Subcritical Bifurcation Research Articles

The effect of aspect ratio and mass ratio on the flow-induced flutter of a thin flexible sheet

This study experimentally investigates the flow-induced flutter of a thin flexible sheet, focusing on how the sheet's aspect ratio and mass ratio affect its stability and flutter characteristics in the post-critical regime. The flutter frequency of the sheet was obtained using hotwire measurements, while flutter amplitude and mode shape were acquired through high-speed imaging. The flowfield around the flapping sheet was analyzed using particle image velocimetry (PIV). Based on experimental observations, we report the onset of flutter as a subcritical bifurcation with hysteresis. The dynamic characteristics of the sheet play a significant role in its flutter instability, with the onset and cessation of flutter occurring at a frequency close to the sheet's second-mode natural frequency. The results show that both aspect ratio and mass ratio significantly affect the critical wind speed and flutter characteristics in the post-critical regime. Both flutter frequency and amplitude decrease as the aspect ratio decreases. PIV measurements in various planes reveal the highly three-dimensional nature of the flow. Results from off-axis PIV show a pair of counter-rotating spiral vortices in the wake that oscillate and change orientation with the sheet's movement. Additionally, a theoretical analysis was conducted to derive an approximate analytical relationship between the aspect ratio and critical wind speed. Experimental results aligned well with theoretical predictions for sheets with low aspect ratios (aspect ratio ≤1) but deviated for sheets with higher aspect ratios (aspect ratio >1). The relevant scaling parameters have also been explored to represent the experimental data in a non-dimensional form.

A Framework to Identify Supercritical and Subcritical Turing Bifurcations: Case Study of Soai Reaction Featuring Asymmetric Autocatalysis

A Framework to Identify Supercritical and Subcritical Turing Bifurcations: Case Study of Soai Reaction Featuring Asymmetric Autocatalysis

Instability, bifurcation and nonlinear dynamics of Poiseuille flow in fluid overlying an anisotropic and inhomogeneous porous domain

The present study focuses on the finite amplitude analysis of Poiseuille flow in an anisotropic and inhomogeneous porous domain that underlies a fluid domain. The nonlinear interactions are studied by imposing finite amplitude disturbances to the Poiseuille flow. The former interactions in terms of modal amplitudes dictate the fundamental mode, the distorted mean flow, the second harmonic and the distorted fundamental mode. The harmonics are solved progressively in increasing order of the least stable mode obtained from the linear theory to ascertain the cubic Landau equation, which in turn helps to determine the bifurcation phenomena. The presented weakly nonlinear theory predicts the existence of subcritical transition to turbulence of Poiseuille flow in such superposed systems. In general, on moving away from the bifurcation point, it is found that a decrease in the value of inhomogeneity (in terms of Ai), Darcy number (δ) and an increase in the value of depth ratio (dˆ; the ratio of fluid domain thickness to that of porous domain) favours subcritical bifurcation. For the considered variation of parameters, the bifurcation, either subcritical or supercritical, remains the same irrespective of the value of media anisotropy (ξ) in the vicinity of the bifurcation point except for dˆ=0.2,Ai=1. In such a situation, subcritical (supercritical) bifurcation is witnessed for ξ=0.001,0.01,0.1 (1,3). Furthermore, in contrast to isotropic and homogeneous porous media, both subcritical and supercritical bifurcations are observed when moving away from the bifurcation point. A correspondence between the type of mode via linear theory and the type of bifurcation via nonlinear theory is witnessed, which is further affirmed by the secondary flow patterns. Finally, the presented theoretical results reveal an early onset of subcritical transition to turbulence in comparison with isotropic and homogeneous porous media.

Pattern dynamics of a Lotka-Volterra model with taxis mechanism

This paper deals with the Turing bifurcation and pattern dynamics of a Lotka-Volterra model with the predator-taxis and the homogeneous no-flux boundary conditions. To investigate the pattern dynamics, we first give the occurrence conditions of the Turing bifurcation. It is found that there is no Turing bifurcation when predator-taxis disappears, while the Turing bifurcation occurs as predator-taxis is presented. Next, we establish the amplitude equation by virtue of weakly nonlinear analysis. Our theoretical result suggests the Lotka-Volterra model admits the supercritical or subcritical Turing bifurcation. In this manner, we can determine the stability of the bifurcating solution. Finally, some numerical simulation results verify the validity of the theoretical analysis. The stripe pattern, the mixed patterns, and wave patterns are performed. Interestingly, the stable stripe patterns will be broken and become wave patterns when the predator-taxis parameter is far from the Turing bifurcation critical point.

Effect of colored noise on precursors of thermoacoustic instability in model gas turbine combustors

In this work, we numerically investigate the dynamics of a prototypical thermoacoustic system, the generalized Van der Pol oscillator, in the presence of additive noise of varying color (correlation time) and intensity while the system undergoes supercritical and subcritical Hopf bifurcation. We specifically investigate the influence of noise color on trends in the coherence factor and the Hurst exponent in the subthreshold region to assess their reliability as instability precursors. The Hurst exponent is found reliable only for correlation times much larger than the time scale of the instability while the coherence factor is found to be reliable for the entire range of noise color investigated. These inferences are found to hold for both supercritical and subcritical bifurcation cases.

EFFECTS OF SYMBIOTIC BACTERIA IN PATHOGENIC INTERACTIONS: THE CASE OF BATRACHOCHYTRIUM DENDROBATIDIS AND PSEUDOMONAS SP. IN AMPHIBIAN POPULATIONS

The decline in amphibian populations in recent decades may be linked to the occurrence of infectious diseases such as chytridiomycosis, which is caused by the chytrid fungus Batrachochytrium dendrobatidis (Bd). It is known that symbiotic bacteria protect the host due to their inhibitory nature. However, how the population dynamics of amphibians is affected by additional effects provided by symbiotic bacteria has not been analyzed in depth. In this paper, a model is proposed to describe the interaction among susceptible amphibians, susceptible amphibians with symbiotic bacteria and amphibians with chytrid fungus. When the modeling takes into account the additional reproductive benefits that the symbiont Pseudomonas sp. grants to the host, multiple endemic equilibrium points can exist if [Formula: see text] ([Formula: see text] is the basic reproduction number for Bd). In this scenario, the existence of a subcritical bifurcation at [Formula: see text], which can occur in two different disease-free equilibrium points, gives rise to complex dynamics and stability scenarios. Particularly, the analysis of the model shows that a sudden increase of fungus-infected amphibians can occur even when [Formula: see text] due to bistability phenomena. In this scenario, the existence of a subcritical bifurcation, which translates for the fungus into colonization even for values of [Formula: see text] less than one, represents an advantage for the chytrid fungus Batrachochytrium dendrobatidis since the pathogen should benefit from remaining as close as possible to an endemic equilibrium. To control the fungal infection, [Formula: see text] must be reduced to a value below one until the endemic equilibrium points disappear. Finally, we show that the amphibian population can reach a critical population level close to an extinction scenario when [Formula: see text] increases.

Effects of nonlinearities and geometric imperfections on multistability and deformation localization in wrinkling films on planar substrates

Compressed elastic films on soft substrates release part of their strain energy by wrinkling, which represents a loss of symmetry, characterized by a pitchfork bifurcation. Its development is well understood at the onset of supercritical bifurcation, but not beyond, or in the case of subcritical bifurcation. This is mainly due to nonlinearities and the extreme imperfection sensitivity. In both types of bifurcations, the energy–displacement diagrams that can characterize an energy landscape are non-convex, which is notoriously difficult to determine numerically or experimentally, let alone analytically. To gain an elementary understanding of such potential energy landscapes, we take a thin beam theory suitable for analyzing large displacements under small strains and significantly reduce its complexity by reformulating it in terms of the tangent rotation angle. This enables a comprehensive analytical and numerical analysis of wrinkling elastic films on planar substrates, which are effective stiffening and/or softening due to either geometric or material nonlinearities. We also validate our findings experimentally. We explicitly show how effective stiffening nonlinear behavior (e.g., due to substrate or membrane deformations) leads to a supercritical post-bifurcation response, makes the energy landscape non-convex through energy barriers causing multistability, which is extremely problematic for numerical computation. Moreover, this type of nonlinearity promotes uni-modal, uniformly distributed, periodic deformation patterns. In contrast, nonlinear effective softening behavior leads to subcritical post-bifurcation behavior, similarly divides the energy landscape by energy barriers and conversely promotes localization of deformations. With our theoretical model we can thus explain an experimentally observed phenomenon that in structures with effective softening followed by an effective stiffening behavior, the symmetry is initially broken by localizing the deformation and later restored by forming periodic, distributed deformation patterns as the load is increased. Finally, we show that initial imperfections can significantly alter the local or global energy-minimizing deformation pattern and completely remove some energy barriers. We envision that this knowledge can be extrapolated and exploited to convexify extremely divergent energy landscapes of more sophisticated systems, such as wrinkling compressed films on curved substrates (e.g., on cylinders and spheres) and that it will enable elementary analysis and the development of specialized numerical tools.

Transition to oscillatory instability of lean methane–air flames in microchannels

Micro-flow reactors and porous media burners are prone to instability in the form of flame with repetitive extinction and ignition (FREI) near the region of a temperature gradient. Although significant progress has been made in this field, there is still a lack of understanding regarding the nature of such stable-to-unstable transition. Some studies reported a smooth and continuous instability transition in microchannels, interpreting this phenomenon as a supercritical bifurcation. Meanwhile, others have observed that the transition is subcritical, with rapid, stepwise evolution of the stable flame to FREI. This study aims to rigorously determine the transition character theoretically and numerically using a two-dimensional model with a precise resolution of the bifurcation parameter (flow velocity) near the critical point and gradual wall temperature ramp. The results indicate that the transition is a subcritical bifurcation with strong hysteresis. However, the amplitude of the limit cycle born in the bifurcation point could be small compared to FREI-like oscillations. Moreover, further development of the instability depends significantly on the channel width. In relatively wide channels, the stable regime transforms immediately into the extinction/ignition one with rapid growth in amplitude. In moderate channels, complex transitional dynamics were observed with an evident pulsating regime, which evolved into FREI through mixed-mode oscillations. In narrow channels, the amplitude of transitional pulsating flame becomes comparable to the fully-developed extinction/ignition cycles, and the bifurcation diagram has a continuous character with no significant jumps or discontinuities. Such qualitative variation of transition character provides a possible explanation for contradictory interpretations presented in the literature.Novelty and Significance StatementCurrently, there is no consensus about the character of transition between the stable and extinction/ignition regimes in microchannels. Some studies have reported a continuous instability transition with an evident pulsating regime, interpreting it as a supercritical bifurcation, while others have observed subcritical bifurcation with a rapid transition. The novelty of this research lies in the rigorous determination of the bifurcation type and further instability development in microchannels of different widths based on an accurate simulation of flame dynamics near the critical point with very small steps of the flow velocity variation, along with the bifurcation theory. The study contributes to understanding the instability transition nature and may explain the different interpretations of the transitional phenomenon found in the literature. The results offer valuable insights for designing micro-scale and porous media combustion devices.

Synchronization transitions in phase oscillator populations with partial adaptive coupling.

The adaptation underlying many realistic processes plays a pivotal role in shaping the collective dynamics of diverse systems. Here, we untangle the generic conditions for synchronization transitions in a system of coupled phase oscillators incorporating the adaptive scheme encoded by the feedback between the coupling and the order parameter via a power-law function with different weights. We mathematically argue that, in the subcritical and supercritical correlation scenarios, there exists no critical adaptive fraction for synchronization transitions converting from the first (second)-order to the second (first)-order. In contrast to the synchronization transitions previously deemed, the explosive and continuous phase transitions take place in the corresponding regions as long as the adaptive fraction is nonzero, respectively. Nevertheless, we uncover that, at the critical correlation, the routes toward synchronization depend crucially on the relative adaptive weights. In particular, we unveil that the emergence of a range of interrelated scaling behaviors of the order parameter near criticality, manifesting the subcritical and supercritical bifurcations, are responsible for various observed phase transitions. Our work, thus, provides profound insights for understanding the dynamical nature of phase transitions, and for better controlling and manipulating synchronization transitions in networked systems with adaptation.

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.

Adaptive nonlinear damping control of active secondary suspension for hunting stability of high-speed trains

This study employs an active secondary suspension with adaptive nonlinear damping to enhance the hunting stability of high-speed trains. Adjusting passive suspension parameters to optimize ride comfort and hunting stability simultaneously in varied extreme operational conditions poses a significant challenge for high-speed trains. This research integrates high-order displacement-dependent nonlinear damping into the secondary lateral suspension, drawing on insights from field test data. We analyze three typical hunting motion bifurcations: carbody hunting and both subcritical and supercritical bifurcations of bogie hunting. Theoretical analysis demonstrates that (a) the adaptive nonlinear damping narrows the unstable speed range and reduces the amplitude of the limit cycle in carbody hunting, (b) while it does not increase the critical speed for supercritical bogie hunting bifurcation, it substantially reduces the amplitude of the limit cycle, and (c) it increases the nonlinear critical speed for subcritical bogie hunting bifurcation, with the potential to alter the bifurcation from subcritical to supercritical, which holds considerable practical significance. Implementing such nonlinear damping directly through passive structure is challenging. Therefore, in view of this underactuated control problem, the constraint-following control is applied in the active suspension system to inherit the benefits of adaptive nonlinear damping. The simulation results show that the proposed active suspension system can effectively suppress the abnormal vibration caused by hunting motions.

Two-dimensional hydrodynamic simulation for synchronized oscillatory flows in two collapsible channels connected in parallel.

We investigated self-sustained oscillation in a collapsible channel, in which a part of one rigid wall is replaced by a thin elastic wall, and synchronization phenomena in the two channels connected in parallel. We performed a two-dimensional hydrodynamic simulation in a pair of collapsible channels which merged into a single channel downstream. The stable synchronization modes depended on the distance between the deformable region and the merging point; only an in-phase mode was stable for the large distance, in-phase and antiphase modes were bistable for the middle distance, and again only an in-phase mode was stable for the small distance. An antiphase mode became stable through the subcritical pitchfork bifurcation by decreasing the distance. Further decreasing the distance, the antiphase mode became unstable through the subcritical Neimark-Sacker bifurcation. We also clarified the distance dependences of the amplitude and frequency for each stable synchronization mode.

Existence and stability of steady-state bifurcations in a prey–predator system with a nonmonotonic functional response

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.

Existence conditions for bifurcations of homoclinic orbits in a railway wheelset model

This paper investigates the bifurcations of homoclinic orbits to hyperbolic saddle points in a simplified railway wheelset model with cubic and quintuple nonlinear terms. Using Melnikov’s method, the sufficient conditions for the existence of the supercritical and the subcritical pitchfork bifurcations of homoclinic orbits are proven. To determine the integrability of the variational equations around homoclinic orbits in the meaning of differential Galois theory, the corresponding Fuchsian second-order differential equation for the normal variational equation and the Riemann P function are obtained. It is shown that the coefficients of the linear terms and the cubic coupling terms play a very significant role on influencing the existence of homoclinic orbits. While, the cubic coupling terms have little effect on the size of the left-hand and right-hand potential wells of homoclinic orbits. These results are beneficial to explore the key mechanism of hunting stability of a simplified railway wheelset model.

Curvature controls beading in soft coated elastic cylinders: Finite wavemode instability and localized modulations

Axisymmetric beading instabilities in soft, elongated cylinders have been observed in a plethora of scenarios, ranging from cellular nanotunnels and nerves in biology to swollen cylinders and electrospun fibers in polymer physics. One of the common geometrical features that can be seen in these systems is the finite wavelength of the emerging pattern. However, modelling studies often predict that the instability has an infinite wavelength, which can be associated with localized necking or bulging. In this paper, we consider a soft elastic cylinder with a thin coating that resists bending, as described by the Helfrich free energy functional. The bending stiffness and natural mean curvature of the coating are two novel features whose competition against bulk elasticity and capillarity is investigated. For intermediate values of the bending stiffness, a linear stability analysis reveals that the mismatch between the current and natural mean curvature of the coating can lead to patterns emerging with a finite wavelength. This analysis creates a continuous bridge between the classical solutions of the shape equation obtained from the Helfrich functional and a curvature-controlled zero-wavemode instability, similar to the one induced by the competition between bulk elasticity and capillarity. A weakly non-linear analysis predicts that the criticality of the bifurcation depends on the controlling parameter, with both supercritical and subcritical bifurcations possible. When capillarity is introduced, the criticality of the bifurcation changes in a non-trivial way.

Dynamical analysis of a tabu learning neuron through the discrete implicit mapping method

The complex dynamics of neuron model including bifurcation behaviors and firing patterns, especially unstable firing patterns, furnish a favorable point of view for the research of neurons, which may be of great importance in the prevention of brain-based diseases. To study this issue in depth, the discrete implicit mapping method is employed to evaluate a tabu learning neuron system, which is coupled with electromagnetic excitation and applied driving current in this paper. The bifurcation orbits of periodic motion with rich dynamic phenomena are accurately shown with varying the amplitudes of external input. The evolutions of period-5 to period-20 and period-6 to period-12 can be obtained via period doubling bifurcations and saddle bifurcation. The subcritical pitchfork bifurcation is also exhibited through the complex coexistence behavior of stable and unstable periodic orbits in such a model, which cannot be obtained by the traditional numerical method due to its accumulative errors. The corresponding stabilities and bifurcation of the periodic motions are determined by means of eigenvalues. Therefore, using such a method, the neural spiking events can be developed through phase diagram and time history of membrane potential. Moreover, the tabu learning system is implemented via FPGA (i.e., field programmable gate array), and the unstable and stable phase diagrams are observed completely from oscilloscope, where the experiments results verify the feasibility of semi-analytical solutions. Such investigation will also benefit the development of artificial intelligence and the advances in Brain Medicine.

Supercritical and subcritical aeroelastic behaviors of a three-dimensional wing coupled with a nonlinear energy sink

This paper studies the flutter control of a three-dimensional wing using a nonlinear energy sink (NES), which has a purely cubic stiffness and a linear damper. The structural model is derived by Lagrange's equations and the aerodynamics is modeled using the auto regressive with exogenous input method. Both models are solved in a tightly coupling manner to obtain time domain results. Then, the structural describing function and the aerodynamic continuous-time state space model are used to derive the equivalent linearized aeroelastic system, so that eigenvalue analyses can be used to obtain stable and unstable limit cycle oscillations (LCOs). The LCOs obtained by time domain simulations agree well with the equivalent linearization method, while differences are observed for aperiodic motions which are only predicted in time domain for small damping coefficients. Results show the square of aeroelastic amplitude is inversely proportional to the NES stiffness coefficient for both periodic and aperiodic motions. The NES can induce LCOs below flutter speed for small damping coefficients, indicating a subcritical bifurcation which can change to a supercritical one by increasing the damping coefficient. It is discovered that the upper bound of partial suppression region is quantified by a turning point speed. The regions of complete suppression, partial suppression and no suppression are clarified.

Analysis of Local Bifurcations in One-Dimensional Systems

This article delves deeper into the study of local bifurcation in maps, although only in the context of a single dimension. The various local bifurcation types common to one-dimensional maps each have their own specific conditions that must be fulfilled. An object divides into two when it bifurcates. After a bifurcation, a family of one-parameter functions retains its stable or cyclic point structure. A bifurcation in the iterative process happens when a parameter is altered, causing a qualitative change in behavior. The phenomena of local splitting can be induced by adjusting just one parameter. Both the norm form of the transcritical split and the stability of the saddle node are highlighted. As with supercritical and subcritical bifurcations, the stability of the pitchfork bifurcation is assessed, as is its norm form. Jagannath University Journal of Science, Volume 10, Number II, Dec. 2023, pp. 108-123

Hopf bifurcation and normal form in a delayed oncolytic model

In this paper, we investigate the mathematical analysis of a mathematical model describing the virotherapy treatment of a cancer with logistic growth and the effect of viral cycle presented by a time delay. The cancer population size is divided into uninfected and infected compartments. Depending on time delay, we prove the positivity and boundedness and the stability of equilibria. We give conditions on which the viral cycle leads to “Jeff’s phenomenon” observed in laboratory and causes oscillations in cancer size via Hopf bifurcation theory. We establish an algorithm that determines the bifurcation elements via center manifold and normal form theories. We give conditions which lead to a supercritical or subcritical bifurcation. We end with numerical simulations illustrating our theoretical 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.