Bifurcation Threshold Research Articles

Ferrofluidic Taylor Couette flow in between finite length porous cylinders with radial through-flow

We study ferrofluidic Taylor Couette flow under the influence of radial inflow and outflow in combination with an external applied magnetic field in a finite-length cavity via direct numerical simulations. As is the case for no magnetic field, the base state (cylindrical Couette flow and modified circular-Couette flow in presence of a transverse magnetic field component, respectively) with an external magnetic field is stabilized for any radial inflow and strong radial outflow, while the system becomes slightly destabilized for weak radial outflow. The particular parameter range for destabilization depends on the field strength of the applied magnetic field. Slightly increasing the field strength shrinks the range, while it grows for larger field strengths. In general, a larger field strength tends to minimize and compensate the effect of any radial flow, resulting in bifurcation thresholds (critical Reynolds number vs. radial flow) which have less curvature, i.e. they are more flat. We elucidate the origin of this effect to be in the symmetry breaking nature of the transverse magnetic field itself. Azimuthal velocity isocontours are shifted different strong due to radial flow, either in the part of the annulus that is aligned with the direction of the applied magnetic field or perpendicular to it. In particular, the modulation amplitude in the isocontours perpendicular to the field increase. As a result the flow is locally stabilized with different strength, so that the overall stabilization is weaker relative to the situation without any applied field. This diminishing curvature effect with variation of the radial flow becomes more pronounced with stronger applied magnetic fields.

Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments

The collective behaviour of stochastic multi-agents swarms driven by Gaussian and non-Gaussian environments is analytically discussed in a mean-field approach. We first exogenously implement long range mutual interactions rules with strengths that are weighted by the real-time distance separating each agent with the swarm barycentre. Depending on the form of this barycentric modulation, a transition between two drastically different collective behaviours can be unveiled. A behavioural bifurcation threshold due to the tradeoff between the desynchronisation effects of the stochastic environment and the synchronising interactions is analytically calculated. For strong enough interactions, the emergence of a soliton propagating wave is established. Alternatively, weaker interactions cannot overcome the environmental noise and evanescent diffusive waves result. In a second and complementary approach, we show that the emergent solitons can alternatively be interpreted as being the optimal equilibrium of mean-field games (MFG) models with ad-hoc running cost functions which are here exactly determined. These MFG's soliton equilibria are therefore endogenously generated. Hence for the classes of models here proposed, an explicit correspondence between exogenous and endogenous interaction rules leading to similar collective effects is explicitly constructed. For non-Gaussian environments our results offer a new class of exactly solvable mean-field games dynamics.

Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

The present work investigates the effects of maturation and dispersal delays on dynamics of single species populations. Both delays have been incorporated in a single species nonlocal hyperbolic-parabolic population model, which admits traveling and stationary wave solutions. We reduce the model into various forms and obtain the corresponding analytical solutions. Analysis of the reduced models indicates that the dispersal delay can result in loss of monotonicity, where the solutions oscillate as they converge to a positive equilibrium. The stability analysis of the general model reveals that the maturation time delay admits a Hopf bifurcation threshold, which is expressed as a function of the dispersal delay. The numerical simulations of the general model suggest that the global stability of the stationary wave solutions is lost when the dispersal delay is increased from zero. In conclusion, population models with maturation and dispersal delays can give new insights into the complex dynamics of single species.

Local and global bifurcations in magnetic resonance force microscopy

The focus of this paper is on the investigation of local and global bifurcations in a continuum mechanics-based resonator model proposed for measurement of electron spin via magnetic resonance force microscopy (MRFM). The resonator model, derived using the extended Hamilton’s principle incorporating the Bloch equations for magnetization, is investigated analytically and numerically. Analysis of both adiabatic and non-adiabatic equilibrium configurations enables formulation of the dynamical system bifurcation structure and identification of the parameter space required for stable MRFM operation. A multiple-scales analysis of the limiting adiabatic model enables estimation of the local bifurcation thresholds for bistable solutions and prediction of the frequency shift that enables spin detection. Orbital instabilities of the adiabatic model reveal a global bifurcation structure where lengthy chaotic transients occur below a homoclinic jump-to-contact threshold which is determined via a Melnikov–Holmes analysis. Both local and global bifurcations are verified numerically in the non-adiabatic model and reveal a dense power spectra for the magnetic moments. The computation of the parameter space governing the model orbital instabilities enables a consistent estimation of robust MRFM operation conditions.

Cops-on-the-dots: The linear stability of crime hotspots for a 1-D reaction-diffusion model of urban crime

In a singularly perturbed limit, we analyse the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci., 20, Suppl., (2010)], which models urban crime with police intervention. In our extended RD model, the field variables are the attractiveness field for burglary, the criminal population density and the police population density. Our model includes a scalar parameter that determines the strength of the police drift towards maxima of the attractiveness field. For a special choice of this parameter, we recover the ‘cops-on-the-dots’ policing strategy of Jones et. al., where the police mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocal eigenvalue problems (NLEPs) that characterise the linear stability of these hotspot steady states to ${\cal O}$(1) timescale instabilities. For a cops-on-the-dots policing strategy, we prove that a multi-hotspot steady state is linearly stable to synchronous perturbations of the hotspot amplitudes. Alternatively, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical–numerical method is used to construct linear stability phase diagrams in the police vs. criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model, this provides a parameter range where the effect of a cops-on-the-dots policing strategy is to only displace crime temporally between neighbouring spatial regions. Our hybrid approach to study the NLEPs combines rigorous spectral results with a numerical parameterisation of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.

Strategies for the existence of spatial patterns in predator–prey communities generated by cross-diffusion

Fear of predators is an important drive for predator–prey interactions, which increases survival probability but cost the overall population size of the prey. In this paper, we have extended our previous work spatiotemporal dynamics of predator–prey interactions with fear effect by introducing the cross-diffusion. The conditions for cross-diffusion-driven instability are obtained using the linear stability analysis. The standard multiple scale analysis is used to derive the amplitude equations for the excited modes near Turing bifurcation threshold by taking the cross-diffusion coefficient as a bifurcation parameter. From the stability analysis of amplitude equations, the conditions for the emergence of various ecologically realistic Turing patterns such as spot, stripe, and mixture of spots and stripes are identified. Analytical results are verified with the help of numerical simulations. Turing bifurcation diagrams are plotted taking diffusion coefficients as control parameters. The effect of the cross-diffusion coefficients on the homogeneous steady state and pattern structures of the self-diffusive model is illustrated using the simulation techniques. It is also observed that the level of fear has stabilizing effect on the cross-diffusion induced instability and spot patterns change to stripe, then a mixture of spots and stripes and finally to the labyrinthine type of patterns with an increase in the level of fear.

Non-uniform plane waves (ghost waves) in general anisotropic medium

In this paper, we explicitly demonstrate that general anisotropic (in our case dielectric magnetic anisotropic) medium supports ghost wave (non-uniform plane wave), that possess the characteristics of the both, propagating and evanescent wave. Solutions for ghost waves exist, below the threshold of tangent bifurcation in the dispersion diagram when the system dynamics are extended to higher dimension. Subsequently, we also indicate the ghost regions in the wavevector plane (phase space) of general anisotropic medium, within which these ghost waves can exist. As the name implies, general anisotropic is a general medium, therefore, previous case in which these ghost waves exist becomes a special case.

IMITATION, PROXIMITY, AND GROWTH — A COLLECTIVE SWARM DYNAMICS APPROACH

This paper is based on the premise that economic growth is driven by an interplay between innovation and imitation in an economy composed of interacting firms operating in a stochastic environment. A novel approach to modeling imitation is presented based on range-dependent processes that describe how firms consider proximity when imitating peers who are found in a given neighborhood in terms of productivity. Using a particularly tractable approach, we are able to analyze how drastically different economic growth scenarios emerge from different imitation strategies. These emerging scenarios range from diffusive growth where the variance of productivity grows indefinitely, to balanced growth described by a traveling wave with fixed variance. The latter scenario is sustained only when imitation strength among firms exceeds a critical bifurcation threshold.

Stability of Hopf-bifurcating limit cycles in a diffusion-driven prey-predator system with Allee effect and time delay.

In this paper, we investigate the effect of the gestation delay on the spatiotemporal pattern formation in a prey-predator system with monotonic functional response with saturation and intraspecific competition among the predator population in presence of the additive Allee effect in prey growth. In this regard, we present rigorous analytical results for determining the delay-induced Hopf bifurcation threshold and the associated properties of the Hopf-bifurcating periodic solutions and verify them with the help of numerical simulations. We derive analytically the delay-induced Hopf bifurcation threshold by employing the linear stability analysis about the unique spatially uniform coexistence steady state. Also, we provide the expressions for determining the direction and stability of the Hopfbifurcating periodic solutions by using the normal form theory and center manifold reduction. The numerical simulation results reveal that the Hopf bifurcation can potentially lead to spatially homogeneous periodic in time distribution of the populations which will eventually settles to chaotic in space and time distribution for sufficiently large value of the time delay. Further, our numerical investigations reveal that the time delay can change one stationary pattern to another through the loss of monotonicity property of the spatially averaged densities.

Swarms on the 3-sphere with adaptive synapses: Hebbian and anti-Hebbian learning rule

We introduce and analyze several models of swarm dynamics on the sphere S3 with adaptive (state-dependent) interactions between agents. The equations describing the interaction dynamics are variations of the classical Hebbian principle from Neuroscience. We study asymptotic behavior in models with various realizations of Hebbian and anti-Hebbian learning rules. The swarm with the Hebbian rule and strictly nonnegative (attractive) interactions evolves towards consensus. If the Hebbian rule allows both attractive and repulsive interactions the swarm converges to bipolar configuration. The most interesting is the model with anti-Hebbian learning rule with both attractive and repulsive interactions. This model displays a rich variety of dynamical regimes and stationary formations, depending on the number of agents and system parameters. We prove that the model with such anti-Hebbian rule evolves towards a stable stationary configuration if the system parameter is above a certain bifurcation threshold. Finally, some simulation results are presented demonstrating how these theoretical results can be applied to coordination of rotating bodies in 3D space; this is done by mapping the trajectories from S3 to special orthogonal group SO(3).

Nonlinear dynamics of infectious diseases via information-induced vaccination and saturated treatment

The present study of a nonlinear compartmental SVIR model accounts for the dynamics of infectious diseases in population. In this model, the effect of information on vaccination coverage is quantified when medical resources are limited during the epidemic outbreak. Model analysis is performed and the global asymptotic stability of the disease free equilibrium is established when treatment is available for all infective individuals. However, the existence of multiple endemic equilibria is observed due to saturation in medical treatment and information-induced vaccination coverage. If the basic reproduction number is greater than unity, a unique endemic equilibrium for the model is obtained under some parametric conditions. Using the geometric approach, the global asymptotic stability of this endemic equilibrium is established under a parametric constraint. Further, when medical resources are limited, the existence Hopf bifurcation is shown analytically, which infers the oscillatory persistence of disease within the population. Occurrence of Hopf–Hopf bifurcation is also investigated numerically. At first Hopf bifurcation threshold, an endemic equilibrium loses its stability and produces oscillations, and further, it regains its stability at another Hopf bifurcation threshold when oscillations disappear. Hence, multiple stability switches are observed due to the saturated treatment and information-induced vaccination. The effect of the fading of memory also plays a crucial role and gives rise to complex dynamical behaviour and also affects the stability properties. Numerically, different types of backward bifurcation are also examined. Extensive numerical experimentations are also carried out to further explore the model system.

Pattern selection in the 2D FitzHugh–Nagumo model

We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.

Refined stability thresholds for localized spot patterns for the Brusselator model in

In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in $\mathbb{R}^2$, for the Brusselator reaction–diffusion model $$ \begin{equation*} v_t = \epsilon^2 \Delta v + \epsilon^2 - v + fu v^2 \,, \qquad \tau u_t = D \Delta u + \frac{1}{\epsilon^2}\left(v - u v^2\right) \,, \end{equation*} $$ where the parameters satisfy 0 < f < 1, τ > 0 and D > 0. A previous leading-order linear stability theory characterizing the onset of spot amplitude instabilities for the parameter regime D = ${\mathcal O}$(ν−1), where ν = −1/log ϵ, based on a rigorous analysis of a non-local eigenvalue problem (NLEP), predicts that zero-eigenvalue crossings are degenerate. To unfold this degeneracy, the conventional leading-order-in-ν NLEP linear stability theory for spot amplitude instabilities is extended to one higher order in the logarithmic gauge ν. For a multi-spot pattern on a finite domain under a certain symmetry condition on the spot configuration, or for a periodic pattern of spots centred at the lattice points of a Bravais lattice in $\mathbb{R}^2$, our extended NLEP theory provides explicit and improved analytical predictions for the critical value of the inhibitor diffusivity D at which a competition instability, due to a zero-eigenvalue crossing, will occur. Finally, when D is below the competition stability threshold, a different extension of conventional NLEP theory is used to determine an explicit scaling law, with anomalous dependence on ϵ, for the Hopf bifurcation threshold value of τ that characterizes temporal oscillations in the spot amplitudes.

A new model of the L–H transition and H-mode power threshold

In order to understand the mechanism of the confinement bifurcation and H-mode power threshold in magnetically confined plasma, a new dynamical model of the L–H transition based on edge instability phase transition (EIPT) has been developed. With the typical plasma parameters of the EAST tokamak, the self-consistent turbulence growth rate is analyzed using the simplest case of pressure-driven ballooning-type instability, which indicates that the L–H transition can be caused by the stabilization of the edge instability through EIPT. The weak E × B flow shear in L-mode is able to increase the ion inertia of the electrostatic motion by increasing the radial wave number of the tilted turbulence structures, which play an important role for accelerating the trigger process of EIPT rather than directly to suppress the turbulent transport. With the acceleration mechanism of E × B flow shear, fast L–H and H–L transitions are demonstrated under the control of the input heating power. Due to the simplified scrape-off-layer boundary condition applied, the ratio between the heating powers at the H–L and L–H transition respectively differs from the ratio by Nusselt number. The results of the modeling reveal a scaling of the power threshold of the L–H transition, PL−H ∝ n0.76B0.8 for deuterium plasma. It is found finite Larmor radius induces an isotope effect of the H-mode power threshold.

High order asymptotic dynamics of a nonlinearly coupled electromechanical system

A nonlinearly coupled mathematical model of an electro-magneto-mechanical system is studied via the multiple scale approach in order to investigate its weakly nonlinear dynamics and analytically predict its salient features. The obtained amplitude modulation equations up to the third order perturbation allow to analytically describe the mechanical and electrical responses in terms of frequency-response curves and stability scenarios. A critical threshold of Hopf bifurcation is detected and analyzed as a function of the main system parameters. The subsequent extension of the asymptotic scheme up to the fifth order proves to grasp also the post-critical behavior, providing with the accurate identification of the amplitude of the quasiperiodic responses characterizing the unstable region of the ordinary differential equations system.

Transverse single-file diffusion and enhanced longitudinal diffusion near a subcritical bifurcation.

A quasi-one-dimensional system of repelling particles undergoes a configurational phase transition when the transverse confining potential decreases. Below a threshold, it becomes energetically favorable for the system to adopt one of two staggered raw patterns, symmetric with respect to the system axis. This transition is a subcritical pitchfork bifurcation for short range interactions. As a consequence, the homogeneous zigzag pattern is unstable in a finite zigzag amplitude range [h_{C1},h_{C2}]. We exhibit strong qualitative effects of the subcriticality on the thermal motions of the particles. When the zigzag amplitude is close enough to the limits h_{C1} and h_{C2}, a transverse vibrational soft mode occurs which induces a strongly subdiffusive behavior of the transverse fluctuations, similar to single-file diffusion. On the contrary, the longitudinal fluctuations are enhanced, with a diffusion coefficient which is more than doubled. Conversely, a simple measurement of the thermal fluctuations allows a precise determination of the bifurcation thresholds.

Backward bifurcation and hysteresis in models of recurrent tuberculosis.

An epidemiological model is presented that provides a comprehensive description of the transmission pathways involved for recurrent tuberculosis (TB), whereby cured individuals can become reinfected. Our main goal is to determine conditions that lead to the appearance of a backward bifurcation. This occurs when an asymptotically stable infection free equilibrium concurrently exists with a stable non-trivial equilibria even though the basic reproduction number R0 is less than unity. Although, some 10-30% cases of TB are recurrent, the role of recurrent TB as far as the formation of backward bifurcation is concerned, is rarely if ever studied. The model used here incorporates progressive primary infection, exogenous reinfection, endogenous reactivation and recurrent TB as transmission mechanisms that contribute to TB progression. Unlike other studies of TB dynamics that make use of frequency dependent transmission rates, our analysis provides exact backward bifurcation threshold conditions without resorting to commonly applied approximations and simplifying assumptions. Exploration of the model through analytical and numerical analysis reveal that recurrent TB is sometimes capable of triggering hysteresis effects which allow TB to persist when R0 < 1 even though there is no backward bifurcation. Furthermore, recurrent TB can independently induce backward bifurcation phenomena if it exceeds a certain threshold.

Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in Two Dimensions

For three specific singularly perturbed two-component reaction diffusion systems in a bounded two-dimensional domain admitting localized multispot patterns, we provide a detailed analysis of the pa...

Stability and instability analysis of the substrate supported panels in the forming process based on perturbation growth and bifurcation threshold models

It is highly appreciated the using reinforced layers in supporting metallic parts due to increased performance and formability of the sheets. Firstly, the effect of reinforcing layer will be investigated on the expanding of rings as a benchmark stability problem and then considered when the composite panels undergo the axial and biaxial stretches near instability threshold. In the ring expanding case; process will be explored by the wavelength and wave number changes through the linear perturbation analysis. The linear perturbation or linear stability is established on the elastic-visco-plastic fragmentation of the rings that includes the effect of significant deformation, geometric softening and hydrostatic pressure. Alternately, the vertex model will be used in prediction of localized necking instability that, considers the discontinuity of stress and strain rates inside the neck band. Here, the DYC-angle and quadratic Hill yield criterion help us to predict the localized necking band and consider the anisotropy effect, respectively. Also, the forming limit diagrams will be used for presenting the vertex analysis results. In the vertex theory, the non-linear reinforced layer will be considered among the neo-Hookean materials. Results will be validated with the other theoretical and experimental data available in the literature.

Periodic pulsating dynamics of slow–fast delayed systems with a period close to the delay

We consider slow–fast delayed systems and discuss pulsating periodic solutions, which are characterised by specific properties that (a) the period of the periodic solution is close to the delay, and (b) these solutions are formed close to a bifurcation threshold. Such solutions were previously found in models of mode-locked lasers. Through a case study of population models, this work demonstrates the existence of similar solutions for a rather wide class of delayed systems. The periodic dynamics originates from the Hopf bifurcation on the positive equilibrium. We show that the continuous transformation of the periodic orbit to the pulsating regime is simultaneous with multiple secondary almost resonant Hopf bifurcations, which the equilibrium undergoes over a short interval of parameter values. We derive asymptotic approximations for the pulsating periodic solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realisation of the bifurcation scenario is highlighted.