Sequential activity and multistability in an ensemble of coupled Van der Pol oscillators

In this paper collective dynamics of an ensemble of inhibitory coupled Van der Pol oscillators are studied. It was found that a stable heteroclinic contour and a stable heteroclinic channel between saddle cycles exist. These heteroclinic structures are responsible for the sequential activity of different oscillations. The corresponding bifurcations leading to the appearance of heteroclinic trajectories are analyzed.     

An elementary model of torus canards

We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend long times near a repelling branch of slowly varying limit cycles. In this article, we carry out a study of torus canards in an elementary third-order system that consists of a rotated planar system of van der Pol type in which the rotational symmetry is broken by including a phase-dependent term in the slow component of the vector field. In the regime of fast rotation, the torus canards behave much like their planar counterparts. In the regime of slow rotation, the phase dependence creates rich torus canard dynamics and dynamics of mixed mode type. The results of this elementary model provide insight into the torus canards observed in a higher-dimensional neuroscience model.     

具有时滞状态反馈的随机Van der Pol系统的动力学研究

研究了时滞及时滞反馈控制参数对Van der Pol系统极限环幅值的影响. 运用自适应的平均场近似方法给出了系统的线性化近似及系统参数Lyapunov稳定性的边界条件, 同时给出了Van der Pol系统的关联时间和功率谱密度的数值模拟结果. 通过与平均场近似下的解析结果比较后发现, 数值模拟结果和理论结果符合.进一步讨论了时滞反馈控制参数、噪声强度以及时滞对关联时间和功率谱密度的影响. 关键词： 平均场近似 关联时间 Lyapunov稳定性     

Energy-based analysis of frequency entrainment described by van der Pol and phase-locked loop equations

This paper analyzes frequency entrainment described by van der Pol and phase-locked loop (PLL) equations. The PLL equation represents the dynamics of a PLL circuit that appear in typical phase-locking phenomena. These two equations describe frequency entrainment by a periodic force. The entrainment originates from two different types of limit cycles: libration for the van der Pol equation and rotation for the PLL one. To explore the relationship between the geometry of limit cycles and the mechanism of entrainment, we investigate the entrainment using an energy balance relation. This relation is equivalent to the energy conservation law of dynamical systems with dissipation and input terms. We show response curves for the dc component, harmonic amplitude, phase difference, and energy supplied by a periodic force. The obtained curves indicate that the entrainments for the two equations have different features of supplied energy, and that the entrainment for the PLL equation possibly has the same mechanism as does the regulation of the phase difference for the van der Pol equation.     

Deformation of limit cycle under perturbations

The robustness of limit cycles of nonlinear dynamical systems is investigated by adding a small random velocity field to the famous van der Pol (VDP) equation in its two-dimensional phase plane. Our numerical calculations show that a limit cycle does not change much under a weak random perturbation. Thus it confirms the conjecture that a limit cycle will make only a small deformation under an external perturbation. The idea can be used to understand the ac response of self-sustained oscillations in nonlinear dynamical systems.     

Regular and random dynamics of a Van der Pol oscillator with retarding feedback

A comparative analysis is made of the properties of certain mathematical models that have been proposed to describe the nonlinear dynamics of narrow-band oscillating systems with a lag. A modified model of a Van der Pol oscillator with a retarding feedback loop is studied numerically, and an analysis is made of the ways in which periodic motion in such systems becomes unstable and random oscillations are generated. A physical interpretation is given of the numerical results. Tomsk University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 104–113, February, 1998.     

Linearizability conditions for the Rayleigh-like oscillators

In this work we consider a family of nonlinear oscillators that is cubic with respect to the first derivative. Particular members of this family of equations often appear in numerous applications. We solve the linearization problem for this family of equations, where as equivalence transformations we use generalized nonlocal transformations. We explicitly find correlations on the coefficients of the considered family of equations that give the necessary and sufficient conditions for linearizability. We also demonstrate that each linearizable equation from the considered family admits an autonomous Liouvillian first integral, that is Liouvillian integrable. Furthermore, we demonstrate that linearizable equations from the considered family does not possess limit cycles. Finally, we illustrate our results by two new examples of the Liouvillian integrable nonlinear oscillators, namely by the Rayleigh–Duffing oscillator and the generalized Duffing–Van der Pol oscillator.     

Van der Pol and the history of relaxation oscillations: toward the emergence of a concept

Relaxation oscillations are commonly associated with the name of Balthazar van der Pol via his paper (Philosophical Magazine, 1926) in which he apparently introduced this terminology to describe the nonlinear oscillations produced by self-sustained oscillating systems such as a triode circuit. Our aim is to investigate how relaxation oscillations were actually discovered. Browsing the literature from the late 19th century, we identified four self-oscillating systems in which relaxation oscillations have been observed: (i) the series dynamo machine conducted by Ge?rard-Lescuyer (1880), (ii) the musical arc discovered by Duddell (1901) and investigated by Blondel (1905), (iii) the triode invented by de Forest (1907), and (iv) the multivibrator elaborated by Abraham and Bloch (1917). The differential equation describing such a self-oscillating system was proposed by Poincare? for the musical arc (1908), by Janet for the series dynamo machine (1919), and by Blondel for the triode (1919). Once Janet (1919) established that these three self-oscillating systems can be described by the same equation, van der Pol proposed (1926) a generic dimensionless equation which captures the relevant dynamical properties shared by these systems. Van der Pol's contributions during the period of 1926-1930 were investigated to show how, with Le Corbeiller's help, he popularized the "relaxation oscillations" using the previous experiments as examples and, turned them into a concept.     

The dynamics of two coupled Van der Pol oscillators with attractive and repulsive coupling

We consider a variant of two coupled Van der Pol oscillators with both attractive and repulsive mean-field interactions. In the presence of attractive coupling, the system is in the complete synchrony, while repulsive coupling shows anti-synchronization state leading to suppression of oscillations with increasing interaction strength. The coupled system with both attractive and repulsive interactions shows competitive tendencies of being complete synchronization and anti-synchronization resulting in the stabilization of the fixed point. We have also studied the effect of the damping coefficient of the VdP oscillator on the nature of the transition from oscillatory to a steady-state. These oscillators stabilize to unstable equilibrium point or coupling dependent inhomogeneous steady state via second or first-order transitions respectively depending upon the damping coefficient and coupling strength. These transitions are analyzed in the parameter plane by analytical and numerical studies of the two coupled Van der Pol oscillators.     

Lock-in and quasiperiodicity in hydrodynamically self-excited flames: Experiments and modelling

Hydrodynamically self-excited flames are often assumed to be insensitive to low-amplitude external forcing. To test this assumption, we apply acoustic forcing to a range of jet diffusion flames. These flames have regions of absolute instability at their base and this causes them to oscillate at discrete natural frequencies. We apply the forcing around these frequencies, at varying amplitudes, and measure the response leading up to lock-in. We then model the system as a forced van der Pol oscillator.Our results show that, contrary to some expectations, a hydrodynamically self-excited flame oscillating at one frequency is sensitive to forcing at other frequencies. When forced at low amplitudes, it responds at both frequencies as well as at several nearby frequencies, indicating quasiperiodicity. When forced at high amplitudes, it locks into the forcing. The critical forcing amplitude for lock-in increases both with the strength of the self-excited instability and with the deviation of the forcing frequency from the natural frequency. Qualitatively, these features are accurately predicted by the forced van der Pol oscillator. There are, nevertheless, two features that are not predicted, both concerning the asymmetries of lock-in. When forced below its natural frequency, the flame is more resistant to lock-in, and its oscillations at lock-in are stronger than those of the unforced flame. When forced above its natural frequency, the flame is less resistant to lock-in, and its oscillations at lock-in are weaker than those of the unforced flame. This last finding suggests that, for thermoacoustic systems, lock-in may not be as detrimental as it is thought to be.     

Local Dynamics of a Chain of Coupled Van der Pol Equations

Radiophysics and Quantum Electronics - We consider the problem of local dynamics of a system of a diffusive-coupled chain of the Van der Pol equations. A transition to the spatially distributed...     

Fronts between rhythms: spatiotemporal dynamics of extended polyrhythmic media

We study the propagation of fronts in extended oscillatory reaction-diffusion systems that contain several coexisting limit cycles. In contrast with the variational behavior, fronts between regions oscillating in two different limit cycles are found to propagate not necessarily towards the region of the less stable limit cycle, but towards the regions of the largest amplitudes, provided that the frequency mismatch between the cycles is sufficiently large. In other words, the smaller oscillations can always be made to control the whole system.     

经由脉冲式爆炸连接的复合式张弛振荡

张弛振荡现象普遍存在于自然科学以及工程技术的各个领域,探索张弛振荡的可能路径是张弛振荡研究的重要问题之一.最近,一种名为"脉冲式爆炸"(pulse-shaped explosion,PSE)的可以诱发张弛振荡的新机制被相继报道.PSE意味着平衡点和极限环表现出了与参数变化相关的脉冲式急剧量变,这导致系统出现急剧转迁现象,进而诱发张弛振荡.本文以多频激励Mathieu-van der Pol-Duffing系统为例,探讨了复合式的张弛振荡现象.当参数激励和外部激励存在相位差时,快子系统包含了两个不同的向量场部分,由此得到了系统的双稳定特性.特别地,在狭小的参数范围内,分岔会随着PSE的产生而产生,这使得PSE更具复杂性.基于此,揭示了两种复合式的张弛振荡,其特征是每一周期的演化过程包含了由PSE连接的两个张弛振荡簇.我们的研究深化了对PSE及张弛振荡复杂动力学行为的理解.     

Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems

We report a systematic two-parameter study of the organization of mixed-mode oscillations and period-adding sequences observed in an extended Bonhoeffer-van der Pol and in a FitzHugh-Nagumo oscillator. For both systems, we construct isospike diagrams and show that the number of spikes of their periodic oscillations are organized in a remarkable hierarchical way, forming a Stern-Brocot tree. The Stern-Brocot tree is more general than the Farey tree. We conjecture the Stern-Brocot tree to also underlie the hierarchical structure of periodic oscillations of other systems supporting mixed-mode oscillations.     

Zeno dynamics for open quantum systems

In this paper, we formulate limit Zeno dynamics of general open systems as the adiabatic elimination of fast components. We are able to exploit previous work on adiabatic elimination of quantum stochastic models to give explicitly the conditions under which open Zeno dynamics will exist. The open systems formulation is further developed as a framework for Zeno master equations, and Zeno filtering (that is, quantum trajectories based on a limit Zeno dynamical model). We discuss several models from the point of view of quantum control. For the case of linear quantum stochastic systems, we present a condition for stability of the asymptotic Zeno dynamics.     

Type-I intermittency with noise versus eyelet intermittency

In this Letter we compare the characteristics of two types of the intermittent behavior (type-I intermittency in the presence of noise and eyelet intermittency taking place in the vicinity of the chaotic phase synchronization boundary) supposed hitherto to be different phenomena. We show that these effects are the same type of dynamics observed under different conditions. The correctness of our conclusion is confirmed by the consideration of different sample systems, such as quadratic map, Van der Pol oscillator and Rössler system. Consideration of the problem concerning the upper boundary of the intermittent behavior also confirms the validity of the statement on the equivalence of type-I intermittency in the presence of noise and eyelet intermittency observed in the onset of phase synchronization.     

强Van der Pol振子的谐波与修正算法

引入谐波平衡近似解的符号运算与同伦延拓法,获得了强Van der Pol振子的解析近似极限环与稳定响应.为提高其近似精度,给出了谐波解的修正算法,并与数值模拟比较.结果表明,该算法非常精确可靠,是研究强非线性动力学系统的有效分析方法.     

Phase compactons in chains of dispersively coupled oscillators

We study the phase dynamics of a chain of autonomous oscillators with a dispersive coupling. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons: solitary waves with a compact support and kovatons which are compact formations of glued together kink-antikink pairs that may assume an arbitrary width. These robust objects seem to collide elastically and, together with wave trains, are the building blocks of the dynamics for typical initial conditions. Numerical studies of the complex Ginzburg-Landau and Van der Pol lattices show that the presence of a nondispersive coupling does not affect kovatons, but causes a damping and deceleration or growth and acceleration of compactons.     

Exciton-plasmon quantum metastates: self-induced oscillations of plasmon fields in the absence of decoherence in nanoparticle molecules

We investigate formation of unique quantum states (metastates) in quantum dot-metallic nanoparticle systems via self-induced coherent dynamics generated by interaction of these systems with a visible and an infrared laser fields. In such metastates, the quantum decoherence rates of the quantum dots can become zero and even negative while they start to rapidly change with time. Under these conditions, the energy dissipation rates and plasmon fields of the nanoparticle systems undergo undamped oscillations with gigahertz frequency, while the amplitudes of both visible and the infrared laser fields are considered to be time-independent. These dynamics also lead to variation of the plasmon absorption of the metallic nanoparticles between high and nearly zero values, forming electromagnetically induced transparency oscillations. We show that under these conditions, the effective transition energies and broadening of the quantum dots undergo oscillatory dynamics, highlighting the unique aspects of the metastates. These results extend the horizon for investigation of light-matter interaction in the presence of zero or negative polarization dephasing rates with strong time dependency.     

Synchronization of diffusively coupled oscillators near the homoclinic bifurcation

It has been known that a diffusive coupling between two limit cycle oscillations typically leads to the in-phase synchronization and also that it is the only stable state in the weak-coupling limit. Recently, however, it has been shown that the coupling of the same nature can result in the distinctive dephased synchronization when the limit cycles are close to the homoclinic bifurcation, which often occurs especially for the neuronal oscillators. In this paper we propose a simple physical model using the modified van der Pol equation, which unfolds the generic synchronization behaviors of the latter kind and in which one may readily observe changes in the sychronization behaviors between the distinctive regimes as well. The dephasing mechanism is analyzed both qualitatively and quantitatively in the weak-coupling limit. A general form of coupling is introduced and the synchronization behaviors over a wide range of the coupling parameters are explored to construct the phase diagram using the bifurcation analysis.