An experiment of dynamical behaviours in an erbium-doped fibre-ring laser with loss modulation

This paper reports a systematic experimental investigation on the dynamics in the low-frequency region in an erbium-doped fibre-ring laser with loss modulation. A rich variety of bifurcation is analyzed through the bifurcation diagram and structured with the concept of the winding numbers. The coexistence of multiple attractors and the crisis that appear in the saddle-node bifurcations, and an interesting structure of bifurcation which is similar to the bifurcations in high-frequency range, have been observed.     

Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator

This paper is concerned with numerical continuation and analytical investigations of sliding bifurcations in Filippov systems. In particular, a methodology developed for the continuation of grazing bifurcations in impacting systems is used to continue sliding bifurcations in Filippov systems. A dry-friction oscillator is investigated from a sliding bifurcations point of view and a complex two-parameter bifurcation diagram of sliding bifurcations is presented. A number of codimension-two sliding bifurcation points that act as organising centres for codimension-one sliding bifurcations are revealed. Two representative codimension-two points are analysed and unfolded, and the analysis is used to explain the dynamics of the dry-friction oscillator in the neighbourhood of these points.     

Bifurcations of mixed-mode oscillations in a stellate cell model

Experimental recordings of the membrane potential of stellate cells within the entorhinal cortex show a transition from subthreshold oscillations (STOs) via mixed-mode oscillations (MMOs) to relaxation oscillations under increased injection of depolarizing current. Acker et al. introduced a 7D conductance based model which reproduces many features of the oscillatory patterns observed in these experiments. For the first time, we present a comprehensive bifurcation analysis of this model by using the software package AUTO. In particular, we calculate the stable MMO branches within the bifurcation diagram of this model, as well as other MMO patterns which are unstable. We then use geometric singular perturbation theory to demonstrate how the bifurcations are governed by a 3D reduced model introduced by Rotstein et al. We extend their analysis to explain all observed MMO patterns within the bifurcation diagram. A key role in this bifurcation analysis is played by a novel homoclinic bifurcation structure connecting to a saddle equilibrium on the unstable branch of the corresponding critical manifold. This type of homoclinic connection is possible due to canards of folded node (folded saddle-node) type.     

Bifurcation diagram globally underpinning neuronal firing behaviors modified by SK conductance

Neurons in the brain utilize various firing trains to encode the input signals they have received.Firing behavior of one single neuron is thoroughly explained by using a bifurcation diagram from polarized resting to firing,and then to depolarized resting.This explanation provides an important theoretical principle for understanding neuronal biophysical behaviors.This paper reports the novel experimental and modeling results of the modification of such a bifurcation diagram by adjusting small conductance potassium(SK)channel.In experiments,changes in excitability and depolarization block in nucleus accumbens shell and medium-spiny projection neurons are explored by increasing the intensity of injected current and blocking the SK channels by apamin.A shift of bifurcation points is observed.Then,a Hodgkin–Huxley type model including the main electrophysiological processes of such neurons is developed to reproduce the experimental results.The reduction of SK channel conductance also shifts the bifurcations,which is in consistence with experiment.A global bifurcation paradigm of this shift is obtained by adjusting two parameters,intensity of injected current and SK channel conductance.This work reveals the dynamics underpinning modulation of neuronal firing behaviors by biologically important ionic conductance.The results indicate that small ionic conductance other than that responsible for spike generation can modify bifurcation points and shift the bifurcation diagram and,thus,change neuronal excitability and adaptation.     

An analytical study of a periodically driven laser with a saturable absorber

We consider the rate equations for a laser with an intracavity saturable absorber and subject to a periodically modulated pump. By deriving simplified equations for a map valid for strongly pulsating regimes, analytical conditions are determined that specify the properties of both frequency-locked and unlocked behaviors. As the strength of the modulation is increased, quasiperiodic and period-doubling bifurcations are predicted. However, only the transition from locking to non-locking through a quasiperiodic bifurcation is possible for realistic values of the parameters. Our results are consistent with previous numerical and experimental studies of modulated lasers with a saturable absorber. Received 10 March 2001     

Oscillatory nonequilibrium Nambu systems: the canonical-dissipative Yamaleev oscillator

We study the emergence of oscillatory self-sustained behavior in a nonequilibrium Nambu system that features an exchange between different kinetical and potential energy forms. To this end, we study the Yamaleev oscillator in a canonical-dissipative framework. The bifurcation diagram of the nonequilibrium Yamaleev oscillator is derived and different bifurcation routes that are leading to limit cycle dynamics and involve pitchfork and Hopf bifurcations are discussed. Finally, an analytical expression for the probability density of the stochastic nonequilibrium oscillator is derived and it is shown that the shape of the density function is consistent with the oscillator properties in the deterministic case.     

摩擦力对非弹性蹦球倍周期运动的影响

分析了摩擦力对竖直振动台面上完全非弹性蹦球动力学行为的影响.当控制参数Γ由1逐渐增大时,作用在蹦球上的恒定摩擦力不会改变倍周期分岔的序列,但会使倍周期分岔点的数值变大.与无摩擦力时的情况相比,在飞行时间的分岔图中也存在倍周期分岔密集区,只是被横向拉伸纵向压缩,且具有不同的分形特性.与受振颗粒体系中的倍周期分岔过程做了比较,发现当摩擦力取值为颗粒总重量的20%—30%时两者符合很好.     

Modelling of a Nd:YAG laser Q-switched by a scanning interferometric mirror

We have modelled a continuously pumped Nd:YAG actively Q-switched by a variable interferometric mirror made up of a scanning Michelson or Fabry-Pérot mirror. We have characterised the three-mirror laser dynamics by using a bifurcation diagram constructed from the plot of peak power-enhancement factor as a function of mirror speed. One observes different chaotic windows separated by period-doubling bifurcations, and stable periodic regime. It is demonstrated that the best performance of the Q-switched laser is obtained rather for low than for high mirror speed (pulse width of 20 ns, and high peak power up to 400 times greater than the continuous emission).     

Effects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of Non-degenerate Cascade Two-Photon Lasers

Based on the cascade two-photon laser dynamic equation derived with the technique of quantum Langevin operators with the considerations of coherently prepared three-level atoms and the classical field injected into the cavity, we numerically study the effects of atomic coherence and classical field on the chaotic dynamics of a two-photon laser. Lyapunov exponent and bifurcation diagram calculations show that the Lorenz chaos and hyperchaos can be induced or inhibited by the atomic coherence and the classical field via crisis or Hopf bifurcations.     

Effects of Atomic Coherence and Injected Classical Field on Chaotic Dynamics of Non-degenerate Cascade Two-Photon Lasers

Based on the cascade two-photon laser dynamic equation derived with the technique of quantum Langevin operators with the considerations of coherently prepared three-level atoms and the classical field injected into the cavity, we numerically study the effects of atomic coherence and classical field on the chaotic dynamics of a two-photon laser. Lyapunov exponent and bifurcation diagram calculations show that the Lorenz chaos and hyperchaos can be induced or inhibited by the atomic coherence and the classical field via crisis or Hopf bifurcations.     

一类可变禁区的不连续系统的加周期分岔

研究了一类可变禁区不连续系统的加周期分岔行为, 发现由可变禁区导致不同类型的加周期分岔. 研究表明, 系统的迭代轨道和禁区的上下两个边界均可发生边界碰撞, 从而产生加周期分岔. 基于边界碰撞分岔理论, 定义基本的迭代单元, 解析推导出了相应的分岔曲线, 在全参数空间中给出了不同加周期所出现的范围. 与数值模拟结果比较, 理论分析结果与数值结果高度一致.     

Slip effects on streamline topologies and their bifurcations for peristaltic flows of a viscous fluid

We discuss the effects of the surface slip on streamline patterns and their bifurcations for the peristaltic transport of a Newtonian fluid. The flow is in a two-dimensional symmetric channel or an axisymmetric tube. An exact expression for the stream function is obtained in the wave frame under the assumptions of long wavelength and low Reynolds number for both cases. For the discussion of the particle path in the wave frame, a system of nonlinear autonomous differential equations is established and the methods of dynamical systems are used to discuss the local bifurcations and their topological changes. Moreover, all types of bifurcations and their topological changes are discussed graphically. Finally, the global bifurcation diagram is used to summarize the bifurcations.     

Dynamics of homogeneous magnetizations in strong transverse driving fields

Spatially homogeneous solutions of the Landau-Lifshitz-Gilbert equation are analysed. The different cases of conservative as well as dissipative motion are considered separately. For the linearly polarized driven Hamiltonian system we apply a global perturbation theory to uncover the main resonances as well as the phase space structure. The case of circularly polarized driven dissipative motion is studied in detail. We present the complete bifurcation diagram including bifurcations up to codimension three.     

Dynamics of homogeneous magnetizations in strong transverse driving fields

Spatially homogeneous solutions of the Landau-Lifshitz-Gilbert equation are analysed. The different cases of conservative as well as dissipative motion are considered separately. For the linearly polarized driven Hamiltonian system we apply a global perturbation theory to uncover the main resonances as well as the phase space structure. The case of circularly polarized driven dissipative motion is studied in detail. We present the complete bifurcation diagram including bifurcations up to codimension three.     

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.     

On bifurcations in a chaotically stirred excitable medium

We study a one-dimensional filamental model of a chaotically stirred excitable medium. In a numerical simulation we systematically explore its rich bifurcation scenarios involving saddle-nodes, Hopf bifurcations and hysteresis loops. The bifurcations are described in terms of two parameters signifying the excitability of the reacting medium and the strength of the chaotic stirring, respectively. The solution behaviour, in particular at the bifurcation points, is analytically described by means of a nonperturbative variational method. Using this method we reduce the partial differential equations to either algebraic equations for stationary solutions and bifurcations, or to ordinary differential equations in the case of non-stationary solutions and bifurcations. We present numerical simulations corroborating our analytical results.     

Symmetry breaking via global bifurcations of modulated rotating waves in hydrodynamics

The combined experimental and numerical study finds a complex mechanism of Z(2) symmetry breaking involving global bifurcations for the first time in hydrodynamics. In addition to symmetry breaking via pitchfork bifurcation, the Z(2) symmetry of a rotating wave that occurs in Taylor-Couette flow is broken by a global saddle-node-infinite-period (SNIP) bifurcation after it has undergone a Neimark-Sacker bifurcation to a Z(2)-symmetric modulated rotating wave. Unexpected complexity in the bifurcation structure arises as the curves of cyclic pitchfork, Neimark-Sacker, and SNIP bifurcations are traced towards their apparent merging point. Instead of symmetry breaking due to a SNIP bifurcation, we find a more complex mechanism of Z(2) symmetry breaking involving nonsymmetric two-tori undergoing saddle-loop homoclinic bifurcations and complex dynamics in the vicinity of this global bifurcation.     

On the scenario of transition to the broadband oscillation regime in the prototype of a low-voltage vircator

This study is devoted to experimental investigation of the scenario for a transition from the narrow-band (single-frequency) oscillation regime to the broadband noise-like oscillation in a prototype of a low-voltage oscillator operating with a virtual cathode. The experimental setup and the control instrumentation are considered. Detailed experimental analysis of the dynamics of variation of temporal realizations of the signals generated by the laboratory prototype upon the variation of the chosen control parameters is performed. Analysis of the power spectra, temporal realizations of output signals, phase portraits, and the bifurcation diagram constructed upon a change of the chosen control parameters shows that a transition from the single-frequency oscillation regime to broadband oscillations occurs via a cascade of period-doubling bifurcations (classical Feugenbaum scenario is implemented).     

Bifurcations of a parametrically excited oscillator with strong nonlinearity

A parametrically excited oscillator with strong nonlinearity, including van der Pol and Duffing types, is studied for static bifurcations. The applicable range of the modified Lindstedt－Poincaré method is extended to 1/2 subharmonic resonance systems. The bifurcation equation of a strongly nonlinear oscillator, which is transformed into a small parameter system, is determined by the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analysed.     

Hyperchaos--chaos--Hyperchaos Transition in a Class of On--Off Intermittent Systems Driven by a Family of Generalized Lorenz Systems

Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems＇ dynamics. After the bifurcation the phenomenon of on-off intermittency appears. It is observed that the systems undergo a symmetric hyperchaos-chaos-hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.