Linear Stability Analysis Method Research Articles

Multibistability and self-pulsation in nonlinear high-Qsilicon microring resonators considering thermo-optical effect

Optical bistability (BI) and self-pulsation (SP) in high-$Q$ silicon microring resonators (MRRs) induced by thermo-optical (TO) effect and other nonlinear effects are theoretically studied with coupled mode theory and linear stability analysis method. It is found that the boundaries for both BI and SP are mainly restricted by two counteracting effects: free carrier dispersion effect and TO effect. If the refractive index changes of a MRR caused by these two effects are on the same order of magnitude, the output power will exhibit much more complicated dependence on the input power and wavelength, namely, input-power-dependent multi-BI and multi-SP regions will exist at certain input wavelength range. The controllability of multi-BI and multi-SP phenomena by the input power and input wavelength could be very useful in all-optical nonlinear devices.

Regulation of tumour intracellular pH: A mathematical model examining the interplay between H+ and lactate

Non-invasive measurements of pH have shown that both tumour and normal cells have intracellular pH (pHi) that lies on the alkaline side of neutrality (7.1–7.2). However, extracellular pH (pHe) is reported to be more acidic in some tumours compared to normal tissues. Many cellular processes and therapeutic agents are known to be tightly pH dependent which makes the study of intracellular pH regulation of paramount importance. We develop a mathematical model that examines the role of various membrane-based ion transporters in tumour pH regulation, in particular, with a focus on the interplay between lactate and H+ ions and whether the lactate/H+ symporter activity is sufficient to give rise to the observed reversed pH gradient that is seen is some tumours. Using linear stability analysis and numerical methods, we are able to gain a clear understanding of the relationship between lactate and H+ ions. We extend this analysis using perturbation techniques to specifically examine a rapid change in H+-ion concentrations relative to variations in lactate. We then perform a parameter sensitivity analysis to explore solution robustness to parameter variations. An important result from our study is that a reversed pH gradient is possible in our system but for unrealistic parameter estimates—pointing to the possible involvement of other mechanisms in cellular pH gradient reversal, for example acidic vesicles, lysosomes, golgi and endosomes.

Constructive method of linear systems with delay stability analysis

In this paper, the finite constructive method for linear differential-difference systems stability and instability analysis is proposed, and its convergence is proved. This method is based on obtaining of the quadratic lower bound for the quadratic Lyapunov – Krasovskii functionals on some special set of functions. The method proposed is applied to the stability domain construction in a parameter space, and to the critical delay values numerical determination.

Instabilities of vortex rings generated by surface-tension gradients between co-axial disks

Instabilities of vortex rings generated by surface-tension gradients between co-axial disks with Prandtl numbers of 0.001 and 0.01 are investigated by using the linear stability analysis (LSA) method. The continuity-vortex-energy equations are used as the perturbation equations for stability analysis and discretized using a Chebyshev-collocation method. The critical Reynolds numbers and the angular wavenumbers of the unstable mode are obtained for vortex rings with a range of aspect ratios between 0.05 and 1.2. From stability analyses, it is found that the product of the critical model and the aspect ratio approaches a constant when the aspect ratio decreases. The critical mode is m=16 when the aspect ratio is 0.05. Analyses also indicate that the vortex rings must be of certain energy for the perturbations of flow to grow in amplitude. The viscosity of fluid can dampen perturbations if the magnitude of the stream function of the basic flow is below the critical value. The vortex rings generated by surface-tension gradients become unstable when the magnitude of the stream function is above the critical value.

Interface instability of methane hydrate stability zone in permafrost deposits

The method of linear stability analysis is applied to determine the necessary conditions of existence of natural hydrate deposits under Arctic permafrost. It is found that the existence is limited to relatively small thicknesses of the hydrate stability zone and the underlying methane-saturated layer, low-to-moderate permeability of the deposits, and low-to-moderate hydrate saturation. At larger thicknesses, higher permeability, or higher saturation, the interface between the two layers is unstable to small-amplitude perturbations. The results do not support the hypothesis that the interface instability may lead to accelerating meltdown of natural methane hydrates in response to increase in global temperature.

A domain decomposition matrix-free method for global linear stability

This work is dedicated to the presentation of a matrix-free method for global linear stability analysis in geometries composed of multi-connected rectangular subdomains.An Arnoldi technique using snapshots in subdomains of the entire geometry combined with a multidomain linearized Direct Numerical Finite difference simulations based on an influence matrix for partitioning are adopted. The method is illustrated by three benchmark problems: the lid-driven cavity, the square cylinder and the open cavity flow.The efficiency of the method to extract large-scale structures in a multidomain framework is emphasized. The possibility to use subset of the full domain to recover the perturbation associated with the entire flow field is also highlighted. Such a method appears thus a promising tool to deal with large computational domains and three-dimensionality within a parallel architecture.

非線形波動方程式の線形安定性解析に対する Hill の方法の収束解析

非線形波動方程式の線形安定性解析に対する Hill の方法の収束解析

Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects

Bistability (BS) and self-pulsation (SP) phenomena in silicon microring resonators (MRR) with intense CW light injection are studied. Several nonlinear optical effects including Kerr effect, two-photon absorption, free carrier absorption and free carrier dispersion are taken into account. The threshold optical intensity of BS and SP is derived from the coupled mode theory and a linear stability analysis method. The influences of MRR's parameters (carrier lifetime, linear loss and radius) and light injection conditions (input power, wavelength detuning) on the characteristics of SP (modulation depth and oscillating frequency) are analyzed and discussed. It is shown that, SP occurs only if the carrier lifetime ranges from several ps to several-hundred ps and the input light intensity is higher than 10⁶W/cm². The modulation depth of SP can be as large as 8dB and the associated oscillating frequency is in the range from several GHz to beyond 10 GHz.

Stability and sustained oscillations in a ventricular cardiomyocyte model

The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. The associated stationary problem was reduced to a nonlinear system in only two unknowns, the transmembrane potential V and the intracellular calcium concentration [Ca]( i ). By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For a variable steady depolarizing or hyperpolarizing current (I (st)), the corresponding projection of the static bifurcation diagram in the (I (st), V) plane is complex, featuring three branches of stationary solutions joined by two limit points. On the upper branch oscillations can occur, being either damped at a stable focus or diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle, thus resulting in early after-depolarizations (EADs). The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K(+) current maximal conductances (g (si) and g (K)), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances sustained oscillations tending to a limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods, leading to identification of Hopf bifurcation points, limit points of cycles and period doubling bifurcations. In particular settings, e.g. one-fifth-of-normal g (si), EADs and sustained high amplitude oscillations due to an unstable resting state may occur simultaneously.

Intrinsic Noise Propagation in Two-Step Series Enzymatic Cascades

Series enzymatic cascades such as MAPK cascades act as signal amplifiers and play a key role in processing information during signal transduction in cells. Cell-to-cell variability or noise, consisting of both intrinsic and extrinsic noise, propagate along with the signal through these cascades and thereby strongly affect the cellular response. Characterization of noise propagation in such cascades can provide insights on the ability of the cells to effectively process information in the presence of noise. Accurate estimation of noise in the cascade is essential for noise propagation studies. Based on the stochastic simulations and global sensitivity analysis, we develop a systematic methodology to characaterize noise propagation in the cascade. Using this method, we show that the noise propagation depends on the total upstream enzyme concentration and the total substrate concentrations.We prove that the conventional linear stability analysis method applied to the Langevin type stochastic model of the cascade fails to accurately predict the intrinsic noise propagation. This is due to the fact that the effect of nonlinearity on intrinsic noise propagation is ignored in the linear stability analysis method of noise estimation. We prove this by conducting extensive stochastic simulations of the Langevin type model equation and Gillespie simulations of the chemical master equation. Based on the global sensitivity analysis and Gillespie simulations, we show that this conclusion is valid for a wide range of parameters.

Efficient numerical method for linear stability analysis of solitary waves

In this paper we use a numerical relaxation algorithm to improve and generalize the obtainment of the perturbation eigenstates of nonlinear systems. As a model problem we consider the linear stability analysis of the vortex eigenstates of the cubic–quintic nonlinear Schrödinger equation. It is shown by numerical calculations that the relaxation algorithm permits accurate tracing of complex perturbation eigenvalues.

3-로브 베어링으로 지지된 중형 고속 터보차저 로터의 선형 및 비선형 동적 안정성 특성 규명

In this study linear and nonlinear dynamic stability characteristics of a medium-size high-speed turbocharger, whose rotor is supported by two 3-lobe journal bearings, are analyzed to evaluate and identify the effects of its bearing design variables. The rotor has the rated speed of 40,500 rpm and maximum continuous speed of 45,000 rpm. At first, utilizing the linear stability analysis method, bearing designs of yielding stable or unstable LogDecs as small as possible are searched by manipulating with machined bearing clearances and preloads. As next, utilizing the nonlinear analysis method, limit cycles of the rotor responses at the rated and maximum continuous speeds are simulated to check their acceptances. Results have shown that for the turbocharger rotor-bearing system considered, the 3-lobe journal bearing design with a smaller machined clearance and a larger preload are preferred for the stable rotor responses. More importantly, since there exists a good correlation between the linear and nonlinear stability analysis results, it is concluded that firstly the linear stability analysis method may be applied to screen quickly the ranges of bearing designs for stable or least unstable solutions and then, lastly the nonlinear stability analysis method may be deployed to check an absolute motion stability in terms of the limit cycle.

A new matching method for linear stability analysis of MHD modes close to the ideal MHD marginal stability

A new matching method has been invented for linear stability analysis of magnetohydrodynamics (MHD) modes for plasmas marginally stable against ideal MHD. An inner region with a finite width is utilized as in our previous study [M. Furukawa, S. Tokuda, and L.-J. Zheng, Phys. Plasmas 17, 052502 (2010)]. An ordering scheme for the outer region has been newly developed, thereby it succeeds to include effects of small plasma inertia and resistivity perturbatively in the outer region. The corresponding boundary condition requires direct, not asymptotic, matching of the outer and inner solutions, which assumes nothing special for the behavior of parallel electric field across the matching points. The union of the ordering scheme and the boundary condition enables us to apply our matching method even for plasmas marginally stable against ideal MHD. Because our matching method is not asymptotic, it is easy to implement numerically. The stability analysis of resistive MHD modes, such as internal kink and tearing modes, is satisfactory.

Stability and Self-Sustained Oscillations in a Ventricular Cardiomyocyte Model

The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For different values of a steady depolarizing/hyperpolarizing current (Ist), the corresponding projection of the static bifurcation diagram in the (Ist, V) plane is complex, featuring three branches of stationary solutions delimited by two saddle-node bifurcation points. In addition, on the upper branch oscillations can occur within an Ist range [−4.45, −0.51 μA/cm2] where the Jacobian of the linearized system features two complex conjugated eigenvalues. Oscillations are either damped at a stable focus or amplified until the system trajectory is diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle. The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K+ current maximal conductances (gsi and gK), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances self-sustained oscillations tending to a stable limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods based on numerical continuation algorithms. Both unsustained oscillations, resulting in early after-depolarizations, and sustained oscillations may trigger dangerous ventricular arrhythmias by multiple mechanisms. In particular settings, e.g. for a normal gK and one-fifth-of-normal gsi, these two arrhythmia-threatening conditions may occur simultaneously.

Onset of buoyancy-driven convection in isotropic porous media heated from below

A theoretical analysis of buoyancy-driven instability under transient basic fields is conducted in an initially quiescent, fluid-saturated, horizontal, isotropic porous layer. Darcy’s law is employed to explain characteristics of fluid motion, and Boussinesq approximation is used to consider the density variation. Under the principle of exchange of stabilities, a stability analysis is conducted based on the linear stability analysis and energy method and their modifications. The critical condition of onset of buoyancy-driven convection is obtained as a function of the Darcy-Rayleigh number. The propagation theory and the modified energy method under the self-similar coordinate suggest reasonable stability criteria and support each other. The former one based on the linear stability theory predicts more stable results than the latter based on the energy method. The growth period for disturbances to grow seems to be required until the instabilities are detected experimentally.

Surface patterning of soft polymer film-coated cylinders via an electric field

Using the linear stability analysis method, we investigate the surface wrinkling of a thinpolymer coating on a cylinder in an externally applied electric field. It is demonstrated thatenergy competition between surface energy, van der Waals interactive potential energy andelectrostatic interaction energy may lead to ordered patterns on the film surface. Theanalytical solutions are derived for the critical conditions of both longitudinal andcircumferential instabilities. The wavelengths of the generated surface patternscan be mediated by changing the magnitude of the electric field. Our analysisshows that the surface morphology is sensitive to the curvature radius of the fiber,especially in the micrometer and nanometer length scales. Furthermore, we suggesta potential approach for fabricating hierarchical patterns on curved surfaces.

Effect of dust particles on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium

The problem of the effect of dust particles on the thermal convection in micropolar ferromagnetic fluid saturating a porous medium subject to a transverse uniform magnetic field has been investigated theoretically. Linear stability analysis and normal mode analysis methods are used to find an exact solution for a flat micropolar ferromagnetic fluid layer contained between two free boundaries. In case of stationary convection, the effect of various parameters like medium permeability, dust particles, non-buoyancy magnetization, coupling parameter, spin-diffusion parameter and micropolar heat conduction parameter are analyzed. For sufficiently large values of magnetic parameter M 1 , the critical magnetic thermal Rayleigh number for the onset of instability is determined numerically and results are depicted graphically. It is also observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and dust particles.

Numerical modeling of dimethyl ether (DME) bubble growth and breakup

A numerical program is written to simulate the process of vapor bubble growth with spherical symmetry from the thermodynamic critical radius in an initially uniformly superheated liquid. The program is validated by the experimental data of superheated water. The calculated results agree with those of experiments well. The program takes into account the variations of properties with temperature precisely to simulate the DME bubble growth under flash boiling conditions. Considering the influences of pressure, surface tension and viscous stress, the linear stability analysis method is adopted to deduce the dispersion equation to represent the disturbance development during the bubble growth, and a new criterion for bubble breakup is established. The results show the bubble becomes more unstable with the increase of bubble Weber number and void fraction, and that with the increase of bubble growth rate or the decrease of initial radius ration of droplet to bubble, the breakup time of bubble becomes shorter.

STABILITY OF FLOW PAST TWO SIDE-BY-SIDE CIRCULAR CYLINDERS

The global stability of flow past two side-by-side circular cylinders is investigated by a linear global stability analysis method based on a spectral/element Navier-Stokes solver. The special attention is paid to the parameter regime of 1<T/D<3, where T is the spacing between two cylinder centers and D the cylinder diameter. A remarkable feature we found is that there exist two critical Reynolds numbers associated with twice bifurcations, first a pitchfork and second a Hopf one, for about 1.55<T/D<1.65. A new flow pattern namely biased steady pattern is identified. In addition, the transition from in-phase lock-in to anti-phase lock-in pattern for larger separation distance T/D>1.65 is proved to be related to the switch between two leading eigenmodes.

A parallel adaptive unstructured finite volume method for linear stability (normal mode) analysis of viscoelastic fluid flows

A parallel unstructured finite volume method is presented for analysis of the stability of two-dimensional steady Oldroyd-B fluid flows to small amplitude three-dimensional perturbations. A semi-staggered dilation-free finite volume discretization with Newton’s method is used to compute steady base flows. The linear stability problem is treated as a generalized eigenvalue problem (GEVP) in which the rightmost eigenvalue determines the stability of the base flow. The rightmost eigenvalues associated with the most dangerous eigenfunctions are computed through the use of the shift-invert Arnoldi method. To avoid fine meshing in the regions where the flow variables are changing slowly, a local mesh refinement technique is used in order to increase numerical accuracy with a lower computational cost. The CUBIT mesh generation environment has been used to refine the quadrilateral meshes locally. In order to achieve higher performance on parallel machines the algebraic systems of equations resulting from the steady problem and the GEVP have been solved by implementing the MUltifrontal Massively Parallel Solver ( MUMPS). The proposed method is applied to the linear stability analysis of the flow of an Oldroyd-B fluid past a linear periodic array of circular cylinders in a channel and a linear array of circular half cylinders placed on channel walls. Two different leading eigenfunctions are identified for close and wide cylinder spacing for the periodic array of cylinders. The numerical results indicate good agreement with the numerical and experimental results available in the literature.