Supercritical Pitchfork Bifurcation Research Articles

Enhancing the performance of coherent Ising machines in the large-noise regime with a fifth-order nonlinearity

Coherent Ising machines (CIMs), leveraging the bistable physical properties of coherent light to emulate Ising spins, exhibit great potential as hardware accelerators for tackling complex combinatorial optimization problems. Recent advances have demonstrated that the performance of CIMs can be enhanced either by incorporating large random noise or higher-order nonlinearities, yet their combined effects on CIM performance remain mainly unexplored. In this work, we develop a numerical CIM model that utilizes a tunable fifth-order polynomial nonlinear dynamic function under large noise levels, which has the potential to be implemented in all-optical platforms. We propose a normal form of a CIM model that allows for both supercritical and subcritical pitchfork bifurcation operational regimes, with fifth-order nonlinearity and tunable hyperparameters to control the Ising spin dynamics. In the benchmark studies, we simulate various sets of MaxCut problems using our fifth-order polynomial CIM model. The results show a significant performance improvement, achieving an average of 59.5% improvement in median time-to-solution (TTS) and an average of 6 times improvement in median success rate (SR) for dense Maxcut problems in the BiqMac library, compared to the commonly used third-order polynomial CIM model with low noise. The fifth-order polynomial CIM model in the large-noise regime also shows better performance trends as the problem size scales up. These findings reveal the enhancements on the computational performance of Ising machines in the large-nose regime from fifth-order nonlinearity, showing important implications for both simulation and hardware perspectives.

Hierarchical Symmetry-Breaking Model for Stem Cell Differentiation

Waddington envisioned stem cell differentiation as a marble rolling down a hill, passing through hierarchically branched valleys representing the cell’s temporal state. The terminal valleys at the bottom of the hill indicate the possible committed cells of the multicellular organism. Although originally proposed as a metaphor, Waddington’s hypothesis establishes the fundamental principles for characterizing the differentiation process as a dynamic system: the generated equilibrium points must exhibit hierarchical branching, robustness to perturbations (homeorhesis), and produce the appropriate number of cells for each cell type. This article aims to capture these characteristics using a mathematical model based on two fundamental hypotheses. First, it is assumed that the gene regulatory network consists of hierarchically coupled subnetworks of genes (modules), each modeled as a dynamical system exhibiting supercritical pitchfork or cusp bifurcation. Second, the gene modules are spatiotemporally regulated by feedback mechanisms originating from epigenetic factors. Analytical and numerical results show that the proposed model exhibits self-organized multistability with hierarchical branching. Moreover, these branches of equilibrium points are robust to perturbations, and the number of different cells produced can be determined by the system parameters.

Solar Influences on Winter Atmospheric Circulation Over the Ural, North Atlantic and North Pacific Regions: Pitchfork‐Like Relationships

AbstractLi's previous study found a one‐to‐two correspondence relationship between the total solar irradiance (TSI) and monthly anomalies of the Ural atmospheric circulation during winter months. This relationship roughly meets the so‐called supercritical pitchfork bifurcation model, hence termed a pitchfork‐like (PL) relationship. Here, we extend this study and provide more evidence to support previous findings. By analyzing data from 192 winter months and estimating the joint probability density function, we find that over the Ural, North Atlantic, and North Pacific regions, there are all PL relationships between the TSI and geopotential height anomalies at pressure levels no higher than 700 hPa. When the TSI gradually decreases and passes a critical threshold of 1360.9 W m−2, the preferred circulation patterns undergo a regime transition from one regime to two others. When TSI < 1360.9 W m−2, positive and negative geopotential height anomalies occur at nearly equal frequencies within the same TSI range. This quantitative relation is met in almost the whole troposphere in each region. We suggest that the occurrence of PL relationships may be associated with the interaction between the westerly flow and topographic barriers; The decrease in TSI can regulate this interaction by reducing the stability of background westerly flow over all three regions, via increasing the land‐ocean thermal contrast and altering the planetary wavenumber‐2 wave at the mid‐high latitudes. Our work highlights the same quantitative relation between the TSI and the monthly mean geopotential height anomalies of three regions in winter, which has important implications for seasonal climate prediction.

A novel entanglement functions-based 4D fractional-order chaotic system and its bifurcation analysis

A novel 4D fractional-order chaotic entanglement system based on sinusoidal functions is established in this paper. We aim to reveal the relationship between the dynamical behavior of the new system and its entanglement coefficients. It is found that the equilibrium point of the system varies regularly with the successive change of the entanglement coefficient. The supercritical pitchfork bifurcation phenomenon of the new system is discussed based on the fractional-order stability theory. Furthermore, sufficient conditions and threshold for supercritical Hopf bifurcation caused by the entanglement coefficient are provided. Finally, the route to chaos of the new system is explored utilizing multiple numerical indicators, such as spectral entropy complexity, bifurcation diagrams, Lyapunov exponential spectrum, phase portraits, and 0–1 test curves. The results indicate that in addition to various chaotic attractors, there are phenomena such as period-doubling bifurcations, period windows, and coexisting symmetric attractors (periodic or chaotic).

Bifurcation of equilibrium positions for ellipsoidal particles in inertial shear flows between two walls

We conducted a systematic numerical investigation of spherical, prolate and oblate particles in an inertial shear flow between two parallel walls, using smoothed particle hydrodynamics (SPH). It was previously shown that above a critical Reynolds number, spherical particles experience a supercritical pitchfork bifurcation of the equilibrium position in shear flow between two parallel walls, namely that the central equilibrium position becomes unstable, leading to the emergence of two new off-centre stable positions (Fox et al., J. Fluid Mech., vol. 915, 2021). This phenomenon was unexpected given the symmetry of the system. In addition to confirming this finding, we found, surprisingly, that ellipsoidal particles can also return to the centre position from the off-centre positions when the particle Reynolds number is further increased, while spherical particles become unstable under this increased Reynolds number. By utilizing both SPH and the finite element method for flow visualization, we explained the underlining mechanism of this reverse of bifurcation by altered streamwise vorticity and symmetry breaking of pressure. Furthermore, we expanded our investigation to include asymmetric particles, a novel aspect that had not been previously modelled, and we observed similar trends in particle dynamics for both symmetric and asymmetric ellipsoidal particles. While further validation through laboratory experiments is necessary, our research paves the road for development of new focusing and separation methods for shaped particles.

Spontaneous locomotion of a symmetric squirmer

The squirmer is a popular model to analyse the fluid mechanics of a self-propelled object, such as a micro-organism. We demonstrate that some fore–aft symmetric squirmers can spontaneously self-propel above a critical Reynolds number. Specifically, we numerically study the effects of inertia on spherical squirmers characterised by an axially and fore–aft symmetric ‘quadrupolar’ distribution of surface-slip velocity; under creeping-flow conditions, such squirmers generate a pure stresslet flow, the stresslet sign classifying the squirmer as either a ‘pusher’ or ‘puller’. Assuming axial symmetry, and over the examined range of the Reynolds number $Re$ (defined based upon the magnitude of the quadrupolar squirming), we find that spontaneous symmetry breaking occurs in the puller case above $Re \approx 14.3$ , with steady swimming emerging from that threshold consistently with a supercritical pitchfork bifurcation and with the swimming speed growing monotonically with $Re$ .

Nonlinear dynamic morphing of conical bistable dielectric elastomer actuator

The bistable dielectric elastomer actuator (BDEA) possesses two stable positions which offers notable advantages of stable-state self-maintenance, fast response, and threshold snap-through characteristic in comparison with conventional dielectric elastomers. However, the strong nonlinearity induced by the coupling among materials, structure, and electrostatic fields greatly affect the dynamic response and gives rise to stability issues. Hence, a novel BDEA is proposed by introducing DEA film centrally connected with one mass block and linear spring, and the bistability can be adjusted by applying external voltage. A nonlinear dynamical model considering the electro-mechanical coupling effects is established using the Euler-Lagrange method, with which the snap-through procedure is theoretically analyzed and validated through the analytic method and finite element method. The influences of the electric actuation and structural parameters on the number of stable states and natural frequency are analyzed. Additionally, the supercritical pitchfork bifurcation and saddle-node bifurcation are investigated through dynamic analysis under forced vibration. Furthermore, the ranges of electrical actuation parameters can be determined for preventing the bifurcation phenomena under parametric excitations. Moreover, an active morphing strategy for achieving nonlinear dynamic morphing between steady states of BDEA using drive voltage is obtained, thereby enhancing the versatility of conical BDEA.

Coriolis acceleration and critical slowing-down: A quantitative laboratory experiment

We experimentally investigate the motion of a pendulum on a turntable. The dynamics of this conical pendulum experiment are very rich and can be studied both at the undergraduate and graduate levels. At low rotational frequency of the turntable, we measure the Coriolis acceleration. Increasing the rotational frequency, we experimentally demonstrate a supercritical pitchfork bifurcation: above a critical rotational frequency, the pendulum arm spontaneously rises up. Beyond the characterization of the equilibrium pendulum angle, we evidence the so-called critical slowing down corresponding to the increase in the pendulum period when approaching the critical rotational frequency. Bifurcation and critical slowing down are key concepts in the study of critical phenomena that are seldom illustrated experimentally. All our experimental measurements are in excellent quantitative agreement with the theory we provide.

Flow stability and sequence of bifurcations in a cubic cavity driven by a constant shear stress

The progressive destabilisation of the incompressible flow in a cubical cavity driven by a constant shear stress is investigated numerically. To that end, one of the square faces of the cube is subjected to a constant shear stress parallel to two opposite edges of that face. The three-dimensional steady basic flow loses its mirror symmetry through a supercritical pitchfork bifurcation leading to a pair of steady stable non-symmetric flow states that are antisymmetric to each other. Upon increase of the strength of the driving, these non-symmetric equilibria become unstable via a Hopf bifurcation resulting in two limit cycles. The bifurcations are investigated using classical linear stability analyses as well as nonlinear simulations. Upon a further increase of the driving shear stress, the limit cycles destabilise through bursts triggering a complex interplay between the unstable equilibria. The transition to turbulence resembles the Pomeau–Manneville scenario.

Pitchfork bifurcation and vibrational resonance in a neutral delayed system

Abstract The pitchfork bifurcation and vibrational resonance phenomenon are analytically and numerically investigated in the neutral delayed system. The results show that the neutral delayed time induces the pitchfork bifurcation transiting between supercritical and subcritical cases while the neutral delayed strength may induce supercritical pitchfork bifurcation. The neutral delayed time can also induce the periodic or quasi-periodic vibrational resonance. This period simultaneously contains the period of the low-frequency harmonic excitation and period of the high-frequency harmonic excitation. The results extend the investigations of the pitchfork bifurcation and vibrational resonance.

Inertial migration of a sphere in plane Couette flow

We study the inertial migration of a torque-free neutrally buoyant sphere in wall-bounded plane Couette flow over a wide range of channel Reynolds numbers $Re_c$ in the limit of small particle Reynolds number ( $Re_p\ll 1$ ) and confinement ratio ( $\lambda \ll 1$ ). Here, $Re_c = V_{wall}H/\nu$ , where $H$ denotes the separation between the channel walls, $V_\text {wall}$ denotes the speed of the moving wall, and $\nu$ is the kinematic viscosity of the Newtonian suspending fluid. Also, $\lambda = a/H$ , where $a$ is the sphere radius, with $Re_p=\lambda ^2 Re_c$ . The channel centreline is found to be the only (stable) equilibrium below a critical $Re_c$ ( $\approx 148$ ), consistent with the predictions of earlier small- $Re_c$ analyses. A supercritical pitchfork bifurcation at the critical $Re_c$ creates a pair of stable off-centre equilibria, located symmetrically with respect to the centreline, with the original centreline equilibrium becoming unstable simultaneously. The new equilibria migrate wall-ward with increasing $Re_c$ . In contrast to the inference based on recent computations, the aforementioned bifurcation occurs for arbitrarily small $Re_p$ provided that $\lambda$ is sufficiently small. An analogous bifurcation occurs in the two-dimensional scenario, that is, for a circular cylinder suspended freely in plane Couette flow, with the critical $Re_c$ being approximately $110$ .

Dynamical behavior of a particle-doped multi-segment dielectric elastomer minimal energy structure

The dynamic behavior of dielectric elastomers (DEs) has significant influence on their performance. The present study investigates the nonlinear dynamics of particle-doped multi-segmented DE minimum energy structures (DEMESs). To simulate the multi-segment DEMES, we consider each segment as a combination of hyperelastic film and elastic beam and obtain the ordinary differential equations governing the system dynamics based on the Euler–Lagrange equations. Due to the difficulty in measuring various physical parameters of DEs in practice, we utilize experimental data from a single-segment DE and employ a physics-informed neural network to predict the unknown parameters of the DE and the framework, such as stiffness K bb and doping volume fraction ϕ. Based on these predictions, nonlinear analysis is performed for the multi-segment system. Stability analyses of the motion equations reveal that the system exhibits a supercritical pitchfork bifurcation with hyperelastic thin film pre-stretching as the bifurcation parameter. For the three-segment DEMES, there are eight stable modes, but only four are illustrated in the bifurcation diagram due to the identical parameter settings for each segment. The amplitude-frequency curves under different AC voltage loads indicate the presence of harmonic, superharmonic, and subharmonic resonances in the system, with varying frequencies and magnitudes depending on the applied load. The Poincaré maps of the time response demonstrate that the system response is predominantly quasiperiodic. Under low voltage loads, the system exhibits periodic oscillations, while under certain high voltage loads, chaotic behavior emerges, characterized by strong nonlinearity in the time-dependent curves and non-periodicity in the Poincaré maps. This study provides insights into the present mathematical model in the motion control of DEMES.

Exploring multi-stability in three-dimensional viscoelastic flow around a free stagnation point

Fluid elements passing near a stagnation point experience finite strain rates over long persistence times, and thus accumulate large strains. By the numerical optimization of a microfluidic 6-arm cross-slot geometry, recent works have harnessed this flow type as a tool for performing uniaxial and biaxial extensional rheometry (Haward et al., 2023 [5,6]). Here we use the microfluidic ‘Optimized-shape Uniaxial and Biaxial Extensional Rheometer’ (OUBER) geometry to probe an elastic flow instability which is sensitive to the alignment of the extensional flow. A three-dimensional symmetry-breaking instability occurring for flow of a dilute polymer solution in the OUBER geometry is studied experimentally by leveraging tomographic particle image velocimetry. Above a critical Weissenberg number, flow in uniaxial extension undergoes a supercritical pitchfork bifurcation to a multi-stable state. However, for biaxial extension (which is simply the kinematic inverse of uniaxial extension) the instability is strongly suppressed. In uniaxial extension, the multiple stable states align in an apparently random orientation as flow joining from four neighbouring inlet channels passes to one of the two opposing outlets; thus forming a mirrored asymmetry about the stagnation point. We relate the suppression of the instability in biaxial extension to the kinematic history of flow under the context of breaking the time-reversibility assumption.

Modeling of heat transfer at local convection over a vertical throughflow in a two-layered air-heat generating porous system

Convective heat transfer in an air layer partially filled with a heat-generating granular porous medium is studied. There is a slow seepage of air through the layer in the vertical direction with a constant velocity. Equal temperatures are maintained at the outer solid permeable boundaries, and the heat source strength is constant within the porous sublayer and is proportional to the solid volume fraction. The permanent heat generation within the porous sublayer, combined with the vertical throughflow, causes a nonlinear thermal profile which is conducive for convection to occur. The Boussinesq approximation and Darcy's law are used to describe this convection. Numerical solution of the nonlinear convective problem is obtained on the basis of Newton's method. In the limiting case, the numerical data for the onset of convection are compared with the results of the earlier paper of the authors, where a linear stability theory and a method for constructing of the fundamental system of partial solution vectors have been applied, and with the data by other authors. The stationary regimes of local convection, which occurs in an “air – heat-generating porous medium-air” system over the basic vertical throughflow, and its effect on the heat transfer from the porous air sublayer with the growth of supercriticality are studied. It is shown that, depending on the velocity of the basic throughflow (the Peclet number), convection excitation can be both soft (due to supercritical pitchfork bifurcation) and hard (when the loss of stability of the basic throughflow is accompanied by subcritical pitchfork bifurcation that gives rise to an unstable secondary convective regime). This secondary regime is replaced by a stable tertiary convective regime with increasing supercriticality. It has been found that the total heat transfer rate for the upward basic throughflow exceeds that for the downward basic throughflow significantly, and that local convection at any direction of this throughflow increases the heat transfer rate in the system. An increase in the Nusselt number with the growth of supercriticality is recorded. However, a noticeable contribution of local convection to the total heat transfer is observed only when all values of the Pecklet number are negative and its positive values are lower than 2.

Symmetry-breaking bifurcations and excitations of solitons in linearly coupled NLS equations with PT-symmetric potentials

We address symmetry-breaking bifurcations (SBBs) in the ground state (GS) and dipole-mode (DM) solitons of the one-dimensional linearly coupled NLS equations, modelling the propagation of light in a dual-core planar waveguide with the Kerr nonlinearity and two types of P T -symmetric potentials. The P T -symmetric potential is employed to obtain different types of solutions. A supercritical pitchfork bifurcation occurs in families of symmetric solutions of both the GS and DM types. A novel feature of the system is interplay between breakings of the P T and inter-core symmetries. Stability of symmetric GS and DM modes and their asymmetric counterparts, produced by SBBs of both types, is explored via the linear-stability analysis and simulations. It is found that the instability of P T -symmetric solutions takes place prior to the inter-core symmetry breaking. Surprisingly, stable inter-core-symmetric GS solutions may remain stable while the P T symmetry is broken. Fully asymmetric GS and DM solitons are only partially stable. Moreover, we construct symmetric and asymmetric GS solitons under the action of a pure imaginary localized potential, for which the SBB is subcritical. These results exhibit that stable solitons can still be found in dissipative systems. Finally, excitations of symmetric and asymmetric GS solitons are investigated by making the potential’s parameters or the system’s coupling constant functions, showing that GS solitons can be converted from an asymmetric shape onto a symmetric one under certain conditions. These results may pave the way for the study of linear and nonlinear phenomena in a dual-core planar waveguide with P T potential and related experimental designs.

The static and dynamic regimes of the bistable asymmetric cross-ply composite laminated plates

For the static and dynamic regimes of the bistable asymmetric cross-ply composite laminated plates, the theoretical model and the finite element model (FEM) are established. The 2D and 3D diagrams for the supercritical pitchfork bifurcation, which are dominated by the length, the width, the aspect ratio, the ratio of side to thickness, the number of layers and the temperature difference, can be used to describe the static regimes. The restoring force curves (RFCs) and the potential energy curves (PECs) are shown. The single-well vibration, the limit-cycle oscillations, the multiple-period snap-through, and the chaotic snap-through are the dynamic regimes that predominate.

Bifurcation and chaos detection of a fractional Duffing–van der Pol oscillator with two periodic excitations and distributed time delay

This paper analytically and numerically investigates the dynamical characteristics of a fractional Duffing–van der Pol oscillator with two periodic excitations and the distributed time delay. First, we consider the pitchfork bifurcation of the system driven by both a high-frequency parametric excitation and a low-frequency external excitation. Utilizing the method of direct partition of motion, the original system is transformed into an effective integer-order slow system, and the supercritical and subcritical pitchfork bifurcations are observed in this case. Then, we study the chaotic behavior of the system when the two excitation frequencies are equal. The necessary condition for the existence of the horseshoe chaos from the homoclinic bifurcation is obtained based on the Melnikov method. Besides, the parameters effects on the routes to chaos of the system are detected by bifurcation diagrams, largest Lyapunov exponents, phase portraits, and Poincaré maps. It has been confirmed that the theoretical predictions achieve a high coincidence with the numerical results. The techniques in this paper can be applied to explore the underlying bifurcation and chaotic dynamics of fractional-order models.

Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.

Nonlinear parametric resonances of a rotating composite thin-walled beam under aerodynamic force and hygrothermal environment

Considering combined effects of aerodynamic force and hygrothermal environment, the nonlinear parametric resonance of a composite thin-walled beam subjected to a time-dependent rotating speed is investigated in this work. The rotating speed is presumed to comprise a small disturbance harmonically changing one with a steady one. The dynamical model of the system is obtained with the help of established dynamic model in existing research. Then the nonlinear partial differential equation of the system is discretized by the Galerkin method. Afterwards, the low-frequency and high-frequency resonances of the system are analyzed via the multiple scales method to discuss the combined effect of rotating speed, pitch angle, setting angle, fiber orientation angle, temperature-humidity and inflow ratio on the amplitude-frequency responses and the unstable region. In addition, the Routh-Hurwitz criterion is employed to determine the stability conditions of the trivial and non-trivial solutions. The results show that both sub- and super-critical pitchfork bifurcations will occur to the system in parametric resonances regime, accompanied with stable and unstable non-trivial solutions simultaneously. Particularly, the mechanism of the influence of hygrothermal environment is deeply studied, which reveals a competitive relation between the change of mechanical properties and additional stress brought by the change of temperature and humidity.

Non-Oberbeck-Boussinesq effects and sub-critical primary bifurcations in porous media convection

Large temperature variations and highly unstable density stratification may exist between the cold and hot boundaries of the porous non-vacuum gas filled insulation layers of cryogenic storage tanks. For example, the density can vary by about a factor 3 in N2 filled liquefied natural gas (LNG) tank insulation layers and factor 15 in H2 filled liquid hydrogen (LH2) tank insulation layers. In such cases, the bifurcation of the convective states from the conduction state could be sub-critical leading to sudden jumps and hysteresis effects in heat transfer and boil-off rates. This work examines the impact of large physical property variation and non-Oberbeck-Boussinesq (non-OB) effects on the nature of bifurcation of convective states in gas filled porous insulation layers. Specifically, we show that the well-known super-critical (pitchfork) bifurcation of convective states that appear under the OB approximation changes to sub-critical bifurcation when the temperature difference exceeds a critical value. Further, under such large temperature variations, the emerging convective solutions could be highly asymmetric with center of circulation much closer to the cold boundary. We present linear stability results, local bifurcation analysis as well as numerical simulations of the full model equations for the two selected LNG and LH2 cases.