A note on the stability of pencils of plane curves

The stability of the MHD tangential discontinuity is studied in compressible plasmas in the presence of anisotropic viscosity and thermal conductivity. The general dispersion equation is derived, and solutions to this dispersion equation and stability criteria are obtained for the limiting cases of incompressible and cold plasmas. In these two limiting cases the effect of thermal conductivity vanishes, and the solutions are only influenced by viscosity. The stability criteria for viscous plasmas are compared with those for ideal plasmas, where stability is determined by the Kelvin—Helmholtz velocity VKH as a threshold for the difference in the equilibrium velocities. Viscosity turns out to have a destabilizing influence when the viscosity coefficient takes different values at the two sides of the discontinuity. Viscosity lowers the threshold velocity V below the ideal Kelvin—Helmholtz velocity VKH, so that there is a range of velocities between V and VKH where the overstability is of a dissipative nature.

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Intrinsic Thermal Stability of Anisotropic Thin-Film Superconductors

Journal of Heat Transfer ◽

10.1115/1.2910330 ◽

1990 ◽

Vol 112 (1) ◽

pp. 10-15 ◽

Cited By ~ 21

Author(s):

M. I. Flik ◽

C. L. Tien

Keyword(s):

Thin Film ◽

Thermal Stability ◽

Stability Criterion ◽

High Temperature Superconductors ◽

Stability Criteria ◽

Anisotropic Thermal Conductivity ◽

Operating Current ◽

Released Energy ◽

The Stability ◽

Thermal Stability Of

Intrinsic thermal stability denotes a situation where a superconductor can carry the operating current without resistance at all times after the occurrence of a localized release of thermal energy. This novel stability criterion is different from the cryogenic stability criteria for magnets and has particular relevance to thin-film superconductors. Crystals of ceramic high-temperature superconductors are likely to exhibit anisotropic thermal conductivity. The resultant anisotropy of highly oriented films of superconductors greatly influences their thermal stability. This work presents an analysis for the maximum operating current density that ensures intrinsic stability. The stability criterion depends on the amount of released energy, the Biot number, the aspect ratio, and the ratio of the thermal conductivities in the plane of the film and normal to it.

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The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

Mathematical Problems in Engineering ◽

10.1155/2015/348235 ◽

2015 ◽

Vol 2015 ◽

pp. 1-7

Author(s):

Rui Zhang ◽

Yinjing Guo ◽

Xiangrong Wang ◽

Xueqing Zhang

Keyword(s):

Optimal Control ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Stochastic Stability ◽

Stochastic Systems ◽

Stability Criteria ◽

Two Measures ◽

The Mean ◽

The Stability

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

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The Stability Criteria with Compound Matrices

Abstract and Applied Analysis ◽

10.1155/2013/930576 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5

Author(s):

Yazhuo Zhang ◽

Baodong Zheng

Keyword(s):

Dynamical Systems ◽

Spectral Property ◽

Discrete Dynamical Systems ◽

Asymptotical Stability ◽

Hopf Bifurcations ◽

Stability Criteria ◽

Bifurcation Problem ◽

Compound Matrices ◽

The Stability ◽

Schur Stability

The bifurcation problem is one of the most important subjects in dynamical systems. Motivated by M. Li et al. who used compound matrices to judge the stability of matrices and the existence of Hopf bifurcations in continuous dynamical systems, we obtained some effective methods to judge the Schur stability of matrices on the base of the spectral property of compound matrices, which can be used to judge the asymptotical stability and the existence of Hopf bifurcations of discrete dynamical systems.

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Asymptotic Stability of Caputo Type Fractional Neutral Dynamical Systems with Multiple Discrete Delays

Abstract and Applied Analysis ◽

10.1155/2014/138124 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Hai Zhang ◽

Daiyong Wu ◽

Jinde Cao

Keyword(s):

Asymptotic Stability ◽

Matrix Theory ◽

Sufficient Conditions ◽

Algebraic Approach ◽

Stability Criteria ◽

Differential Systems ◽

Discrete Delays ◽

Transcendental Equations ◽

The Stability ◽

Theoretical Results

We discuss the delay-independent asymptotic stability of Caputo type fractional-order neutral differential systems with multiple discrete delays. Based on the algebraic approach and matrix theory, the sufficient conditions are derived to ensure the asymptotic stability for all time-delay parameters. By applying the stability criteria, one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and convenient. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed theoretical results.

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Effect of Low-Frequency Modulation on Thermal Convection Instability

Zeitschrift für Naturforschung A ◽

10.1515/zna-2001-0614 ◽

2001 ◽

Vol 56 (6-7) ◽

pp. 509-522 ◽

Cited By ~ 8

Author(s):

P. K. Bhatia ◽

B. S. Bhadauria

Keyword(s):

Asymptotic Solution ◽

Temperature Difference ◽

Frequency Modulation ◽

Thermal Convection ◽

Low Frequency ◽

Horizontal Layer ◽

Time Dependent ◽

Stability Criteria ◽

Steady Temperature ◽

The Stability

Abstract The stability of a horizontal layer of fluid heated from below is examined when, in addition to a steady temperature difference between the horizontal walls of the layer a time-dependent low-frequency per turbation is applied to the wall temperatures. An asymptotic solution is obtained which describes the be haviour of infinitesimal disturbances to this configuration. Possible stability criteria are analyzed and the results are compared with the known experimental as well as numerical results.

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Predicting Collapse of Complex Ecological Systems: Quantifying the Stability-Complexity Continuum

10.1101/713578 ◽

2019 ◽

Author(s):

Susanne Pettersson ◽

Van M. Savage ◽

Martin Nilsson Jacobi

Keyword(s):

Single Species ◽

Ecological Systems ◽

Ecological Research ◽

Stability Criteria ◽

Numerical Techniques ◽

Intermediate Regime ◽

Limits Of Stability ◽

Species Abundances ◽

The Stability ◽

Degree Of Instability

Dynamical shifts between the extremes of stability and collapse are hallmarks of ecological systems. These shifts are limited by and change with biodiversity, complexity, and the topology and hierarchy of interactions. Most ecological research has focused on identifying conditions for a system to shift from stability to any degree of instability—species abundances do not return to exact same values after perturbation. Real ecosystems likely have a continuum of shifting between stability and collapse that depends on the specifics of how the interactions are structured, as well as the type and degree of disturbance due to environmental change. Here we map boundaries for the extremes of strict stability and collapse. In between these boundaries, we find an intermediate regime that consists of single-species extinctions, which we call the Extinction Continuum. We also develop a metric that locates the position of the system within the Extinction Continuum—thus quantifying proximity to stability or collapse—in terms of ecologically measurable quantities such as growth rates and interaction strengths. Furthermore, we provide analytical and numerical techniques for estimating our new metric. We show that our metric does an excellent job of capturing the system behaviour in comparison with other existing methods—such as May’s stability criteria or critical slowdown. Our metric should thus enable deeper insights about how to classify real systems in terms of their overall dynamics and their limits of stability and collapse.

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Mathematical Modelling of Deforestation Due to Population Density and Industrialization

Jurnal Varian ◽

10.30812/varian.v5i1.1412 ◽

2021 ◽

Vol 5 (1) ◽

pp. 9-16

Author(s):

Didiharyono D. ◽

Irwan Kasse

Keyword(s):

Numerical Simulation ◽

Population Density ◽

Mathematical Modelling ◽

Equilibrium Point ◽

Equilibrium Points ◽

Stability Criteria ◽

Hurwitz Stability ◽

Stable Conditions ◽

Linear Differential ◽

The Stability

The focus of the study in this paper is to model deforestation due to population density and industrialization. To begin with, it is formulated into a mathematical modelling which is a system of non-linear differential equations. Then, analyze the stability of the system based on the Routh-Hurwitz stability criteria. Furthermore, a numerical simulation is performed to determine the shift of a system. The results of the analysis to shown that there are seven non-negative equilibrium points, which in general consist equilibrium point of disturbance-free and equilibrium points of disturbances. Equilibrium point TE7(x, y, z) analyzed to shown asymptotically stable conditions based on the Routh-Hurwitz stability criteria. The numerical simulation results show that if the stability conditions of a system have been met, the system movement always occurs around the equilibrium point.

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