Multiple scales and normal forms in a ring of delay coupled oscillators with application to chaotic Hindmarsh–Rose neurons

We study the appearance and stability of spatiotemporal periodic patterns like phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or antiphase oscillations, and coexistence of multiple patterns, in a ring of bidirectionally delay coupled oscillators. Hopf bifurcation, Hopf–Hopf bifurcation, and the equivariant Hopf bifurcation are studied in the viewpoint of normal forms obtained by using the method of multiple scales which is a kind of perturbation technique, thus a clear bifurcation scenario is depicted. We find time delay significantly affects the dynamics and induces rich spatiotemporal patterns. With the help of the unfolding system near Hopf–Hopf bifurcation, it is confirmed in some regions two kinds of stable oscillations may coexist. These phenomena are shown for the delay coupled limit cycle oscillators as well as for the delay coupled chaotic Hindmarsh–Rose neurons.     

Hopf bifurcation and spatio-temporal patterns in?delay-coupled van der Pol oscillators

In this paper, the dynamics of a pair of van der Pol oscillators with delayed velocity coupling is studied by taking the time delay as a bifurcation parameter. We first investigate the stability of the zero equilibrium and the existence of Hopf bifurcations induced by delay, and then study the direction and stability of the Hopf bifurcations. Then by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations. We find that there are different in-phase and anti-phase patterns as the coupling time delay is increased. The analytical theory is supported by numerical simulations, which show good agreement with the theory.     

Using delay to quench undesirable vibrations

A van der Pol type system with delayed feedback is explored by employing the two variable expansion perturbation method. The perturbation scheme is based on choosing a critical value for the delay corresponding to a Hopf bifurcation in the unperturbed ε=0 system. The resulting amplitude–delay relation predicts two Hopf bifurcation curves, such that in the region between these two curves oscillations will be quenched. The perturbation results are verified by comparison with numerical integration.     

Rich dynamics in a non-local population model over three patches

We consider a system of nonlinear delay differential equations that describes the growth of the mature population of a species with age-structure living over three patches. We analyze existence of non-negative homogeneous equilibria and their stability and discuss possible Hopf bifurcation from these equilibria. More precisely, by employing both the standard Hopf bifurcation theory and the symmetric bifurcation theory for functional differential equations, we obtain very rich dynamics for the system, including bistable equilibria, transient oscillations, synchronous periodic solutions, phase-locked periodic solutions, mirror-reflecting waves and standing waves.     

Dynamics of microbubble oscillators with delay coupling

Two vibrating bubbles submerged in a fluid influence each others’ dynamics via sound waves in the fluid. Due to finite sound speed, there is a delay between one bubble’s oscillation and the other’s. This scenario is treated in the context of coupled nonlinear oscillators with a delay coupling term. It has previously been shown that with sufficient time delay, a supercritical Hopf bifurcation may occur for motions in which the two bubbles are in phase. In this work, we further examine the bifurcation structure of the coupled microbubble equations, including analyzing the sequence of Hopf bifurcations that occur as the time delay increases, as well as the stability of this motion for initial conditions which lie off the in-phase manifold. We show that in fact the synchronized, oscillating state resulting from a supercritical Hopf is attracting for such general initial conditions.     

Multiple scales for two-parameter bifurcations in a neutral equation

Van der Pol??s equation with extended delay feedback is investigated as a neutral differential-difference equation. Normal forms near codimension two bifurcations, including Hopf?Cpitchfork and Hopf?CHopf bifurcation, are determined by the method of multiple scales. Through analyzing the associated amplitude equations, we obtain the detailed bifurcation sets and find some interesting phenomena such as quasi-periodic oscillations and strange attractor, which are confirmed by several numerical simulations.     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Hopf and Bogdanov�CTakens bifurcations in a coupled FitzHugh�CNagumo neural system with delay

In this paper, Hopf bifurcation and Bogdanov?CTakens bifurcation with codimension 2 in a coupled FitzHugh?CNagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov?CTakens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov?CTakens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of Bogdanov?CTakens bifurcation indicates that the neuron signals can transit between resting and spiking.     

Effect of amplitude and frequency of limit cycle oscillators on their coupled and forced dynamics

The occurrence of synchronization and amplitude death phenomena due to the coupled interaction of limit cycle oscillators (LCO) has received increased attention over the last few decades in various fields of science and engineering. Studies pertaining to these coupled oscillators are often performed by studying the effect of various coupling parameters on their mutual interaction. However, the effect of system parameters (i.e., the amplitude and frequency) on the coupled interaction of such LCO has not yet received much attention, despite their practical importance. In this paper, we investigate the dynamical behavior of time-delay coupled Stuart–Landau (SL) oscillators exhibiting subcritical Hopf bifurcation for the variation of amplitude and frequency of these oscillators in their uncoupled state. For identical SL oscillators, a gradual increase in the amplitude of LCO shrinks the amplitude death regions observed between the regions of in-phase and anti-phase synchronization leading to its eventual disappearance, resulting in the occurrence of phase-flip bifurcations at higher amplitudes of LCO. We also observe an alternate existence of in-phase and anti-phase synchronization regions for higher values of time delay, whose prevalence of occurrence increases with an increase in the frequency of the oscillator. With the introduction of frequency mismatch, the region of amplitude death. The forced response of SL oscillator shows an asymmetry in the Arnold tongue and the manifestation of asynchronous quenching of LCO. An increase in the amplitude of LCO narrows the Arnold tongue and reduces the region of asynchronous quenching observed in the system. Finally, we compare the coupled and forced response of SL oscillators with the corresponding experimental results obtained from laminar thermoacoustic oscillators and the numerical results from van der Pol (VDP) oscillators. We show that the SL model qualitatively displays many features observed experimentally in coupled and forced thermoacoustic oscillators. In contrast, the VDP model does not capture most of the experimental results due to the limitation in the independent variation of system parameters.

     

反馈时滞对van der Pol振子张弛振荡的影响

研究反馈控制环节时滞对van derPol振子张弛振荡的影响. 首先, 通过稳定性切换分析, 得到了系统的慢变流形的稳定性和分岔点分布图, 结果表明, 当时滞大于某临界值时, 系统慢变流形的结构发生本质的变化.其次, 基于几何奇异摄动理论, 分析了慢变流形附近解轨线的形状, 发现时滞反馈会引起张弛振荡中的慢速运动过程中存在微幅振荡, 其中微幅振荡来自于内部层引起的振荡和Hopf分岔产生的振荡两个方面; 同时, 时滞对张弛振荡的周期也具有显著的影响. 实例分析表明理论分析结果与数值结果相吻合.      

Multiple scales analysis for double Hopf bifurcation with?1:3?resonance

The dynamical behavior of a general n-dimensional delay differential equation (DDE) around a 1:3 resonant double Hopf bifurcation point is analyzed. The method of multiple scales is used to obtain complex bifurcation equations. By expressing complex amplitudes in a mixed polar-Cartesian representation, the complex bifurcation equations are again obtained in real form. As an illustration, a system of two coupled van der Pol oscillators is considered and a set of parameter values for which a 1:3 resonant double Hopf bifurcation occurs is established. The dynamical behavior around the resonant double Hopf bifurcation point is analyzed in terms of three control parameters. The validity of analytical results is shown by their consistency with numerical simulations.     

负反馈诱发神经电振荡的反常现象的复杂动力学

传统观念认为,负反馈容易使系统达到稳定平衡点而正反馈容易引起振荡.本研究基于神经元理论模型,提出了负反馈可以诱发稳定平衡点、也就是静息、变为振荡、也就是放电的新观点.在Hopf分岔点附近,作用在静息上的一次足够大的负向脉冲电流的抑制性刺激,能够引起一个动作电位及随后的衰减振荡的后电位;而能够在后电位上诱发出动作电位的负脉冲电流强度阈值也是衰减振荡的.在模型中,引入具有时滞($\tau$)的负反馈来模拟抑制性自突触,一个动作电位诱发的负反馈自突触电流会作用到比动作电位延迟$\tau$的后电位上.随时滞增加,能够诱发出放电的负反馈增益强度阈值呈现出具有衰减振荡特点的类似多重相干共振的特性,衰减振荡的周期与电流阈值曲线的周期以及分岔点附近的放电周期相关.另外,负反馈还能诱发出放电与静息共存的复杂行为.本研究的结果不仅揭示了负反馈的新的反常调控作用,还有助于理解在现实神经系统中存在的慢抑制性自突触的潜在功能.      

Control of cross-flow-induced vibrations of square cylinders using linear and nonlinear delayed feedbacks

We investigate the effectiveness of linear and nonlinear time-delay feedback controls to suppress high amplitude oscillations of an elastically mounted square cylinder undergoing galloping oscillations. A representative model that couples the transverse displacement and the aerodynamic force is used. The quasi-steady approximation is used to model the galloping force. A linear analysis is performed to investigate the effect of linear time-delay controls on the onset speed of galloping and natural frequencies. It is demonstrated that a linear time-delay control can be used to delay the onset speed of galloping. The normal form of the Hopf bifurcation is then derived to characterize the type of the instability (supercritical or subcritical) and to determine the effects of the linear and nonlinear time-delay parameters on their outputs near the bifurcation. The results show that the nonlinear time-delay control can be efficiently implemented to significantly reduce the galloping amplitude and suppress any dangerous behavior by converting any subcritical Hopf bifurcation into a supercritical one.     

The impact of delayed feedback on the pulsating oscillations of class-B lasers

Class-B laser systems can be described by a set of slow-fast differential equations with a small parameter, and they exhibit pulsating oscillations in common. In this paper, the impact of the delayed feedback on the pulsating solutions is investigated. At first, a careful analysis of the local stability and bifurcation shows the existence of a series of Hopf bifurcations and double Hopf bifurcations when the strength of the delayed feedback or the delay increases, by means of stability switches. Then, an application of the geometric singular perturbation theory reveals the evolutional mechanism of the pulsating solutions from transients to stationary states, and the effect of the delayed feedback upon the pulsating solutions is examined.     

Double Hopf bifurcation of time-delayed feedback control for maglev system

This paper undertakes an analysis of a double Hopf bifurcation of a maglev system with time-delayed feedback. At the intersection point of the Hopf bifurcation curves in velocity feedback control gain and time delay space, the maglev system has a codimension 2 double Hopf bifurcation. To gain insight into the periodic solution which arises from the double Hopf bifurcation and the unfolding, we calculate the normal form of double Hopf bifurcation using the method of multiple scales. Numerical simulations are carried out with two pairs of feedback control parameters, which show different unfoldings of the maglev system and we verify the theoretical analysis.     

Flow-structure interactions of two parallel inverted flags with small separation distances in a water tunnel

In this study, the dynamics and flow fields of two parallel inverted flags are investigated using particle image velocimetry technology. The separation distance between two flags is less than two times the length of the flag, and the length ratio of these two flags is considered in the investigation. The results show that for the dynamic behaviours of two identical flags with a larger separation distance, the anti-phase and in-phase modes occur successively in the periodic oscillation as the flow velocity increases. The anti-phase and in-phase oscillations occur according to the formation position of the low-pressure and recirculation areas at different flow velocities. Moreover, a novel coupled flapping mode is observed at smaller separation distances: the contact anti-phase flapping mode, in which one flag oscillates with a large symmetric amplitude, and the other flag oscillates with a single-side large amplitude. As the separation distance further decreases, the in-phase mode appears for a larger range of flow velocity values, to avoid contact for the largest possible amplitude oscillation. Finally, as the length ratio decreases to 0.75, the oscillation frequency of the shorter flag becomes twice that of the longer flag, causing the in-phase and anti-phase oscillations to occur simultaneously in one cycle (i.e., the multi-phase flapping state). Interestingly, the two flags oscillate out of phase in the flapping apart process to avoid contact at a higher flow velocity. In general, the lower amplitude of the longer flag and two contact flags relative to that of an isolated flag clearly indicates the importance of two equal-length and non-contact flags for energy harvesting.     

Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis

The standard approach to study symmetric Hopf bifurcation phenomenon is based on the usage of the equivariant singularity theory developed by M. Golubitsky et?al. In this paper, we present the equivariant degree theory based method which is complementary to the equivariant singularity approach. Our method allows systematic study of symmetric Hopf bifurcation problems in non-smooth/non-generic equivariant settings. The exposition is focused on a network of eight identical van der Pol oscillators with hysteresis memory, which are coupled in a cube-like configuration leading to S ₄-equivariance. The hysteresis memory is the source of non-smoothness and of the presence of an infinite dimensional phase space without local linear structure. Symmetric properties and multiplicity of bifurcating branches of periodic solutions are discussed in the context showing a direct link between the physical properties and the equivariant topology underlying this problem.     

Stochastic Hopf bifurcation of quasi-integrable Hamiltonian systems with multi-time-delayed feedback control and wide-band noise excitations

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with multi-time-delayed feedback control subject to wide-band noise excitations is studied. First, the time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay and the system is converted into an ordinary quasi-integrable Hamiltonian system. The averaged It? stochastic differential equations are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then the expression for average bifurcation parameter of the averaged system is obtained approximately and a criterion for determining the stochastic Hopf bifurcation induced by time-delayed feedback control forces in the original system using average bifurcation parameter is proposed. An example is worked out in detail to illustrate the criterion and its validity and to show the effect of time delay in feedback control on stochastic Hopf bifurcation of the system.     

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知，平面自治系统即使具有光滑非线性存在，系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时，时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈，得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理，研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔，产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子，这些共存的吸引子相碰是导致系统复杂的动力学行为，包括概周期和混沌运动。     

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

We investigate the resonant double Hopf bifurcation in a diffusive complex Ginzburg–Landau model with delayed feedback and phase shift. The conditions for the existence of resonant double Hopf bifurcation are obtained by analyzing the roots’ distribution of the characteristic equation, and a general formula to determine the bifurcation point is given. For the cases of 1:2 and 1:3 resonance, we choose time delay, feedback strength and phase shift as bifurcation parameters and derive the normal forms which are proved to be the same as those in non-resonant cases. The impact of cubic terms on the unfolding types is discussed after obtaining the normal form till 3rd order. By fixing phase shift, we find that varying time delay and feedback strength simultaneously can induce the coexistence of two different periodic solutions, the existence of quasi-periodic solutions and strange attractors. Also, the effects on the existence of transient quasi-periodic solution exerted by the phase shift are illustrated.