tangencies.     

Experimental unfolding of the nonlinear dynamics of a cable-mass suspended system around a divergence-Hopf bifurcation

Systematic experimental investigation of the finite amplitude dynamics of a multiple internally resonant suspended cable-mass, subjected to anti-phase support motion at primary resonance, is accomplished. Upon getting hints from a basic system configuration assumed as reference setup about the multiple bifurcation event possibly governing transition to complex dynamics, an improved experimental apparatus is used to make it technically accessible. Results obtained by varying three control parameters, namely the frequency and amplitude of excitation and the temperature of a thermostatic chamber embedding the experimental system, allow us to characterize in-depth various occurring classes of motion in terms of time and spatial complexity, to describe peculiar and/or persistent features of transition to nonregular dynamics, and to trace them back to a canonical scenario from bifurcation theory. Variable response paths are detected via bifurcation diagrams and spectra of singular values of measurement results, and overall behaviour charts are built in the excitation parameter space. Considering the temperature as a controllable parameter shows to be fundamental for: (i) indirectly setting cable material properties to values for which the conjectured codimension 2 bifurcation becomes apparent, (ii) qualitatively referring the experimental unfolding of regular and nonregular cable dynamics to the theoretical unfolding of the divergence-Hopf bifurcation normal form, and (iii) determining system response not only in the strict neighbourhood of the organizing divergence-Hopf bifurcation but also in the ensuing postcritical regions where the dependence of material damping on temperature affects secondary bifurcations to low-dimensional homoclinic chaos.     

Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust ‘slow switching’ oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate the implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking).For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S₃-transcritical homoclinic bifurcation, (b) a saddle-node/heteroclinic bifurcation and (c) a Z₃-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry.For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions.     

Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models

We present a qualitative analysis of a generic model structure that can simulate the bursting and spiking dynamics of many biological cells. Four different scenarios for the emergence of bursting are described. In this connection a number of theorems are stated concerning the relation between the phase portraits of the fast subsystem and the global behavior of the full model. It is emphasized that the onset of bursting involves the formation of a homoclinic orbit that travels along the route of the bursting oscillations and, hence, cannot be explained in terms of bifurcations in the fast subsystem. In one of the scenarios, the bursting oscillations arise in a homoclinic bifurcation in which the one-dimensional (1D) stable manifold of a saddle point becomes attracting to its whole 2D unstable manifold. This type of homoclinic bifurcation, and the complex behavior that it can produce, have not previously been examined in detail. We derive a 2D flow-defined map for this situation and show how the map transforms a disk-shaped cross-section of the flow into an annulus. Preliminary investigations of the stable dynamics of this map show that it produces an interesting cascade of alternating pitchfork and boundary collision bifurcations. Received 24 June 1999 and Received in final form 17 February 2000     

周期切换下Chen系统的振荡行为与非光滑分岔分析

研究了不同参数Chen系统之间进行周期切换时的分岔和混沌行为.基于平衡态分析,考虑Chen系统在不同稳态解时通过周期切换连接生成的复合系统的分岔特性,得到系统的不同周期振荡行为.在演化过程中,由于切换导致的非光滑性,复合系统不仅仅表现为两子系统动力特性的简单连接,而且会产生各种分岔,导致诸如混沌等复杂振荡行为.通过Poincaré映射方法,讨论了如何求周期切换系统的不动点和Floquet特征乘子.基于Floquet理论,判定系统的周期解是渐近稳定的.同时得到,随着参数变化,系统既可以由倍周期分岔序列进入混沌,也可以由周期解经过鞍结分岔直接到达混沌.研究结果揭示了周期切换系统的非光滑分岔机理.     

广义BVP电路系统的振荡行为及其非光滑分岔机理

探讨了具有分段线性特性的广义BVP电路系统随参数变化的复杂动力学演化过程. 其非光滑分界面将相空间划分成不同的区域, 分析了各区域中平衡点的稳定性, 得到其相应的简单分岔和Hopf分岔的临界条件. 给出了不同分界面处广义Jacobian矩阵特征值随辅助参数变化的分布情况, 讨论了分界面处系统可能存在的分岔行为, 指出当广义特征值穿越虚轴时可能引起Hopf分岔, 导致系统由周期振荡转变为概周期振荡, 而当出现零特征值时则导致系统的振荡在不同平衡点之间转换. 针对系统的两种典型振荡行为, 结合数值模拟验证了理论分析的结果.     

兴奋性作用诱发神经簇放电个数不增反降的分岔机制

非线性动力学在识别神经放电的复杂现象、机制和功能方面发挥了重要作用.不同于传统观念,本文提出了兴奋性作用可以降低而不是增加簇内放电个数的新观点.在簇放电模式休止期的适合相位施加强度合适的脉冲或自突触电流,能诱发簇内放电个数降低;电流的施加相位越早,所需的强度阈值越大,簇内放电个数越少.进一步,利用快慢变量分离获得的簇放电的动力学性质进行了理论解释.簇放电模式表现出低电位的休止期和高电位的放电的交替,存在于快子系统的鞍结分岔点和同宿轨分岔点之间;放电起始于鞍结分岔、结束于同宿轨分岔;越靠近同宿轨分岔从休止期跨越到放电所需的电流强度越大.因此,电流在休止期上的作用相位越早,就越靠近同宿轨分岔,因而从休止期跨越到放电需要的电流强度阈值越大,放电起始相位到同宿轨分岔之间的区间变小导致放电个数变少.研究结果丰富了非线性现象及机制,对兴奋性作用提出了新看法,给出了调控簇放电模式的新途径.     

“Hopf/homoclinic”簇放电和“SubHopf/homoclinic"簇放电之间的同步

以修正过的Morris-Lecar神经元模型为例,讨论了“Hopf/homoclinic”簇放电和“SubHopf/homoclinic"簇放电之间的同步行为.首先,分别考察了同一拓扑类型的两个耦合簇放电神经元的同步行为,发现“Hopf/homoclinic”簇放电比“SubHopf/homoclinic”簇放电达到膜电位完全同步所需要的耦合强度小,即前者比后者更容易达到膜电位完全同步.其次,对这两个不同拓扑类型的簇放电神经元的耦合同步行为进行了讨论.通过数值分析发现随着耦合强度的增加,两种不同类型的簇放电首先达到簇放电同步,然后当耦合强度足够大时甚至可以达到膜电位完全同步,并且同步后的放电类型更接近容易同步的簇放电类型,即“Hopf/homoclinic”簇放电.然而令人奇怪的是此时慢变量并没有达到完全同步,而是相位同步;慢变量之间呈现为一种线性关系.这一点和现有文献的结果截然不同.     

Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle

A detailed asymptotic study of the effect of small Gaussian white noise on a relaxation oscillator undergoing a supercritical Hopf bifurcation is presented. The analysis reveals an intricate stochastic bifurcation leading to several kinds of noise-driven mixed-mode oscillations at different levels of amplitude of the noise. In the limit of strong time-scale separation, five different scaling regimes for the noise amplitude are identified. As the noise amplitude is decreased, the dynamics of the system goes from the limit cycle due to self-induced stochastic resonance to the coherence resonance limit cycle, then to bursting relaxation oscillations, followed by rare clusters of several relaxation cycles (spikes), and finally to small-amplitude oscillations (or stable fixed point) with sporadic single spikes. These scenarios are corroborated by numerical simulations.