Bifurcation Transition Research Articles

Classical Bifurcation Diagrams by Quantum Means

Asymptotic state of an open quantum system may change substantially upon variations of system parameters. These changes can be often identified as bifurcation transitions in the classical mean‐filed equations describing evolution of the relevant observables. We demonstrate that these bifurcations can be made visible as changes in the structure of the asymptotic density matrix of the original quantum system. By using an N‐boson dimer, we present quantum ‘bifurcation diagrams’ for the pitchfork and saddle‐node bifurcations (in the stationary dimer) and visualize a period‐doubling transition to chaos (in a periodically modulated dimer).

New higher-order transition in causal dynamical triangulations

We reinvestigate the recently discovered bifurcation phase transition in Causal Dynamical Triangulations (CDT) and provide further evidence that it is a higher order transition. We also investigate the impact of introducing matter in the form of massless scalar fields to CDT. We discuss the impact of scalar fields on the measured spatial volumes and fluctuation profiles in addition to analysing how the scalar fields influence the position of the bifurcation transition.

Two Dynamical Regimes of the Substrate Radical Rearrangement Reaction in B12-Dependent Ethanolamine Ammonia-Lyase Resolve Contributions of Native Protein Configurations and Collective Configurational Fluctuations to Catalysis.

The kinetics of the substrate radical rearrangement reaction step in B12-dependent ethanolamine ammonia-lyase (EAL) from Salmonella typhimurium are measured over a 92 K temperature range. The observed first-order rate constants display a piecewise-continuous Arrhenius dependence, with linear regions over 295 → 220 K (monoexponential) and 214 → 203 K (biexponential) that are delineated by a kinetic bifurcation and kinks at 219 and 217 K, respectively. The results are interpreted by using a free energy landscape model and derived microscopic kinetic mechanism. The bifurcation and kink transitions correspond to the effective quenching of two distinct sets of native collective protein configurational fluctuations that (1) reconfigure the protein within the substrate radical free energy minimum, in a reaction-enabling step, and (2) create the protein configurations associated with the chemical step. Below 217 K, the substrate radical decay reaction persists. Increases in activation enthalpy and entropy of both the microscopic enabling and reaction steps indicate that this non-native reaction coordinate is conducted by local, incremental fluctuations. Continuity in the Arrhenius relations indicates that the same sets of protein groups and interactions mediate the rearrangement over the 295 to 203 K range, but with a repertoire of configurations below 217 K that is restricted, relative to the native configurations accessible above 219 K. The experimental features of a culled reaction step, first-order kinetic measurements, and wide room-to-cryogenic temperature range, allow the direct demonstration and kinetic characterization of protein dynamical contributions to the core adiabatic, bond-making/bond-breaking reaction in EAL.

Unidirectional reflectionless light propagation at exceptional points

Abstract In this paper, we provide a comprehensive review of unidirectional reflectionless light propagation in photonic devices at exceptional points (EPs). EPs, which are branch point singularities of the spectrum, associated with the coalescence of both eigenvalues and corresponding eigenstates, lead to interesting phenomena, such as level repulsion and crossing, bifurcation, chaos, and phase transitions in open quantum systems described by non-Hermitian Hamiltonians. Recently, it was shown that judiciously designed photonic synthetic matters could mimic the complex non-Hermitian Hamiltonians in quantum mechanics and realize unidirectional reflection at optical EPs. Unidirectional reflectionlessness is of great interest for optical invisibility. Achieving unidirectional reflectionless light propagation could also be potentially important for developing optical devices, such as optical network analyzers. Here, we discuss unidirectional reflectionlessness at EPs in both parity-time (PT)-symmetric and non-PT-symmetric optical systems. We also provide an outlook on possible future directions in this field.

Hysteresis, resonant oscillations and bifurcation mode of a system modeled by a forced modified Van der Pol-Duffing oscillator

Nonlinear oscillations and its applications in physics, chemistry, engineering, biophysics, communications are studied with some analytical, numerical and experimental methods. In the present paper, hysteresis, resonant oscillations and bifurcation mode of a system modeled by a forced modified Van der Pol-Duffing oscillator are considered. The plasma oscillations are considered and are described by a nonlinear differential equation. By using the harmonic balance technique and the multiple time scales methods, the amplitudes of the forced harmonic, superharmonic and subharmonic oscillatory states are obtained. Then, we derived admissible values of the amplitude of the external strength. Some bifurcation structures and transition to chaos of the model have been investigated. The model presented several dynamics motions which are influenced by nonlinear parameters. It can be concluded that the nonlinear parameters have a real impact on the dynamics of the model.

A peculiar bifurcation transition route of thermocapillary convection in rectangular liquid layers

Transition routes of thermocapillary convection are complex and diversiform in rectangular layers. In general, three types of transition routes were observed in thermocapillary convection, including quasi-periodic bifurcations, period-doubling bifurcations and tangent bifurcations. The tangent bifurcation is a peculiar type of bifurcations, which often appears accompanied with other types of bifurcation sequences. In our experiments, a detailed research on tangent bifurcations of thermocapillary convection in rectangular liquid layers was conducted through the measurement of single-point temperature oscillation in liquid. There are different tangent bifurcation series for various of experimental conditions, including different Prandtl numbers of silicone oil and different aspect ratios. It has certain occasionality for appearance of the tangent bifurcation, mainly in the experimental condition of Prandtl number of silicone oil equal to 16 (1cSt) or Prandtl number equal to 25 (1.5cSt). Chaotic characteristics of bifurcations are investigated by the method of phase-space reconstruction, maximum Lyapunov exponent and permutation entropy.

Bifurcation of a Newtonian-fluid flow in a planar channel with sudden contraction and expansion

The features of a Newtonian-fluid flow in a two-dimensional channel with sudden contraction and expansion are investigated by numerical modeling. The kinetics of the bifurcation transition from the symmetric mode to steady-state asymmetric flow on the outlet from the zone of contraction of the channel is analyzed. The linear dependence of the degree of asymmetry of flow on the Reynolds number is established.

Low-Dimensional Analysis and Numerical Simulation of the Incompressible Flow Between Two Concentric Rotating Cylinders

There has been a very large number of experimental and theoretical studies of flow between concentric rotating cylinders in the century, since these pioneering works, the instability of Couette flow and Taylor vortex flow, now known as the “Couette–Taylor problem” is a paradigm of nonlinear problem. In order to explore the transition way of Couette–Taylor flow from laminar to turbulence and the characteristics of chaotic attractors in the turbulent regime of Couette–Taylor flow, dynamical behaviors and numerical simulation of Couette–Taylor flow have been studied by using low-dimensional model analysis method in this paper. Dynamical behaviors of a three-model Lorenz-like system of Couette–Taylor flow have been discussed, such as the stability of equilibrium, presence of limit cycles, occurrence of bifurcation and chaos, as well as the analysis of global stability etc. Moreover, using these results we explain successive transitions of Couette–Taylor flow from Laminar flow to turbulence in the experiment. By means of numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum and Poincare map we analyze complex dynamic behaviors of the system from the bifurcation transition to chaos.

Bifurcation transitions in gap-junction-coupled neurons.

Here we investigate transitions occurring in the dynamical states of pairs of distinct neurons electrically coupled, with one neuron tonic and the other bursting. Depending on the dynamics of the individual neurons, and for strong enough coupling, they synchronize either in a tonic or a bursting regime, or initially tonic transitioning to bursting via a period doubling cascade. Certain intrinsic properties of the individual neurons such as minimum firing rates are carried over into the dynamics of the coupled neurons affecting their ultimate synchronous state.

Dynamic of one and two elliptical particles settling in oscillatory flow: Period bifurcation and resonance state

The inertial migrations of one and two elliptical particles in a two-dimensional channel under oscillatory pressure driven flow are investigated by a finite element arbitrary Lagrangian-Eulerian (ALE) method. The effects of oscillation frequency on rotation behavior, vortex structure, pressure distribution and relative displacement during sedimentation have been studied. The results show that the oscillation can accelerate the eddy formation and shedding behind the particle, and cause a strong modification of the turning couples on the ellipse with lateral displacement. As the frequency increases, the initial ‘zigzag migration’ at the channel side goes through a period bifurcation transition to the resonance state with ‘anomalous rotation’ near the wall, and further induces ‘rotation shift’ near the channel center. The angular velocity increases before the resonance state, while the average vertical velocity decreases with the rising frequency. Moreover, the hydrodynamic interaction between the particles has a close association with the oscillatory effect. In the period bifurcation regions, the higher the oscillation frequency, the bigger the horizontal distance of the opposite rotation particles, and the particles separate fastest at the resonance state. In contrast, an attractor is formed between the pairs in the high frequency, and the two particles are exchanging the lead as settling forward. The final configuration of the pairs is shown to be caused by the Magnus type of lift balancing the wall repulsion and the interplay between particles, and is sensitive to the dynamic drag statistically corresponding to the oscillation in the wake and the periodic discharge of vorticity.

Mechanisms of appearance of amplitude and phase chimera states in ensembles of nonlocally coupled chaotic systems

We explore the bifurcation transition from coherence to incoherence in ensembles of nonlocally coupled chaotic systems. It is firstly shown that two types of chimera states, namely, amplitude and phase, can be found in a network of coupled logistic maps, while only amplitude chimera states can be observed in a ring of continuous-time chaotic systems. We reveal a bifurcation mechanism by analyzing the evolution of space-time profiles and the coupling function with varying coupling coefficient and formulate the necessary and sufficient conditions for realizing the chimera states in the ensembles.

Dynamic bifurcations on financial markets

We provide evidence that catastrophic bifurcation breakdowns or transitions, preceded by early warning signs such as flickering phenomena, are present on notoriously unpredictable financial markets. For this we construct robust indicators of catastrophic dynamical slowing down and apply these to identify hallmarks of dynamical catastrophic bifurcation transitions. This is done using daily closing index records for the representative examples of financial markets of small and mid to large capitalisations experiencing a speculative bubble induced by the worldwide financial crisis of 2007-08.

Bifurcation Transition and Nonlinear Response in a Fractional-Order System

We extend a typical system that possesses a transcritical bifurcation to a fractional-order version. The bifurcation and the resonance phenomenon in the considered system are investigated by both analytical and numerical methods. In the absence of external excitations or simply considering only one low-frequency excitation, the system parameter induces a continuous transcritical bifurcation. When both low- and high-frequency forces are acting, the high-frequency force has a biasing effect and it makes the continuous transcritical bifurcation transit to a discontinuous saddle-node bifurcation. For this case, the system parameter, the high-frequency force, and the fractional-order have effects on the saddle-node bifurcation. The system parameter induces twice a saddle-node bifurcation. The amplitude of the high-frequency force and the fractional-order induce only once a saddle-node bifurcation in the subcritical and the supercritical case, respectively. The system presents a nonlinear response to the low-frequency force. The system parameter and the low-frequency can induce a resonance-like behavior, though the high-frequency force and the fractional-order cannot induce it. We believe that the results of this paper might contribute to a better understanding of the bifurcation and resonance in the excited fractional-order system.

Magnetic field effects on bistability and bifurcation phenomena in lipid peroxidation.

A simple model based on physically evident assumptions was constructed that may predict the strong effect of low intensity magnetic fields (MFs) on living systems. As a model system, we studied MF effect on lipid peroxidation reaction under steady state conditions. It was assumed that external MF affects rate constants of radical reactions taking place in elementary stages of lipid peroxidation reaction. It was shown that under certain conditions, low intensity MF may provoke a bifurcation transition in the system followed by a significant change in its characteristics. This means that, in principle, an external MF may play a critical role in lipid peroxidation reaction and a cell life.

Signature change of the metric in CDT quantum gravity?

We study the effective transfer matrix within the semiclassical and bifurcation phases of CDT quantum gravity. We find that for sufficiently large lattice volumes the kinetic term of the effective transfer matrix has a different sign in each of the two phases. We argue that this sign change can be viewed as a Wick rotation of the metric. We discuss the likely microscopic mechanism responsible for the bifurcation phase transition, and propose an order parameter that can potentially be used to determine the precise location and order of the transition. Using the effective transfer matrix we approximately locate the position of the bifurcation transition in some region of coupling constant space, allowing us to present an updated version of the CDT phase diagram.

S-shaped motor learning and nonequilibrium phase transitions.

S-Shaped change in performance outcome has long been considered to be a pathway of motor learning, but there is little or no evidence for it. The experiment investigated the hypothesis that S-shaped motor learning as reflected in the task outcome is a product of a transition in the movement coordination dynamics as a function of practice acting as a control parameter. Young adult participants practiced the roller ball task that required learning the transition of a coordination mode to preserve and enhance the motion of a rotating ball to transition from task failure to success. There were 50 practice trials per day for as many practice days (3-20) as required for each participant to reach the task criterion of success that was followed 1 week later by a retention test. All participants improved their task performance with practice but there were subgroups of patterns of change including S-shaped learning. The enhanced variability during the transition supported the interpretation that the S-shaped learning outcome is reflective of a saddle-node bifurcation or first-order nonequilibrium transition. The learning of a new pattern of movement coordination is a different process from learning to scale an already producible coordination mode to new task demands.

Bifurcation of a nonlinear Schrödinger equation with a symmetrical double well

The bifurcation of a one-dimensional Gross–Pitaevskii-type equation with a symmetrical double well is investigated. A general relation for the critical power, above which a kind of bifurcation transition from a supercritical pitchfork to a subcritical pitchfork occurs, is presented in terms of the ratio of the nonlinear tunneling to the nonlinear interaction. When the nonlinearity is very weak, our relation reduces to a universal power law; when the nonlinearity is strong, this universal critical power depends on the nonlinearity, the shape of the trap and even the dimension. By means of a numerical calculation, a possible bifurcation transition induced by nonlinearity is predicted for both the ground state and the first excited state.

Changes of firing rate induced by changes of phase response curve in bifurcation transitions.

We study dynamical mechanisms responsible for changes of the firing rate during four different bifurcation transitions in the two-dimensional Hindmarsh-Rose (2DHR) neuron model: the saddle node on an invariant circle (SNIC) bifurcation to the supercritical Andronov-Hopf (AH) one, the SNIC bifurcation to the saddle-separatrix loop (SSL) one, the AH bifurcation to the subcritical AH (SAH) one, and the SSL bifurcation to the AH one. For this purpose, we study slopes of the firing rate curve with respect to not only an external input current but also temperature that can be interpreted as a timescale in the 2DHR neuron model. These slopes are mathematically formulated with phase response curves (PRCs), expanding the firing rate with perturbations of the temperature and external input current on the one-dimensional space of the phase [Formula: see text] in the 2DHR oscillator. By analyzing the two different slopes of the firing rate curve with respect to the temperature and external input current, we find that during changes of the firing rate in all of the bifurcation transitions, the calculated slope with respect to the temperature also changes. This is largely dependent on changes in the PRC size that is also related to the slope with respect to the external input current. Furthermore, we find phase transition-like switches of the firing rate with a possible increase of the temperature during the SSL-to-AH bifurcation transition.

Changes in Repetitive Firing Rate Related to Phase Response Curves for Andronov—Hopf Bifurcations

We study specific changes in repetitive firing in the two-dimensional Hindmarsh—Rose (2dHR) oscillatory system that undergoes a bifurcation transition from the supercritical Andronov—Hopf (AH) type to the subcritical Andronov—Hopf (SAH) type. We identify dynamical mechanisms which are responsible for changes of the repetitive firing rate during the AH to SAH bifurcation transitions. These include frequency-shift functions in response to small perturbations of a timescale parameter, its multiplicative parameter, and an external input current in the 2dHR oscillatory system. The frequency-shift functions are explicitly represented as functions relating to the phase response curves (PRCs). Then, we demonstrate that when the timescale is normal and relatively fast, the repetitive firing rate slightly increases and decreases respectively during the AH to SAH bifurcation transition with a change of the intrinsic parameter, whereas it decreases during the SAH to AH bifurcation transition with an increase in the timescale. By analyzing the three different frequency-shift functions, we show that such changes of the repetitive firing rate depend largely on changes of the PRC size. The PRC size for the SAH bifurcation shrinks to the PRC size for the AH bifurcation.

Development of instability in the problem of flow around a sphere

Multimoment hydrodynamics equations are used to solve the problem of flow around a quiescent solid sphere. The solutions to the multimoment hydrodynamics equations are found, which enable to interpret of the phenomenon of vortex shedding. The solutions give a pattern of instability development that qualitatively reproduces experimental data over a wide range of Reynolds numbers. The replacement of one unstable flow mode by another unstable mode is governed the tendency of the system to find the fastest way to depart from the state of statistical equilibrium. After stability loss, the system does not reach a new stable state. Such a scenario is at odds with the ideas of classical hydrodynamics, which interprets the development of instability in terms of a bifurcation transition from one stable state to another. This picture presented shows the direction of solving the problems faced by classical hydrodynamics in the interpretation of the phenomenon of vortex shedding.