Homoclinic bifurcations in radiating diffusion flames

We analyse the dynamics of a model describing a planar diffusion flame with radiative heat losses incorporating a single step kinetic using timestepping techniques for Lewis number equal to one. We construct the full bifurcation diagram with respect to the Damköhler number including the branches of oscillating solutions. Based on this analysis we found, for the first time, homoclinic bifurcations that mark the abrupt disappearance of the nonlinear oscillations near extinction as reported in experiments.     

Successive bifurcations leading to stochastic behavior

A model of interacting normal modes in a nonlinear, dissipative system is constructed in order to analyze speculations by Ruelle and Takens. The first bifurcation leads to a periodic state. The second bifurcation leads to phaselocking, if the first mode is sufficiently energetic. A third bifurcation leads to stochastic behavior. Possible relevance of these phenomena for physical systems is discussed.     

Trapped-particle instability leading to bursting in stimulated Raman scattering simulations

Nonlinear, kinetic simulations of stimulated Raman scattering (SRS) under laser-fusion conditions present a bursting behavior. Different explanations for this regime have been given in previous studies: saturation of SRS by increased nonlinear Landau damping [K. Estabrook et al., Phys. Fluids B 1, 1282 (1989)]], and detuning due to the nonlinear frequency shift of the plasma wave [H. X. Vu et al., Phys. Rev. Lett. 86, 4306 (2001)]]. Another mechanism, also assigning a key role to the trapped electrons is proposed here: the breakup of the plasma wave through the trapped-particle instability.     

Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems

Nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated. We study the bifurcation sequences of the mixed-mode oscillations generated by the homoclinicity, which are shown to belong to two different types, depending on the nature of the Liapunov numbers of the basic periodic orbit. A detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation. Chaotic cool flame oscillations are also observed in the same model. When the control parameter crosses a critical value, this chaotic set of trajectories becomes globally unstable and forms a Cantor-like hyperbolic repellor, and the ignition mechanism generates ahomoclinic tangency to the Cantor set of trajectories. The complex bifurcation diagram may be globally reconstructed from a one-dimensional dynamical system, thanks to the strong contractivity of thermokinetics. It is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system.     

A discrete model exhibiting successive bifurcations leading to the onset of turbulence

A simple discrete model which consists ofN limit-cycle oscillators interacting with a linear coupling is numerically investigated in order to study the sequence of oscillatory states leading to the onset of turbulence. The systems withN=2 and 3 are studied. The system ofN=2 does not exhibit a nonperiodic motion, whereas the system ofN=3 does exhibit a nonperiodic motion. It is shown that, as an external parameter changes, the system ofN=3 undergoes a sequence of bifurcations, exhibiting the singly periodic, doubly periodic and nonperiodic motions, successively. This is similar to the bifurcation scheme for the onset of turbulence proposed by Ruelle and Takens and experimentally shown by Gollub and Swinny in a rotating Couette flow. The successive bifurcations are investigated in details and new features are reported.     

Coexisting attractors and bursting oscillations in IFOC of 3-phase induction motor

The European Physical Journal Special Topics - The dynamics of indirect field oriented control (IFOC) of 3-phase induction motor is studied in this paper. The dynamical behaviors of the studied...     

切换电路系统的振荡行为及其非光滑分岔机理

探讨了周期时间开关及控制阈值下在两个Rayleigh型子系统之间切换的电路系统随参数变化的复杂动力学演化过程, 通过对子系统平衡点的分析, 给出了参数空间中Fold分岔和Hopf分岔的条件, 考察了切换面处广义Jacobian矩阵特征值随辅助参数变化的分布情况, 得到了切换面处系统可能存在的各种分岔行为, 进而讨论了系统不同行为的产生机理, 指出系统的相轨迹存在分别由周期开关和控制阈值决定的两类不同的分界点, 而系统轨迹与非光滑分界面的多次碰撞将导致系统由周期倍化分岔导致混沌振荡.     

Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network

Recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators [R. Curtu, A. Shpiro, N. Rubin, J. Rinzel, Mechanisms for frequency control in neuronal competition models, SIAM J. Appl. Dyn. Syst. 7 (2) (2008) 609-649] showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics. Several types of behavior were identified such as: fusion, equivalent to a steady state of identical activity levels for both neural units; oscillations due to either an escape or a release mechanism; and a winner-take-all state of bistability. The model consists of two neural populations interacting through reciprocal inhibition, each endowed with a slow negative-feedback process in the form of spike frequency adaptation. In this paper we report the occurrence of another complex oscillatory pattern, the mixed-mode oscillations (MMOs). They exist in the model at the transition between the relaxation oscillator dynamical regime and the winner-take-all regime. The system distinguishes itself from other neuronal models where MMOs were found by the following interesting feature: there is no autocatalysis involved (as in the examples of voltage-gated persistent inward currents and/or intrapopulation recurrent excitation) and therefore the two cells in the network are not intrinsic oscillators; the oscillations are instead a combined result of the mutual inhibition and the adaptation. We prove that the MMOs are due to a singular Hopf bifurcation point situated in close distance to the transition point to the winner-take-all case. We also show that in the vicinity of the singular Hopf other types of bifurcations exist and we construct numerically the corresponding diagrams.     

Stability, bifurcations, and dynamics of global variables of a system of bursting neurons

An approximate mean field model of an ensemble of delayed coupled stochastic Hindmarsh-Rose bursting neurons is constructed and analyzed. Bifurcation analysis of the approximate system is performed using numerical continuation. It is demonstrated that the stability domains in the parameter space of the large exact systems are correctly estimated using the much simpler approximate model.     

A chemometric method to identify enzymatic reactions leading to the transition from glycolytic oscillations to waves

In the present work we demonstrate that FTIR-spectroscopy is a powerful tool for the time resolved and noninvasive measurement of multi-substrate/product interactions in complex metabolic networks as exemplified by the oscillating glycolysis in a yeast extract. Based on a spectral library constructed from the pure glycolytic intermediates, chemometric analysis of the complex spectra allowed us the identification of many of these intermediates. Singular value decomposition and multiple level wavelet decomposition were used to separate drifting substances from oscillating ones. This enabled us to identify slow and fast variables of glycolytic oscillations. Most importantly, we can attribute a qualitative change in the positive feedback regulation of the autocatalytic reaction to the transition from homogeneous oscillations to travelling waves. During the oscillatory phase the enzyme phosphofructokinase is mainly activated by its own product ADP, whereas the transition to waves is accompanied with a shift of the positive feedback from ADP to AMP. This indicates that the overall energetic state of the yeast extract determines the transition between spatially homogeneous oscillations and travelling waves.     

Simple models of firms emergence

We present a dynamical model of the emergence of firms as opposed to a flat labour market where entrepreneurs would recruit workers for each business opportunity. The model uses a preferential choice of partners based on previous collaborations experience. A sharp transition in the parameter space separates an ordered regime, where preferential links establish, from a disordered regime corresponding to a fast turnover of employees.     

Order parameter oscillations in a bounded solid solution and their bifurcations upon cooling

A nonlinear boundary-value problem for the Landau-Khalatnikov equation, which models the time evolution of the order parameter in a binary solution, is considered. A parameter characterizing the closeness of the alloy temperature to the critical temperature is introduced. This parameter depends on the dimensionless diffusivity. A change in this parameter results in successive bifurcations of solutions; so the order parameter corresponds to an oscillating steady-state structure to which almost all nonsteady-state solutions of the boundary-value problem are drawn after a long period of time.     

Spurious inertial oscillations in shallow-water models

For most of the discretization schemes, the numerical approximation of shallow-water models is a delicate problem. Indeed, the coupling between the momentum and the continuity equations usually leads to the appearance of spurious solutions and to anomalous dissipation/dispersion in the representation of the fast (Poincaré) and slow (Rossby) waves. In order to understand these difficulties and to select appropriate spatial discretization schemes, Fourier/dispersion analyses and the study of the null space of the associated discretized problems have proven beneficial. However, the cause of spurious oscillations and reduced convergence rates, that have been detected for most of mixed-order finite element shallow-water formulations, in simulating classical problems of geophysical fluid dynamics, is still an open question. The aim of the present study is to show that when spurious inertial solutions are present, they are mainly responsible for the aforementioned problems. Further, a criterion is found which determines the existence and the number of spurious inertial solutions. As it is delicate to cure spurious inertial modes, a class of possible discretization schemes is proposed, that is not affected by such spurious solutions.     

Bounding the quality of stochastic oscillations in population models

We analyze general two-species stochastic models, of the kind generally used for the study of population dynamics. Although usually defined a priori, the deterministic version of these models can be obtained as the infinite volume limit of many stochastic models (which are necessarily defined by more parameters than the deterministic one). It is known that damped oscillations in a deterministic model usually correspond to oscillatory-like fluctuations in their deterministic counterparts. The quality of these “oscillations" depends on details of each stochastic model. We show, however, that the parameters of the deterministic system are generally enough to obtain very good bounds for the quality of “oscillations" in any of its stochastic counterparts. These bounds are shown to depend on only one dimensionless parameter.     

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.     

Spontaneous emergence of angular momentum josephson oscillations in coupled annular bose-einstein condensates

We investigate the nonlinear dynamics of two coupled annular Bose-Einstein condensates (BECs). For certain values of the coupling strength the nonrotating state with uniform density is unstable with respect to fluctuations in the higher angular momentum modes. The Bogoliubov spectrum possesses two branches, one of which exhibits distinct regions of instability enabling one to selectively occupy certain angular momentum modes. For sufficiently long evolution times, angular momentum Josephson oscillations spontaneously appear, breaking the initial chiral symmetry of the BECs.     

Zip and velcro bifurcations in competition models in ecology and economics

During the last six years or so, a number of interesting papers discussed systems with line segments of equilibria, planes of equilibria, and with more general equilibrium configurations. This note draws attention to the fact that such equilibria were considered previously by Miklós Farkas (1932–2007), in papers published in 1984–2005. He called zip bifurcations those involving line segments of equilibria, and velcro bifurcations those involving planes of equilibria. We briefly describe prototypical situations involving zip and velcro bifurcations.

     

Odd periodic window and bifurcations on 2D parameter space of low frequency oscillations

Low frequency current oscillations have been usually investigated under the influence of the external applied voltage, temperature and illumination. A parallel applied magnetic field has been used in the present work in order to obtain odd periodic windows and bifurcations inside them in a two dimension parameter space for a semi-insulating GaAs sample grown by molecular beam epitaxy. Two kinds of parameter spaces have been used in order to stabilize the odd periodic windows and the bifurcations, namely, the external applied voltage versus the parallel magnetic field and the illumination versus the parallel magnetic field. We report on a successful observation of stable cycles of periodicity 3, 4, 5, 6, 7 and 8 inside a period-3 window. An example of bifurcation route is presented following the sequence from chaos to 4, 3, 6, 3, 5, 7, 5, and back to chaos in the parameter space of voltage versus magnetic field. For this bifurcation route we will show which branch of the cycle is bifurcating and which are coalescing with the control parameters.     

Book review: Nonlinear oscillations,dynamical systems,and bifurcations of vector fields

Journal of Statistical Physics -