Bifurcation Transition Research Articles

Dynamical transition in controllable quantum neural networks with large depth

Understanding the training dynamics of quantum neural networks is a fundamental task in quantum information science with wide impact in physics, chemistry and machine learning. In this work, we show that the late-time training dynamics of quantum neural networks with a quadratic loss function can be described by the generalized Lotka-Volterra equations, leading to a transcritical bifurcation transition in the dynamics. When the targeted value of loss function crosses the minimum achievable value from above to below, the dynamics evolve from a frozen-kernel dynamics to a frozen-error dynamics, showing a duality between the quantum neural tangent kernel and the total error. In both regions, the convergence towards the fixed point is exponential, while at the critical point becomes polynomial. We provide a non-perturbative analytical theory to explain the transition via a restricted Haar ensemble at late time, when the output state approaches the steady state. Via mapping the Hessian to an effective Hamiltonian, we also identify a linearly vanishing gap at the transition point. Compared with the linear loss function, we show that a quadratic loss function within the frozen-error dynamics enables a speedup in the training convergence. The theory findings are verified experimentally on IBM quantum devices.

Bifurcation and phase transitions in heterogeneous non-lane-discipline-based car-following model integrating cooperative feedback control under automated and human-driven vehicles environment

Automated vehicles (AVs) equipped with adaptive cruise control (ACC) possess different driving performances from human-driven vehicles (HDVs). AVs will have an unusual impact on the traffic flow, especially in non-lane-discipline-based (NLD) environment. Therefore, we create a heterogeneous NLD car-following model (HNLD-CFM) integrating cooperative feedback control under AVs and HDVs environment. The velocity differences between two preceding vehicles and the current vehicle are adopted for the cooperative feedback control signal. The stability condition of the HNLD-CFM is inferred through control theory, which is closely related to the proportion of AVs, lateral separation of vehicles and control signal. Moreover, bifurcation phenomena are been investigated via theory analysis and simulation. Furthermore, numerical simulation clarifies that traffic congestion can be effectively suppressed and energy consumption is reduced by increasing the proportion of AVs and lateral separation of vehicles, besides a control signal.

Horizontal buoyant jets into viscoplastic ambient fluids

This study investigates the horizontal injection of a heavy Newtonian fluid into a lighter viscoplastic ambient fluid, in a large reservoir. The flow dynamics is experimentally captured via camera imaging, laser-induced fluorescence, and particle image velocimetry techniques. The flow parameters include various density differences, injection velocities, and ambient fluid viscoplastic properties. Our analysis identifies two key dimensionless numbers, the Froude number (Fr) and the effective viscosity ratio (m), which includes the rheology of the viscoplastic fluid. Our study also examines the effects of these dimensionless numbers on critical jet characteristics, such as bifurcation length, transition length, deviation length, and jet trajectory, and provides correlations using Fr and m, to predict these characteristic lengths. A regime classification based on the bifurcation phenomenon is also presented in the Fr−m plane.

New estimation of critical insolation–CO2 relationship for triggering glacial inception

Abstract. It has been previously proposed that glacial inception represents a bifurcation transition between interglacial and glacial states and is governed by the nonlinear dynamics of the climate–cryosphere system. To trigger glacial inception, the orbital forcing (defined as the maximum of summer insolation at 65° N and determined by Earth’s orbital parameters) must be lower than a critical level, which depends on the atmospheric CO2 concentration. While paleoclimatic data do not provide a strong constraint on the dependence between CO2 and critical insolation, its accurate estimation is of fundamental importance for predicting future glaciations and the effect that anthropogenic CO2 emissions might have on them. In this study, we use the novel Earth system model of intermediate complexity CLIMBER-X with interactive ice sheets to produce a new estimation of the critical insolation–CO2 relationship for triggering glacial inception. We perform a series of experiments in which different combinations of orbital forcing and atmospheric CO2 concentration are maintained constant in time. We analyze for which combinations of orbital forcing and CO2 glacial inception occurs and trace the critical relationship between them, separating conditions under which glacial inception is possible from those where glacial inception is not materialized. We also provide a theoretical foundation for the proposed critical insolation–CO2 relation. We find that the use of the maximum summer insolation at 65° N as a single metric for orbital forcing is adequate for tracing the glacial inception bifurcation. Moreover, we find that the temporal and spatial patterns of ice sheet growth during glacial inception are not always the same but depend on the critical insolation and CO2 level. The experiments evidence the fact that during glacial inception, ice sheets grow mostly in North America, and only under low CO2 conditions are ice sheets also formed over Scandinavia. The latter is associated with a weak Atlantic Meridional Overturning Circulation (AMOC) for low CO2. We find that the strength of AMOC also affects the rate of ice sheet growth during glacial inception.

Effect of higher-order interactions on chimera states in two populations of Kuramoto oscillators.

We investigate the effect of the fraction of pairwise and higher-order interactions on the emergent dynamics of the two populations of globally coupled Kuramoto oscillators with phase-lag parameters. We find that the stable chimera exists between saddle-node and Hopf bifurcations, while the breathing chimera lives between Hopf and homoclinic bifurcations in the two-parameter phase diagrams. The higher-order interaction facilitates the onset of the bifurcation transitions at a much lower disparity between the inter- and intra-population coupling strengths. Furthermore, the higher-order interaction facilitates the spread of breathing chimera in a large region of the parameter space while suppressing the spread of the stable chimera. A low degree of heterogeneity among the phase-lag parameters promotes the spread of both stable chimera and breathing chimera to a large region of the parameter space for a large fraction of the higher-order coupling. In contrast, a large degree of heterogeneity is found to decrease the spread of both chimera states for a large fraction of the higher-order coupling. A global synchronized state is observed above a critical value of heterogeneity among the phase-lag parameters. We have deduced the low-dimensional evolution equations for the macroscopic order parameters using the Ott-Antonsen Ansatz. We have also deduced the analytical saddle-node and Hopf bifurcation curves from the evolution equations for the macroscopic order parameters and found them to match with the bifurcation curves obtained using the software XPPAUT and with the simulation results.

Weakly nonlinear stability analysis of non-isothermal parallel flow in a vertical porous annulus

The study examines the stability of parallel airflow within a tall vertical coaxial circular annulus filled with a highly permeable porous medium under stable stratification. The linear stability theory predicts that the least unstable mode can be either axisymmetric or non-axisymmetric, depending on the values of the governing parameters. The curvature parameter, defined as the gap between coaxial circular cylinders, contributes to the stabilization of the flow. Depending on various controlling parameter values, the stably stratified non-isothermal annular parallel flow (SNAPF) can manifest three instability modes: thermal-shear, thermal-buoyant, and mixed or interactive. The instability type can change abruptly with Reynolds number, curvature parameter, and media permeability. A finite-amplitude instability analysis is examined using weakly nonlinear stability theory. The analysis demonstrates both subcritical and supercritical bifurcations near the instability boundary. The range of Reynolds numbers for subcritical bifurcation expands when switching from axisymmetric to non-axisymmetric disturbance. Bifurcation transitions between supercritical and subcritical occur due to changes in the least stable azimuthal mode. The study also investigates the equilibrium/threshold amplitude within the supercritical/subcritical instability regime, and it verifies the existence of subcritical/supercritical bifurcation through nonlinear kinetic energy balance. Furthermore, the finite-amplitude analysis anticipates that the lower critical Rayleigh number increases as the Reynolds number, curvature parameter, or media permeability decreases. An increase in the Reynolds number shifts the point of inflection in the velocity profile toward the inner cylinder, which subsequently leads to a reduction in the size of the core vortex region. In the subcritical instability regime, the radius of the primary vortex cell decreased significantly as the Rayleigh number increased. When the curvature is altered, a structural transition occurs in the isotherms from multicellular to unicellular.

Bulging of dielectric elastomer tubes considering residual stress and viscoelasticity

This paper investigates the impacts of residual stress and viscoelasticity on the bifurcation and post-bifurcation behaviors of dielectric elastomer tube actuator (DETA) with finite thickness. A theoretical model with residual stress is developed to analyze the electromechanical (EM) behavior of a DETA under a combined effect of voltage, internal pressure, and axial force loadings. A mathematical derivation of the bifurcation condition for the DETA under EM coupling loadings is presented, and a coexistence phase transition condition for steady bugle propagation is established. The effects of voltage, axial force, and residual stress on the bifurcation and coexistence phase transition conditions are studied. To validate the theoretical predictions, finite element (FE) simulations are performed on a closed-ended DETA subjected to the EM loadings. The results reveal a competitive relationship between the localized bugling and electric breakdown (EB). The large residual stress can inhibit the localized bugling and enhance the stability of the tube. Additionally, we explore the bifurcation behavior of the viscoelastic DETA, revealing that the viscoelastic effect significantly influences the critical pressure of bifurcation and the loading/unloading paths. It is observed that snap-back instability can occur during the unloading process under high loading rates or voltages. This work provides valuable insights into the finite deformation and bifurcation behaviors of the DETA under complex EM coupling loadings.

Repertoire of dynamical states in dissimilarly coupled Van der Pol oscillators

We have considered dissimilarly coupled Van der Pol oscillators with an offset parameter which determines the degree of heterogeneity of the dissimilar coupling strength. Increasing degree of heterogeneity for decreasing values of the offset parameter results in a rich repertoire of bifurcation transitions and dynamical states including epochs of period doubling bifurcation. Two distinct multi-stable states are also observed along with several symmetry breaking dynamical states. We have deduced analytical stability conditions for Hopf and pitch-fork bifurcations through a linear stability analysis of symmetry preserving states, namely, trivial steady state and oscillation death state. The analytical conditions are found to match exactly with the simulation results in the two-parameter phase diagram. In addition to torus bifurcation, crisis and crisis induced intermittency routes to chaos are also observed for an appropriate heterogeneity of the dissimilar coupling strength. The period doubling bifurcation is characterized using the largest Lyapunov exponents of the dissimilarly coupled Van der Pol oscillators.

Global dynamics, thermodynamics and non-equilibrium origin of bifurcations for single neuron dynamics.

The understanding of neural excitability and oscillations in single neuron dynamics remains incomplete in terms of global stabilities and the underlying mechanisms for phase formation and associated phase transitions. In this study, we investigate the mechanism of single neuron excitability and spontaneous oscillations by analyzing the potential landscape and curlflux. The topological features of the landscape play a crucial role in assessing the stability of resting states and the robustness/coherence of oscillations. We analyze the excitation characteristics in Class I and Class II neurons and establish their relation to biological function. Our findings reveal that the average curlflux and associated entropy production exhibit significant changes near bifurcation or phase transition points. Moreover, the curlflux and entropy production offer insights into the dynamical and thermodynamical origins of nonequilibrium phase transitions and exhibit distinct behaviors in Class I and Class II neurons. Additionally, we quantify time irreversibility through the difference in cross-correlation functions in both forward and backward time, providing potential indicators for the emergence of nonequilibrium phase transitions in single neurons.

Multicritical bifurcation and first-order phase transitions in a three-dimensional Blume-Capel antiferromagnet.

We present a detailed study by Monte Carlo simulations and finite-size scaling analysis of the phase diagram and ordered bulk phases for the three-dimensional Blume-Capel antiferromagnet in the space of temperature and magnetic and crystal fields (or two chemical potentials in an equivalent lattice-gas model with two particle species and vacancies). The phase diagram consists of surfaces of second- and first-order transitions that enclose a "volume" of ordered phases in the phase space. At relatively high temperatures, these surfaces join smoothly along a line of tricritical points, and at zero magnetic field we obtain good agreement with known values for tricritical exponent ratios [Y. Deng and H. W. J. Blöte, Phys. Rev. E 70, 046111 (2004)10.1103/PhysRevE.70.046111]. In limited field regions at lower temperatures (symmetric under reversal of the magnetic field), the tricritical line for this three-dimensional model bifurcates into lines of critical endpoints and critical points, connected by a surface of weak first-order transitions inside the region of ordered phases. This phenomenon is not seen in the two-dimensional version of the same model. We confirm the location of the bifurcation as previously reported [Y.-L. Wang and J.D. Kimel, J. Appl. Phys. 69, 6176 (1991)0021-897910.1063/1.348797], and we identify the phases separated by this first-order surface as antiferromagnetically (three-dimensional checker-board) ordered with different vacancy densities. We visualize the phases by real-space snapshots and by structure factors in the three-dimensional space of wave vectors.

Effect of flow bifurcation transitions of shear-thinning fluids on hydrodynamic resistance of channels with sudden contraction and expansion

The aim of this work was to study the relationship between hydrodynamic resistance and flow instability of shear-thinning fluids in channels with sudden contraction and expansion. To this end, the critical conditions of bifurcation transitions from symmetric to asymmetric vortex flow depending on the channel geometry and rheological properties of Carreau model fluid were investigated. It was found that increasing the contraction ratio leads to decreasing the critical Reynold number as a power function, while variation in the length of the channel narrow part has almost no effect on the position of the bifurcation transition point, but changes the corner vortex lengths. It was found that the considered bifurcation transitions in shear-thinning fluids lead to a non-monotonic dependence of the hydrodynamic resistance of channels with sudden contraction and expansion on Reynolds number with a minimum located around the critical Reynolds number.

Synchronization in the two networks-frustrated coupled oscillators with a noisy attractive-repulsive frequencies

We investigate the correlation effects of the two networks combined with an attractive-repulsive frequency altered by noise on the mutual synchronization of the two coupled Kuramoto models with parametric random phase-shift properties. The necessity and significance of collective behavior between ensembles of interacting oscillators and their rich phenomenology offer an idealization of several disciplines in which mutual synchronization competes with force. In this paper, we derive the phase-locked states and identify the significant synchronization transition points analytically with exact boundary conditions for the correlated and uncorrelated joint distributions, their stability, and bifurcation diagrams. We find that a perfect and imperfect supercritical to subcritical Hopf bifurcation transition occurs depending on the synchronic transition points for the correlated cases, characterized by the power scales and the largest eigenvalues of the networks. Moreover, we show the powerful interplay of force, noise, frustration, and network on the synchronization transitions of the two populations and their compromise between the correlated and uncorrelated joint probability distributions. The intensity and transmissibility of noise, in particular, vary within and between populations.

Bifurcation and mode transition of buckled ribbons under oblique compressions

Three-dimensional (3D) buckled ribbon mesostructures have important implications in a variety of engineering fields, because they can afford novel functionalities and unprecedented mechanical properties with optimized structural/material designs. In many scenarios of practical applications, buckled ribbons would undergo oblique compressions, which could potentially lead to functional degradation or even device failure. While extensive studies have been reported on buckled straight ribbons compressed by a horizontal indenter, oblique compressions with an inclined indenter still remain unexplored, where distinct deformation paths and high-order bifurcation would emerge. This work investigates bifurcation and mode transition of buckled ribbons through combined theoretical/computational modeling and experimental observations. Development of a theoretical model allows establishment of phase diagrams for three deformation paths and six deformation modes that could possibly emerge at different assembly or loading parameters. The multi-step bifurcation, which has been verified experimentally, can be attributed mainly to the Euler buckling of flattened segments or the instability of multiple waves. By harnessing the multi-step bifurcation, a tri-stable mesostructure with a buckled ribbon and an assistive position limiting ribbon is demonstrated, and used to design reconfigurable 3D antennas capable of finely tuning the central frequency or the radiation pattern.

Synchronization in the network-frustrated coupled oscillator with attractive-repulsive frequencies.

We investigate the synchronization behavior of a generalized useful mode of the emergent collective behavior in sets of interacting dynamic elements. The network-frustrated Kuramoto model with interaction-repulsion frequency characteristics is presented, and its structural features are crucial to capture the correct physical behavior, such as describing steady states and phase transitions. Quantifying the effect of small-world phenomena on the global synchronization of the given network, the impact of the random phase-shift and their mutual behavior shows particular challenges. In this paper, we derive the phase-locked states and identify the significant synchronization transition points analytically with exact boundary conditions for the correlated and uncorrelated degree-frequency distributions and their full stability analysis. We find that a supercritical to subcritical bifurcation transition occurs depending on the synchronic transition points, characterized by the power scale of the network for the correlated degree frequency and the largest eigenvalue of the network in the uncorrelated case. Furthermore, our frustrated degree-frequency distribution brings us to the classical Kuramoto model with all-to-all coupling, with β=1/2 for the correlated case and λ_{N}=1 for the uncorrelated distribution. In addition, the interplay between the network topology and the frustration forms a powerful alliance, where they control the synchronization ability of the generalized model without affecting its stability.

Topological near fields generated by topological structures.

The central idea of metamaterials and metaoptics is that, besides their base materials, the geometry of structures offers a broad extra dimension to explore for exotic functionalities. Here, we discover that the topology of structures fundamentally dictates the topological properties of optical fields and offers a new dimension to exploit for optical functionalities that are irrelevant to specific material constituents or structural geometries. We find that the nontrivial topology of metal structures ensures the birth of polarization singularities (PSs) in the near field with rich morphologies and intriguing spatial evolutions including merging, bifurcation, and topological transition. By mapping the PSs to non-Hermitian exceptional points and using homotopy theory, we extract the core invariant that governs the topological classification of the PSs and the conservation law that regulates their spatial evolutions. The results bridge singular optics, topological photonics, and non-Hermitian physics, with potential applications in chiral sensing, chiral quantum optics, and beyond photonics in other wave systems.

Order pattern recurrence for the analysis of complex systems

In this paper, order pattern recurrence plot (OPRP) and order pattern recurrence quantification analysis (OPRQA) are proposed to quantify recurrence characteristics of complex systems. The method uses the recurrence of order patterns in symbolic sequence to explore the order structure of the original data, in which the color of the symbol is employed to distinguish different order patterns. The main advantage of the approach is its robustness with respect to non-stationary data and low requirements for the length of the required time series. The method is demonstrated to be effective in synthetic data and real data. As demonstrated in simulation models, the conclusion is that the method can not only make a distinction between chaotic system and random noise but also quantify various bifurcation transition scenarios like period doubling or other phenomena associated with chaos-to-chaos transitions in logistic map. In empirical analysis, this approach helps observe the different performance of the time series before, during and after the financial crisis. In addition, differences of stocks between developing countries and developed countries are reflected in OPRP and quantified by the corresponding OPRQA. The proposed method brings together recurrence plots and symbolic dynamics to empower researchers with effective means to visualize and quantify complex behavior in dynamic system.

Current bifurcation, reversals and multiple mobility transitions of dipole in alternating electric fields

Anomalous transports of dipole in alternating electric fields are investigated by means of numerical calculation of its average angular velocity (or current). Our results show that the alternating electric fields can make the dipole exhibit many interesting transport behaviors. There exist current bifurcation and multiple current reversal phenomena about frequency of the alternating electric fields in the system in the absence of constant bias force, while many platforms appear in the curve of its average angular velocity vs. the force, i.e., multiple mobility transitions phenomenon in the presence of the constant force, dependent on frequencies of the alternating electric fields. Further investigation indicates that the multiple mobility transitions are attributed to the traveling forces on the dipole. Intrinsic physical mechanism and conditions for the characteristic dynamical behaviors to occur are also discussed in detail. These findings will possess crucial significance for optimizing heating control in the alternating electric fields.

Mechanics of tension-induced film wrinkling and restabilization: a review.

Wrinkling of thin films under tension is omnipresent in nature and modern industry, a phenomenon which has aroused considerable attention during the past two decades because of its intricate nonlinear behaviours and intriguing morphology changes. Here, we review recent advancements in the mechanics of tension-induced film wrinkling and restabilization, by identifying three major stages of its progress: small-strain (less than ) wrinkling of stiff sheets, finite-strain (up to ) wrinkling and restabilization (isola-centre bifurcation) of soft films, and the effects of curved configurations and material properties on pattern formation. Growing demand for fundamental understanding, quantitative prediction and precise tracking of secondary bifurcation transitions in morphological evolution of thin films helps to advance finite-strain plate/shell theories and sophisticated modelling methods. This progress not only promotes our insightful understanding of complex instability behaviour but also reveals novel phenomena and sheds light on developing wrinkle-tunable membrane structures and functional surfaces.

Instability of electroconvection in viscoelastic fluids induced by strong unipolar injection between two coaxial cylinders

The effect of electric field on the flow of viscoelastic fluids has been investigated and the bifurcation and flow pattern transition processes have been mainly studied through numerical simulation considering a coaxial cylinder model. Different bifurcations have been found with transition among them and accordingly critical criteria of bifurcation also change. Plus, among these bifurcation paths, the swinging, rotating, splitting, and vanishing behaviors of charge void regions which are quite different from those of Newtonian fluids, have been observed.

Hopf and quasi-periodic Hopf bifurcations and deterministic coherence in coupled Duffing-Holmes and Van der Pol oscillators: the Arnol’d resonance web

A survey on coupled Van der Pol (VDP) and Duffing-Holmes (DH) oscillators, a model widely encountered in various branches of Physics and engineering is done. A bifurcation analysis of the model is presented when both unidirectional and mutual couplings are considered. It is found that stable, unstable and chaotic behaviors appear in the models. Using the method of charts of dynamics regimes in parameter planes, numerical study of the parameters space of the initial differential equations is done. Results from both approaches are compared and discussed. Features of the bifurcation picture are discussed when varying control parameters and analysis of slow-flow equations is presented. We show that the local bifurcation transition from an invariant one-torus (IT1) to an invariant two-torus (IT2) is caused by a Neimark-Saker (NS) bifurcation, also known as a one dimension-higher quasi-periodic Hopf (QH) bifurcation, these by analysing the graph of Lyapunov exponents. We observe that in the mutual coupling case, a complex structure generally-called Arnold’s resonance web phenomenon appears. Numerical simulations are compared to experimental measurements to illustrate the above behaviors. We also note the birth of coherence resonance in the slave oscillator for a certain coupling strength in the case of unidirectional coupling. We also analyze the effect of the coupling strength on the generalized synchronization between the master system and the slave system in the case of unidirectional coupling, through the concept of mutual false nearest neighbors. It allows us to understand when and how closeness in response space implies closeness in driving space.