Continuation of periodic solutions in the waveguide array mode-locked laser

We apply the adjoint continuation method to construct highly accurate, periodic solutions that are observed to play a critical role in the multi-pulsing transition of mode-locked laser cavities. The method allows for the construction of solution branches and the identification of their bifurcation structure. Supplementing the adjoint continuation method with a computation of the Floquet multipliers allows for explicit determination of the stability of each branch. This method reveals that, when gain is increased, the multi-pulsing transition starts with a Hopf bifurcation, followed by a period-doubling bifurcation, and a saddle-node bifurcation for limit cycles. Finally, the system exhibits chaotic dynamics and transitions to the double-pulse solutions. Although this method is applied specifically to the waveguide array mode-locking model, the multi-pulsing transition is conjectured to be ubiquitous and these results agree with experimental and computational results from other models.     

Frequency Switches at Transition Temperature in Voltage-Gated Ion Channel Dynamics of Neural Oscillators

Understanding of the mechanisms of neural phase transitions is crucial for clarifying cognitive processes in the brain. We investigate a neural oscillator that undergoes different bifurcation transitions from the big saddle homoclinic orbit type to the saddle node on an invariant circle type, and the saddle node on an invariant circle type to the small saddle homoclinic orbit type. The bifurcation transitions are accompanied by an increase in thermodynamic temperature that affects the voltage-gated ion channel in the neural oscillator. We show that nonlinear and thermodynamical mechanisms are responsible for different switches of the frequency in the neural oscillator. We report a dynamical role of the phase response curve in switches of the frequency, in terms of slopes of frequency-temperature curve at each bifurcation transition. Adopting the transition state theory of voltagegated ion channel dynamics, we confirm that switches of the frequency occur in the first-order phase transition temperature states and exhibit different features of their potential energy derivatives in the ion channel. Each bifurcation transition also creates a discontinuity in the Arrhenius plot used to compute the time constant of the ion channel.     

First-order Fréedericksz transition and front propagation in a liquid crystal light valve with feedback

Fréedericksz transition can become subcritical in the presence of a feedback mechanism that leads to the dependence of the local electric field onto the liquid crystal re-orientation angle. We have characterized experimentally the first-order Fréedericksz transition in a Liquid Crystal Light Valve with optical feedback. The bistability region is determined, together with the Fréedericksz transition point and the Maxwell point. We show the propagation of fronts connecting the different metastable states and we estimate the front velocity. Theoretically, we derive an amplitude equation, valid close to the Fréedericksz transition point, which accounts for the subcritical character of the bifurcation.Received: 21 October 2003, Published online: 6 January 2004PACS: 05.45.-a Nonlinear dynamics and nonlinear dynamical systems - 64.60.-i General studies of phase transitions     

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.     

Noise-induced precursors of tonic-to-bursting transitions in hypothalamic neurons and in a conductance-based model

The dynamics of neurons is characterized by a variety of different spiking patterns in response to external stimuli. One of the most important transitions in neuronal response patterns is the transition from tonic firing to burst discharges, i.e., when the neuronal activity changes from single spikes to the grouping of spikes. An increased number of interspike-interval sequences of specific temporal correlations was detected in anticipation of temperature induced tonic-to-bursting transitions in both, experimental impulse recordings from hypothalamic brain slices and numerical simulations of a stochastic model. Analysis of the modelling data elucidates that the appearance of such patterns can be related to particular system dynamics in the vicinity of the period-doubling bifurcation. It leads to a nonlinear response on de- and hyperpolarizing perturbations introduced by noise. This explains why such particular patterns can be found as reliable precursors of the neurons' transition to burst discharges.     

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc-dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation.     

Multistability and memory effect in a highly turbulent flow: experimental evidence for a global bifurcation

We report experimental evidence of a global bifurcation on a highly turbulent von Kármán flow. The mean flow presents multiple solutions: the canonical symmetric solution becomes marginally unstable towards a flow which breaks the basic symmetry of the driving apparatus even at very large Reynolds numbers. The global bifurcation between these states is highly subcritical and the system thus keeps a memory of its history. The transition recalls low-dimension dynamical system transitions and exhibits very peculiar statistics. We discuss the role of turbulence in two ways: the multiplicity of hydrodynamical solutions and the effect of fluctuations on the nature of transitions.     

Controlled Hopf bifurcation of a storage-ring free-electron laser

Local bifurcation control is a topic of fundamental importance in the field of nonlinear dynamical systems. We discuss an original example within the context of storage-ring free-electron laser physics by presenting a new model that enables analytical insight into the system dynamics. The transition between the stable and the unstable regimes, depending on the temporal overlapping between the light stored in the optical cavity and the electrons circulating into the ring, is found to be a Hopf bifurcation. A feedback procedure is implemented and shown to provide an effective stabilization of the unstable steady state.     

Nonlinear dynamics in wurtzite InN diodes under terahertz radiation

We carry out a theoretical study of nonlinear dynamics in terahertz-driven n+ nn+ wurtzite InN diodes by using time-dependent drift diffusion equations.A cooperative nonlinear oscillatory mode appears due to the negative differential mobility effect,which is the unique feature of wurtzite InN aroused by its strong nonparabolicity of the Γ 1 valley.The appearance of different nonlinear oscillatory modes,including periodic and chaotic states,is attributed to the competition between the self-sustained oscillation and the external driving oscillation.The transitions between the periodic and chaotic states are carefully investigated using chaos-detecting methods,such as the bifurcation diagram,the Fourier spectrum and the first return map.The resulting bifurcation diagram displays an interesting and complex transition picture with the driving amplitude as the control parameter.     

Theory of direct optical transitions in an optical indirect semiconductor with a superlattice structure

Starting from a model of an indirect optical semiconductor with two bands, the electron states are calculated in the presence of an additional periodic one-dimensional potential (superlattice) in the semiconductor material. These states are used to determine the transition probability connected with the absorption of a photon. This transition corresponds to an optical direct transition — no phonon takes part in this process. The optical direct and optical indirect transitions are compared. For optical frequencies near the band gap one expects only direct transitions, whereby the optical indirect transitions may be neglected.     

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.     

Propagation of vector solitons in a quasi-resonant medium with stark deformation of quantum states

The nonlinear dynamics of a vector two-component optical pulse propagating in quasi-resonance conditions in a medium of nonsymmetric quantum objects is investigated for Stark splitting of quantum energy levels by an external electric field. We consider the case when the ordinary component of the optical pulse induces ?? transitions, while the extraordinary component induces the ?? transition and shifts the frequencies of the allowed transitions due to the dynamic Stark effect. It is found that under Zakharov-Benney resonance conditions, the propagation of the optical pulse is accompanied by generation of an electromagnetic pulse in the terahertz band and is described by the vector generalization of the nonlinear Yajima-Oikawa system. It is shown that this system (as well as its formal generalization with an arbitrary number of optical components) is integrable by the inverse scattering transformation method. The corresponding Darboux transformations are found for obtaining multisoliton solutions. The influence of transverse effects on the propagation of vector solitons is investigated. The conditions under which transverse dynamics leads to self-focusing (defocusing) of solitons are determined.     

Photon echo under superradiance

The dynamics of optical coherence and phase memory in three-level medium in conditions of superradiance pulse emission is considered. In the case of coherent excitation of the upper level in three-level system of lambda configuration it is shown that the superradiance eliminates optical coherence on the adjacent transitions of excitation and superradiance emission and induces optical coherence on the remain transition. This, in turn, makes it possible to observe new effects of photon-echo, simple example of which is described in the present Letter.     

Low frequency fluctuations in a multimode semiconductor laser with optical feedback

We study a multimode semiconductor laser subject to a moderate optical feedback. The steady state is destabilized by either a simple Hopf bifurcation leading to in phase dynamics or by a degenerate Hopf bifurcation leading to antiphase dynamics. The degenerate bifurcation is also a source of multiple coexisting attractors. We show that a simple interpretation of the low frequency fluctuations in the multimode regime is provided by a chaotic itinerancy among the many coexisting unstable attractors produced by the degenerate Hopf bifurcation.     

抽运参数随时间慢变所诱发Laser-Lorenz方程的分岔与光学双稳态

提出了新的简便的方法研究具有入射信号的半经典Laser-Lorenz方程-由于抽运参数随时间慢变,导致分岔滞后,并诱发暂态的光学双稳态-通过量级平衡和用线性化系统的慢变解作近似,给出了分岔转迁的量级关系和暂态光学双稳态发生的条件,并通过数值计算给出分岔转迁区间和暂态光学双稳态的滞后环- 关键词：     

非线性动力系统分岔点邻域内随机共振的特性

研究了叉形分岔系统和FitzHugh Nagumo(FHN)细胞模型两种非线性动力系统分岔点邻域内 随机共振的特性.研究结果表明：这两种系统在分岔发生时具有由一个吸引子变为两个吸引 子或者由两个吸引子变为一个吸引子共同的分岔特性，即在分岔点的邻域内, 系统在分岔点 的两侧有分岔前吸引子和分岔后吸引子存在，在噪声的作用下，系统的运动除了像传统随机 共振的机理那样在分岔点一侧共存的吸引子之间跃迁，还在分岔点两侧三个吸引子（分岔前 一个吸引子和分岔后两个吸引子）之间跃迁，并且这种跃迁单独诱发了随机共振 ；在两种 跃迁都发生的情况下, 在其分岔点的邻域内，由第二种跃迁诱发的随机共振在引起第一种跃 迁噪声的强度很大的范围内变化仍可维持, 而第一种跃迁诱发的随机共振在引起第二种跃迁 噪声的强度很小的范围内变化即迅速消失. 关键词： 随机共振 吸引子 分岔点 跃迁     

Universal occurrence of the phase-flip bifurcation in time-delay coupled systems

Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of pi as a function of the coupling strength or the time delay. This phase-flip is accompanied by discontinuous changes in the frequency of the synchronized oscillators, and in the largest negative Lyapunov exponent or its derivative. Here we illustrate the phenomenology of the bifurcation for several classes of nonlinear oscillators, in the regimes of both periodic and chaotic dynamics. We present extensive numerical simulations and compute the oscillation frequencies and the Lyapunov spectra as a function of the coupling strength. In particular, our simulations provide clear evidence of the phase-flip bifurcation in excitable laser and Fitzhugh-Nagumo neuronal models, and in diffusively coupled predator-prey models with either limit cycle or chaotic dynamics. Our analysis demonstrates marked jumps of the time-delayed and instantaneous fluxes between the two interacting oscillators across the bifurcation; this has strong implications for the performance of the system as well as for practical applications. We further construct an electronic circuit consisting of two coupled Chua oscillators and provide the first formal experimental demonstration of the bifurcation. In totality, our study demonstrates that the phase-flip phenomenon is of broad relevance and importance for a wide range of physical and natural systems.