Bifurcation Transition Research Articles

Regularities of lamellae ordering in the formation of polypropylene membrane porous structure

The lamellar structure of porous polypropylene membranes prepared by the extrusion of the polymer melt, annealing of extruded films, and their uniaxial extension has been investigated using data of computer analysis of scanning electron microscopy images. It has been shown that the formation of pores at the stage of uniaxial extension is accompanied by an ordering of lamellae or their self-organization controlled by the annealing temperature Tann. It has been found that the self-organization of lamellae can occur through two mechanisms depending on the spin draw ratio at the stage of extrusion of the polymer melt: in the first case, an increase in the temperature Tann leads to a gradual transition of the disorder-order type; in the second case, a bifurcation transition occurs with a nonmonotonic change in the order parameter. The conditions of the preparation of regular spatial lattices of lamellae have been discussed.

Width bifurcation and dynamical phase transitions in open quantum systems

The states of an open quantum system are coupled via the environment of scattering wave functions. The complex coupling coefficients ω between system and environment arise from the principal value integral and the residuum. At high-level density where the resonance states overlap, the dynamics of the system is determined by exceptional points. At these points, the eigenvalues of two states are equal and the corresponding eigenfunctions are linearly dependent. It is shown in the present paper that Im(ω) and Re(ω) influence the system properties differently in the surrounding of exceptional points. Controlling the system by a parameter, the eigenvalues avoid crossing in energy near an exceptional point under the influence of Re(ω) in a similar manner as it is well known from discrete states. Im(ω), however, leads to width bifurcation and finally (when the system is coupled to one channel, i.e., to one common continuum of scattering wave functions), to a splitting of the system into two parts with different characteristic time scales. The role of observer states is discussed. Physically, the system is stabilized by this splitting since the lifetimes of some states are longer than before, while that of one state is shorter. In the cross section the short-lived state appears as a background term in high-resolution experiments. The wave functions of the long-lived states are mixed in those of the original ones in a comparably large parameter range. Numerical results for the eigenvalues and eigenfunctions are shown for N=2,4, and 10 states coupled mostly to one channel.

Dynamics of resistive state in thin superconducting channels

We theoretically study how the dynamics of the resistive state in narrow superconducting channels shunted by an external resistor depends on channel's length $L$, the applied current $j$, and parameter $u$ characterizing the penetration depth of the electric field in the nonequilibrium superconductors. We show that changing $u$ dramatically affects both the behaviour of the current-voltage characteristics of the superconducting channels and the dynamics of their order parameter. Previously, it was demonstrated that when $u$ is less than the critical value $u_{c1}$, which does not depend on $L$, the phase slip centers appear simultaneously at different spots of the channel. Herewith, for $u>u_{c1}$ these centres arise consecutively at the same place. In our work we demonstrate that there is another critical value for $u$. Actually, if $u$ does not exceed a certain value $u_{c2}$, which depends on $L$, the current-voltage characteristic exhibits the step-like behaviour. However, for $u>u_{c2}$ it becomes hysteretic. In this case, with increase of $j$ the steady state, which corresponds to the time independent distribution of the order parameter along the channel, losses its stability at switching current value $j_{sw}$, and time periodic oscillations of both the order parameter and electric filed occur in the channel. As $j$ sweeps down, the periodic dynamics ceases at certain retrapping current value $j_r<j_{sw}$. Shunting the channel by a resistor increases the value of $j_r$, while $j_{sw}$ remains unchanged. Thus, for some high enough conductivity of the shunt $j_r$ and $j_{sw}$ eventually coincide, and the hysteretic loop disappears. We reveal dynamical regimes involved in the hysteresis, and discuss the bifurcation transitions between them.

Electronic Implementation of a Repressilator with Quorum Sensing Feedback

We investigate the dynamics of a synthetic genetic repressilator with quorum sensing feedback. In a basic genetic ring oscillator network in which three genes inhibit each other in unidirectional manner, an additional quorum sensing feedback loop stimulates the activity of a chosen gene providing competition between inhibitory and stimulatory activities localized in that gene. Numerical simulations show several interesting dynamics, multi-stability of limit cycle with stable steady-state, multi-stability of different stable steady-states, limit cycle with period-doubling and reverse period-doubling, and infinite period bifurcation transitions for both increasing and decreasing strength of quorum sensing feedback. We design an electronic analog of the repressilator with quorum sensing feedback and reproduce, in experiment, the numerically predicted dynamical features of the system. Noise amplification near infinite period bifurcation is also observed. An important feature of the electronic design is the accessibility and control of the important system parameters.

An experimental investigation and characterization on flame bifurcation and leaning transition behavior of a pool fire in near wake of a square cylinder

This paper is to investigate the fire behaviors in the wake of buildings. Experiments are carried out in a wind tunnel to reveal and to characterize flame behavior of a pool fire in the near wake behind a square cylinder of different cross sectional sizes (12cm, 18cm, 24cm and 30cm) under horizontal wind flow speeds in the range of 0–1.72m/s, with corresponding Reynolds number (Re) in the range of 0.5–3.5×104. Both square (10cm) and rectangular (4cm width and 24cm long) ethanol pool fires are used as fire source. The flame behavior is recorded by two CCD cameras from two directions (normal to the flow or in the direction of the flow). It is found that the flame in the near wake bifurcates and its leaning direction transits, behaving uniquely different from those in the quiescent air or in just a horizontal wind flow without the square cylinder bluff body. The distance from flame bifurcation point (rectangular pool fire) to the cylinder correlates well linearly with the cylinder width (size). Another interesting behavior revealed is that there is a transition of leaning direction of the flame of the square pool fire, from towards the cylinder transisting to the opposite direction. This happens either when increasing the distance (D) between the pool fire and the cylinder for a given cylinder width (L), or when decreasing L for a given D. A parameter Left Flame Area Ratio (LFAR, 2-D projected flame area at the left side of the vertical axial of pool center divided by the total flame area) is brought forward to quantify the flame leaning extent and direction, as to find out quantitatively the critical scale relation between these valuables at the transition state. The critical value of D corresponding to LFAR=0.5 (indicating the transitional turning point of the flame leaning direction) is found to vary remarkably with cylinder size but independent of wind speed. However, the normalized value of this critical distance between the pool center and the cylinder by the characteristic length scale taking as the cylinder width is found to converge to be all about 1.0.

Bifurcation on synchronous full annular rub of rigid-rotor elastic-support system

An aero-engine rotor system is simplified as an unsymmetrical-rigid-rotor with nonlinear-elastic-support based on its characteristics. Governing equations of the rubbing system, obtained from the Lagrange equation, are solved by the averaging method to find the bifurcation equations. Then, according to the two-dimensional constraint bifurcation theory, transition sets and bifurcation diagrams of the system with and without rubbing are given to study the influence of system eccentricity and damping on the bifurcation behaviors, respectively. Finally, according to the Lyapunov stability theory, the stability region of the steady-state rubbing solution, the boundary of static bifurcation, and the Hopf bifurcation are determined to discuss the influence of system parameters on the evolution of system motion. The results may provide some references for the designer in aero rotor systems.

Bifurcation transitions in a photochemical system under low magnetic fields

In the last decades, the effect of low magnetic fields on biochemical and chemical systems has been an urgent problem. By now numerous experimental and theoretical studies have been conducted to demonstrate that commonly this effect is of no essence as it does not exceed 10%. However, there are experimental works which testify that in some systems, magnetic field effects are more significant. Thus, of great interest is an active search for rather simple but realistic models that are based on physically explicit assumptions and able to account for a strong effect of low magnetic fields. The present work not only offers a theoretical study on the simplest photochemical system, describing a reversible reaction of photodissociation, but also shows how a low magnetic field can strongly modify its properties under highly nonequilibrium conditions. It is assumed that external magnetic field can have effect on the rates of radical reactions occurring in a system. This, in turn, leads to bifurcation of the nonequilibrium stationary state and, thus, to a drastic change in the properties of chemical systems (temperature and reagent concentration).

Forced synchronization of a delayed-feedback oscillator

Dynamics of a delayed-feedback oscillator with cubic nonlinearity driven by an external harmonic signal is considered. Regimes of forced synchronization and their stability conditions are studied analytically and numerically for the cases of single-frequency and multiple-frequency (self-modulation) operation of the free-running oscillator. Special attention is paid to synchronization of the oscillator with bistability of steady states which arises near the boundary of generation zone. Results of numerical simulation of bifurcation transitions to synchronization regime for different driving frequencies are presented. Main differences from the classical picture of synchronization for a system with one degree of freedom are discussed.

The study of stability and bifurcation of highline cable of alongside replenishment system

A theoretical model of transverse vibration of highline cable of alongside replenishment system was established, which considers the influence of the concentrated loading. The partial differential equation was asymptotically analysed by Galerkin method and muti-scale method. The bifurcation equation of system with higher order nonlinear terms was obtained by the solvability condition of eliminating the secular terms. Local bifurcation behavior of highline cable was studied with the software Mathematica. Bifurcation models and transition sets of system were obtained. The result shows that there are many kinds of bifurcation behaviors in highline cable of alongside replenishment system.

Template analysis of a Faraday disk dynamo

In a recent paper Moroz [1] returned to a nonlinear three-dimensional model of dynamo action for a self-exciting Faraday disk dynamo introduced by Hide et al. [2]. Since only two examples of chaotic behaviour were shown in [2], Moroz [1] performed a more extensive analysis of the dynamo model, producing a selection of bifurcation transition diagrams, including those encompassing the two examples of chaotic behaviour in [2]. Unstable periodic orbits were extracted and presented in [1], but no attempt was made to identify the underlying chaotic attractor. We rectify that here. Illustrating the procedure with one of the cases considered in [1], we use some of the unstable periodic orbits to identify a possible template for the chaotic attractor, using ideas from topology [3]. In particular, we investigate how the template is affected by changes in bifurcation parameter.

Unstable periodic orbits in a Faraday disk dynamo

AbstractHide et al [2] introduced a system of three nonlinear coupled ordinary differential equations to model a self‐exciting Faraday disk homopolar dynamo. Two examples of chaotic behaviour were shown by them. Moroz [4] performed a more extensive analysis of this dynamo model, including bifurcation transition diagrams and unstable periodic orbits for the two chaotic examples. We now use these unstable periodic orbits to identify a possible template for the chaotic attractor, using ideas from topology [1] and results from a corresponding analysis of the Lorenz attractor. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Strongly nonlinear theory of nanostructure formation owing to elastic and nonelastic strains in crystalline solids

We develop an essentially nonlinear theory of elastic and nonelastic microstrains resulting in the formation of nanostructures. Using the model of mutually penetrating lattices, we generalize the well-known theory of acoustic and optical vibrations to the case of nonlinear interaction between sublattices. This permits treating the sublattice interaction forces as periodic (for example, sinusoidal) functions of the relative displacement of the sublattices. We obtain equations for the macroscopic and microscopic displacement fields containing two characteristic scales of the nanostructure. We find a number of their solutions describing the effects of decrease in the potential interatomic barriers in the external stress field and the formation of defects and domain nanostructure as a result of bifurcation transitions. We prove their stability.

Stochastic bifurcation in FitzHugh–Nagumo ensembles subjected to additive and/or multiplicative noises

We have studied the dynamical properties of finite N -unit FitzHugh–Nagumo (FN) ensembles subjected to additive and/or multiplicative noises, reformulating the augmented moment method (AMM) with the Fokker–Planck equation (FPE) method [H. Hasegawa, J. Phys. Soc. Japan 75 (2006) 033001]. In the AMM, original 2 N -dimensional stochastic equations are transformed to eight-dimensional deterministic ones, and the dynamics is described in terms of averages and fluctuations of local and global variables. The stochastic bifurcation is discussed by a linear stability analysis of the deterministic AMM equations. The bifurcation transition diagram of multiplicative noise is rather different from that of additive noise: the former has the wider oscillating region than the latter. The synchronization in globally-coupled FN ensembles is also investigated. Results of the AMM are in good agreement with those of direct simulations (DSs).

On thermodynamic stability of heteropolymer mesophases formed under weak segregation regime

Conditions of local and global thermodynamic stability of spatially-periodic mesophases formed in an incompressible melt of binary monodisperse copolymers with molecules of arbitrary chemical structure are theoretically scrutinized. For the first time a bifurcation analysis of the Landau free energy in the first harmonic approximation is carried out. Based upon this analysis, feasible scenarios of bifurcation and phase transitions between mesophases of different morphology are discussed. The description of thermodynamic behavior of some systems is presented to exemplify the implementation of the approach proposed.

Bifurcation and synchronization of synaptically coupled FHN models with time delay

This paper presents an investigation of dynamics of the coupled nonidentical FHN models with synaptic connection, which can exhibit rich bifurcation behavior with variation of the coupling strength. With the time delay being introduced, the coupled neurons may display a transition from the original chaotic motions to periodic ones, which is accompanied by complex bifurcation scenario. At the same time, synchronization of the coupled neurons is studied in terms of their mean frequencies. We also find that the small time delay can induce new period windows with the coupling strength increasing. Moreover, it is found that synchronization of the coupled neurons can be achieved in some parameter ranges and related to their bifurcation transition. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behavior are clarified.

Break of information-investment flows in the process of switching over intermittency in the change of the trend of development of the dynamic system

Studies are made of basic regularities of the process of switching over intermittency in the change of the trend of development of the cascade-hierarchical system. It is shown that the transition from the section of growth to the section of fall can be explained as a result of the change of the sign of the feedback (from the positive to the negative one) between the increment of the intermittency exponent Δµ and the increment of the fractal dimension Δd f (the expansion of the "channel" of attraction) for the most rapidly orienting (short-term) dynamic substructures. The subsequent break of the information-investment flow (the flow of commands) from the long-term to short-term substructures makes it impossible to support the dynamic equilibrium in the cascade-hierarchical system, which leads to the bifurcation transition to a new section of the channel of attraction.

The Hide, Skeldon, Acheson dynamo revisited

Hide et al . (Hide, Skeldon &amp; Acheson 1996 Proc. R. Soc. A 452 , 1369–1395) introduced a nonlinear system of three coupled ordinary differential equations to model a self-exciting Faraday disk homopolar dynamo. A very small selection of its possible behaviours was presented in that paper. Subsequent studies have extended the system to incorporate the effects of a nonlinear motor, an external battery and magnetic field, the coupling of two or more identical dynamos together, among other things. In this paper, we return to the original model with a view to perform a more extensive analysis of the Hide et al . dynamo. For the first time, we present bifurcation transition diagrams, so that the two examples of chaotic dynamo action (shown in figure 9 in that paper) can now be placed into context. We exhibit the coexistence of multiple attractors and also identify the lowest order unstable periodic orbits pertaining to some specific cases.

Quantum bifurcations and quantum phase transitions in rotational spectra

An interconnection between quantum bifurcations and quantum phase transitions is investigated. It is shown that the quantum phase transition is the limiting case of the quantum bifurcation in macroscopic systems. Critical phenomena in microscopic many-body systems such as nuclei and molecules are described by quantum bifurcations. They are characterized by a large transition region, which conceals a critical point. Both the quantum bifurcation and quantum phase transition show that a change in the internal structure of a system manifests itself in the modification of its rotational spectrum. Such an intimate connection between internal and rotational motion allows a critical phenomenon to be established.

Oscillators with asymmetric single and double well potentials: transition to chaos revisited

We study the Melnikov criterion for a global homoclinic bifurcation and possible transition to chaos for a single degree of freedom nonlinear oscillator. This provides a systematic method of treatment for an arbitrary potential expressed as a fourth order polynomial. The equation of motion has external excitation and a Duffing type nonlinearity with one or two unsymmetric potential wells.

Growth and Nonlinear Saturation of Fluctuations in a Nonlinear Oscillator at the Threshold of a Bifurcation of Spontaneous Symmetry Breaking

We consider the phenomenon of prebifurcation noise amplification in a nonlinear oscillator at the threshold of a bifurcation of spontaneous symmetry breaking. The studies are based on the model of a nonlinear oscillator in which the potential relief transforms from monostable to symmetric bistable and the noise acting on the system is assumed Gaussian and short-correlated. The fluctuation variance as a function of the regime of the system and the rate at which the bifurcation threshold is reached are examined. Our analytical estimates are in good agreement with the results of numerical simulation for both the linear growth and the nonlinear saturation of fluctuations. It is noted that in the case of fast bifurcation transitions, a loop of noise-dependent hysteresis and breaking of probability symmetry of stable final states are observed in the nonlinear oscillator.