Bifurcation Modes Research Articles

Asymmetrical temporal dynamics of edge modes in Su-Schrieffer-Heeger lattice with Kerr nonlinearity

Optical bistability and oscillating phases exist in a Sagnac interferometer and a single ring resonator made of a χ(3) nonlinear medium where the refractive indices are modulated by the light intensity due to the Kerr nonlinearity. An array of coupled nonlinear ring resonators behave similarly but with more complexity due to the presence of the additional couplings. Here we theoretically demonstrate the bifurcation of edge modes, which leads to optical bistability in the Su-Schrieffer-Heeger lattice with the Kerr nonlinearity. We also demonstrate periodic and chaotic switching behaviors in an oscillating phase resulting from the coupling between the edge mode and bulk modes with different chiralities, i.e., clockwise and counterclockwise circulations. Published by the American Physical Society 2024

Bifurcation and spectral instability of asymptotic quasinormal modes in the modified Pöschl-Teller effective potential

Bifurcation and spectral instability of asymptotic quasinormal modes in the modified Pöschl-Teller effective potential

Experimental study on upstream pressure characteristics of rotating detonation engine with methane and oxygen-enriched air

The interaction mechanism between the rotating detonation engine (RDE) and its upstream is one key issue that needs to be explored when considering its application. In this study, experimental tests with methane and 50% oxygen-enriched air are conducted to investigate the relationship between upstream pressure characteristics and the detonation wave. As a result of operational considerations, a hollow chamber and a rear ignitor are used and the initiation process is discussed. Upstream pressure characteristics are analyzed under varied injection pressures and equivalence ratios. Stable modes including single-wave and double-wave modes are recorded in most cases. Notably, two unstable modes are also captured at low injection pressure in which the detonation quenches quickly. The results highlight that significant feedback pressure pulsations synchronize with the rotation frequency of detonation waves after ignition due to the upstream oblique shock wave (UOSW). But the inherent frequency of injection should also be emphasized at elevated injection pressure. The detonation wave can also cause injection blockage, particularly at low injection pressure, leading to the bifurcation of unstable mode. Such unstable is attributed to an inadequate upstream pressure increase after ignition, which hinders the injection reconstruction. Further analysis reveals that higher injection pressures or lower equivalence ratios diminish the differential between the detonation wave and upstream pressure, and consequently, the pulsation amplitude. An increase in the detonation wave number can also reduce the pulsation amplitude, which can even be lower than 10% in a double-wave mode. The results are beneficial for integrating RDE and its upstream components.

Electromechanical Deformations and Bifurcations in Soft Dielectrics: A Review.

Dielectric elastomers have attracted considerable attention both from academia and industry alike over the last two decades due to their superior mechanical properties. In parallel, research on the mechanical properties of dielectrics has been steadily advancing, including the theoretical, experimental, and numerical aspects. It has been recognized that the electromechanical coupling property of dielectric materials can be utilized to drive deformations in functional devices in a more controllable and intelligent manner. This paper reviews recent advances in the theory of dielectrics, with specific attention focused on the theory proposed by Dorfmann and Ogden. Additionally, we provide examples illustrating the application of this theory to analyze the electromechanical deformations and the associated bifurcations in soft dielectrics. We compared the bifurcations in elastic and dielectric materials and found that only compressive bifurcation modes exist in elastic structures, whereas both compressive and tensile modes coexist in dielectric structures. We summarize two proposed ways to suppress and prevent the tensile bifurcations in dielectric materials. We hope that this literature survey will foster further advancements in the field of the electroelastic theory of soft dielectrics.

Nonlinear in-plane buckling of small-curved and large-curved FG porous microbeams via strain gradient-based isogeometric collocation formulations

In the current investigation, for the first time, the changes in the limit loads and equilibrium branches associated with the nonlinear in-plane stability characteristics of curved microbeams are explored in the presence of different microstructural gradient tensors. In this regard, multiple microsize-dependent equilibria are analyzed relevant to thermomechanical loaded small-curved, medium-curved, and large-curved microbeams made of functionally graded porous (FGP) metal reinforced with nanofillers possessing clamped end supports. To this purpose, based upon the strain gradient elasticity within the framework of the third-order shear flexible curved beam model, the isogeometric collocation formulations incorporating Greville abscissae are constructed resulting in higher-continuity characters as well as remarkable accuracy for higher-order approximations. It is deduced that for the small-curved FGP reinforced microbeam, no limit load can be found due to the absence of the buckling phenomenon, but after rising the temperature by an enough amount, the initial instability mode appears. However, for the medium-curved FGP reinforced microbeam, the limit instability mode of buckling occurs which results in the normalized upper limit load equal to 0.7161 based on the classical theory and 0.7606 based on the strain gradient elasticity (6.21% enhancement). Also, it results in the normalized lower limit load equal to 0.3060 based on the classical theory and 0.3521 based on the strain gradient elasticity (15.07% enhancement). On the other hand, for the large-curved FGP reinforced microbeam, the bifurcation mode of buckling occurs which results in the normalized upper limit load equal to 1.0494 based on the classical theory and 1.1358 based on the strain gradient elasticity (8.23% enhancement). Also, it results in the normalized lower limit load equal to 0.2225 based on the classical theory and 0.2513 based on the strain gradient elasticity (12.94% enhancement).

Plasticity-mediated deformation instabilities in thin film-compliant substrate systems: direct three-dimensional simulations

AbstractSurface wrinkles driven by mechanical instability commonly form in thin-film structures attached to a compliant substrate. In this study, a recently developed computational approach is employed to simulate the formation and transformation of wrinkles involving plastic yielding of the thin film. The three-dimensional (3D) finite element models contain an embedded imperfection at the film-substrate interface, serving to trigger the bifurcation modes. Successful application of this technique to allow for film plasticity is demonstrated, including the evolution of 3D surface patterns and their correlation with the overall load–displacement response. The simulations reveal that plastic yielding transforms the surface instability patterns into more localized forms. Under uniaxial loading, the sinusoidal elastic wrinkles undergo the wrinkle-to-fold transition. With equi-biaxial loading, the initial square-checkerboard array turns into continuous tall ridges along the 45° directions. In both loading modes, the plasticity-induced instability patterns are only partially relieved upon unloading, leaving permanent features on the surface.

Pattern formation of a volume-filling chemotaxis model with a bistable source

<p>In this paper, the pattern formation of a volume-filling chemotaxis model with bistable source terms was studied. First, it was shown that self-diffusion does not induce Turing patterns, but chemotaxis-driven instability occurs. Then, the asymptotic behavior of the chemotaxis model was analyzed by weakly nonlinear analysis with the method of multiple scales. When the chemotaxis coefficient exceeded a threshold value and there was a single unstable mode, the supercritical and subcritical bifurcation of the model was discussed. The amplitude equations and the asymptotic expressions of the patterns were obtained. When the chemotaxis coefficient was large enough, the two-mode competition behavior of the model with two unstable modes was analyzed, and the corresponding amplitude equations and the asymptotic expressions of the patterns were obtained. Finally, numerical simulations were provided to further illuminate the above analytical results.</p>

A novel compound exponential locally active memristor coupled Hopfield neural network

The neural network model coupled with memristors has been extensively studied due to its ability to more accurately represent the complex dynamic characteristics of the biological nervous system. Currently, the mathematical model of memristor used to couple neural networks mainly focuses on primary function, absolute value function, hyperbolic tangent function, etc. To further enrich the memristor-coupled neural network model and take into account the motion law of particles in some doped semiconductors, a new compound exponential local active memristor is proposed and used as a coupling synapse in the Hopfield neural network. Using the basic dynamic analysis method, the system’s dynamic behaviors are studied under different parameters and the coexistence of multiple bifurcation modes under different initial values. In addition, the influence of frequency change of external stimulation current on the system is also studied. The experimental results show that the internal parameters of memristor synapses regulate the system, and the system has a rich dynamic behavior, including symmetric attractor coexistence, asymmetric attractor coexistence, large-scale chaos as shown in attached figure, and bursting oscillation. Finally, the hardware of the system is realized by the STM32 microcontroller, and the experimental results verify the realization of the system.

"Invisible" pulsation and period-doubling bifurcation of harmonic mode locking in a bidirectional fiber laser.

The harmonic mode-locking (HML) "invisible" pulsation (IP) is reported, here, in a bidirectional passively mode-locked fiber laser (BPMLFL). With the help of dispersive Fourier transform (DFT) technology, it is found that due to the alike nonlinear effects experienced by two pulse trains in HML, their evolution is consistent during the IP. Further, as the increase of pump power, period-doubling bifurcations (PDBs) can be observed based on the IP phenomenon in the HML regime, the PDB path experienced by the HML from steady to chaotic is statistically obtained. Finally, the IP and PDB in the bidirectional laser are reproduced and studied through numerical simulations. The effect of IP on the coherence of solitons is further analyzed. We believe our research results will provide new insights into the study of soliton dynamics in fiber lasers.

Remarks on bifurcation of an inflated and extended swellable isotropic tube

We consider a tubular membrane whose mechanical behavior is defined by a strain energy density function that combines the neo-Hookean model with the Demiray model. Both, the neo-Hookean and the Demiray models are isotropic material models, which found their application in the modeling of the mechanical behavior of biological soft tissue. This tubular membrane is subjected to an inner pressure, an axial stretch, and a change in the material volume due to swelling or deswelling. The interplay between the cylinder geometry and the loading conditions and different instability modes, namely, bulging bifurcation, bending bifurcation, and prismatic bifurcation, are studied. It is shown that a change in the material volume has a strong effect on the occurrence of these bifurcation modes because a change in the material volume may stabilize the cylinder against a particular bifurcation mode and may trigger another bifurcation mode at the same time. Despite the restriction to isotropic material behavior, this article shows that the material response to pressure, axial stretch, and material volume change is quite complex.

Multiple equilibrium states of a curved-sided hexagram: Part I—stability of states

The stability of the multiple equilibrium states of a hexagram ring with six curved sides is investigated. Each of the six segments is a rod having the same length and uniform natural curvature. These rods are bent uniformly in the plane of the hexagram into equal arcs of 120o or 240o and joined at a cusp where their ends meet to form a 1-loop planar ring. The 1-loop rings formed from 120o or 240o arcs are inversions of one another and they, in turn, can be folded into a 3-loop straight line configuration or a 3-loop ring with each loop in an “8” shape. Each of these four equilibrium states has a uniform bending moment. Two additional intriguing planar shapes, 6-circle hexagrams, with equilibrium states that are also uniform bending, are identified and analyzed for stability. Stability is lost when the natural curvature falls outside the upper and lower limits in the form of a bifurcation mode involving coupled out-of-plane deflection and torsion of the rod segments. Conditions for stability, or lack thereof, depend on the geometry of the rod cross-section as well as its natural curvature. Rods with circular and rectangular cross-sections will be analyzed using a specialized form of Kirchhoff rod theory, and properties will be detailed such that all four of the states of interest are mutually stable. Experimental demonstrations of the various states and their stability are presented. Part II presents numerical simulations of transitions between states using both rod theory and a three-dimensional finite element formulation, includes confirmation of the stability limits established in Part I, and presents additional experimental demonstrations and verifications.

Mechanism of vorticity amplification by elastic waves in a viscoelastic channel flow

Inertia-less viscoelastic channel flow displays a supercritical nonnormal mode elastic instability due to finite-size perturbations despite its linear stability. The nonnormal mode instability is determined mainly by a direct transition from laminar to chaotic flow, in contrast to normal mode bifurcation leading to a single fastest-growing mode. At higher velocities, transitions to elastic turbulence and further drag reduction flow regimes occur accompanied by elastic waves in three flow regimes. Here, we demonstrate experimentally that the elastic waves play a key role in amplifying wall-normal vorticity fluctuations by pumping energy, withdrawn from the mean flow, into wall-normal fluctuating vortices. Indeed, the flow resistance and rotational part of the wall-normal vorticity fluctuations depend linearly on the elastic wave energy in three chaotic flow regimes. The higher (lower) the elastic wave intensity, the larger (smaller) the flow resistance and rotational vorticity fluctuations. This mechanism was suggested earlier to explain elastically driven Kelvin-Helmholtz-like instability in viscoelastic channel flow. The suggested physical mechanism of vorticity amplification by the elastic waves above the elastic instability onset recalls the Landau damping in magnetized relativistic plasma. The latter occurs due to the resonant interaction of electromagnetic waves with fast electrons in the relativistic plasma when the electron velocity approaches light speed. Moreover, the suggested mechanism could be generally relevant to flows exhibiting both transverse waves and vortices, such as Alfven waves interacting with vortices in turbulent magnetized plasma, and Tollmien-Schlichting waves amplifying vorticity in both Newtonian and elasto-inertial fluids in shear flows.

Formability evaluation based on a constitutive model with stress rate direction dependency considering bifurcation modes

Accurately predicting the fractures that occur in materials during forming not only improves quality, but also reduces the amount of material that is discarded. Therefore, as a key technology for digital transformation to achieve carbon neutrality, it is necessary to improve the accuracy of forming analysis and establish an efficient method for predicting forming limits. Formability assessment is typically performed using ductile fracture criteria or experimental forming limit diagrams (FLDs), which require prior fracture testing. The authors have established a forming evaluation method based on three-dimensional local bifurcation theory as a versatile method that does not require destructive testing. This method can follow abrupt changes in the stress field, such as localized necking, by using a flow rule that takes into account the directionality of the stress rate. In this study, the proposed method is applied to the tensile testing and the forming analysis of a square cup drawing to analytically verify the location and onset time of local bifurcation modes that may lead to fracture.

On the frequency bifurcations of the MHD startup modes in NSTX

The observed bifurcations of the low frequency (<50 kHz) and low toroidal periodicity (n < 5) magnetohydrodynamic (MHD) activity often present in the initial part of the National Spherical Tokamak Experiment (NSTX) discharges can be explained by the evolution of the radial profile of the safety factor (q=rBϕ/RBθ) crossing multiple rational surfaces in the core. Important performance limiting instability mechanisms in the NSTX spherical tokamak are often linked to low frequency and low-n MHD activity. They are quite common in long-pulse NSTX plasmas. They can be present at the beginning of the plasma current flat-top, at the end of the discharge or during the whole duration, and they have been observed to deleteriously impact performance over a wide range of q95. An interesting feature observed in some NSTX discharges is the presence of a bifurcation in the frequency of the low n modes, as low as n = 1, that have frequencies comparable to the plasma core rotation divided by n. Equilibrium reconstructions constrained by magnetic diagnostics data and motional stark effect pitch angle radial profiles suggest that the observed bifurcations are linked to a fast evolving minimum value of q. 3D non-linear resistive MHD simulations show that these modes are ideal and exist as non-resonant before the correspondent rational surface enters the plasma.

Bifurcations of Grid-Following Rectifiers and Routes to Voltage Instability in Weak AC Grids

As indispensable components in modern power systems, grid-following rectifiers would interact with the grid. Such interaction, however, may lead to unexpected collapse of the ac bus voltage. In this paper, three routes to voltage instability are identified, namely, (i) supercritical Hopf bifurcation and successive saddle-node bifurcation of periodic orbits (SNPO); (ii) subcritical Hopf bifurcation; and (iii) departure from a stable basin of attraction. The SNPO and subcritical Hopf bifurcation are reported for the first time in rectifier-based power systems. It is also found that the rectifier-based system may undergo a Bautin bifurcation, which changes a Hopf bifurcation from supercritical to subcritical, or vice versa. The current loop of the grid-following rectifier has been found to play a crucial role in these bifurcation behaviors. Considering the inevitable and immediate voltage breakdown after the onset of a subcritical Hopf bifurcation, it is recommended to increase the bandwidth of the current loop to trigger the Bautin bifurcation, so that only supercritical Hopf bifurcation may occur. Furthermore, the original high-order and nonlinear model of the rectifier-based system can be simplified to a reduced order system by focusing on the center manifold. Based on the reduced system, the occurrence mode of Hopf bifurcation can be confirmed by inspecting the first Lyapunov coefficient. Finally, relevant bifurcation diagrams are derived, providing practical design guidelines to avoid an eventual system collapse.

Coexistence of Periodic, Chaotic and Hyperchaotic Attractors in a System Consisting of a Duffing Oscillator Coupled to a van der Pol Oscillator

Undoubtedly, multistability represents one of the most followed venues for researchers working in the field of nonlinear science. Multistability refers to the situation where a combination of two or more attractors occurs for the same rank of parameters. However, to the best of our knowledge, the situation encountered in the relevant literature is never one where periodicity, chaos and hyperchaos coexist. In this article, we study a fourth-order autonomous dynamical system composing of a Duffing oscillator coupled to a van der Pol oscillator. Coupling consists in disturbing the amplitude of one oscillator with a signal proportional to the amplitude of the other. We exploit analytical and numerical methods (bifurcation diagrams, phase portraits, basins of attraction) to shed light on the plethora of bifurcation modes exhibited by the coupled system. Several ranks of parameters are revealed where the coupled system exhibits two or more competing behaviors. In addition to the transient dynamics, the most gratifying behavior reported in this article concerns the coexistence of four attractors consisting of a limit cycle of period-n, a pair of chaotic attractors and a hyperchaotic attractor. The impact of a fractional-order derivative is also examined. A physical implementation of the coupled oscillator system is performed and the PSpice simulations confirm the predictions of the theoretical study conducted in this work.

On the helical buckling of anisotropic tubes with application to arteries

The helical stability experienced by elastic cylinders is investigated using numerical methods. A doubly fiber-reinforced incompressible nonlinear elastic tube subject to axial loading, internal pressure and twist is examined using a numerical scheme based on the modified Riks (quasi-static) procedure. Under the application of such loadings (qualitatively physiological for arteries) vessels form tortuous shapes and a range of topologically and geometrically complex morphologies. These configurations can be highly unstable owing to the nonlinear interaction (geometry, material and self contact) among the multiple bifurcation modes. The present work attempts to model such complex configurations using a thick-walled cylindrical tube. These complex morphologies are most commonly termed as helical coiling, looping and winding in the biomechanics community. We show that these bifurcations are very sensitive to the applied pressure, axial stretch and the amount of twist. Illustration of the helical buckling is then provided by considering an anisotropic constitutive model that includes both fiber stretching and fiber shearing, as opposed to previous analyses that only consider fiber stretching. The numerical implementation is achieved via user routine in the finite element code Abaqus.

Calculating quasinormal modes of Schwarzschild anti–de Sitter black holes using the continued fraction method

We investigate the scalar, gravitational, and electromagnetic quasinormal mode spectra of Schwarzschild anti-de Sitter black holes using the numerical continued fraction method. The spectra have similar, almost linear structures. With a few exceptions, the low overtone quasinormal modes are consistent with previously obtained results in the literature that use other numerical techniques. The intermediate and high overtone quasinormal modes, in comparison to the Schwarzschild case, converge very quickly to the asymptotic formulas previously obtained by analytic monodromy techniques. In addition, we find a connection between the analytic asymptotic formulas and the purely imaginary modes. In particular, these formulas can be used to predict the bifurcation of the lowest damped electromagnetic modes. Finally, we find no high overtone quasinormal modes with high oscillation frequency and low damping, which had been previously predicted.

Global modes of variable-viscosity two-phase swirling flows and their triadic resonance

The linear and nonlinear dynamics of two-phase swirling flows produced by the Grabowski–Berger profile under the influence of a viscosity stratification are investigated. We perform axisymmetric nonlinear simulations and fully three-dimensional linear global stability analysis of the flow for several swirl numbers $S$ and viscosity ratios $\tilde {\mu }$ . They are accompanied by nonlinear three-dimensional simulations and subsequent modal analysis using the bispectral mode decomposition, recently introduced by Schmidt (Nonlinear Dyn., vol. 102, issue 4, 2020, pp. 2479–2501). We find a pronounced destabilising effect of the viscosity stratification on both the onset of axisymmetric vortex breakdown and helical instability that is linked to the required shear stress continuity across the interface. Consequently, destabilisation is shifted to lower $S$ as compared with an equivalent flow with uniform viscosity. Further, the stability analysis reveals the simultaneous destabilisation of two global modes with wavenumbers $m=1$ and $m=2$ that have harmonic frequencies. The analysis of the nonlinear flow reveals a strong triadic resonance between these modes that governs the nonlinear dynamics and leads to a rapid departure from the linear dynamics. At larger swirl, the bifurcation of additional modes initiates an interaction cascade by means of triadic resonance which is elucidated by the bispectral analysis. It leads to the emergence of a variety of additional modes in the nonlinear flow. This study contributes to an improved understanding of the influence of viscosity stratification on the onset of vortex breakdown and the destabilisation of global modes. Further, it provides a clear picture of the dynamics of swirling flows with a codimension-two point and related triadic interaction of two global modes at harmonic frequencies and wavenumbers.

Periodical Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model with Feedback Control

The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.