Proof of the principle of linearized stability for one-variable systems

1995 ◽  
pp. 230-233
Keyword(s):  
Linearized Stability

scholarly journals Reduction principle at work

2021 ◽  
Vol 8 (1) ◽  
pp. 46-74
Author(s):  
Christian Pötzsche ◽  
Evamaria Russ
Keyword(s):  
Center Manifold ◽  
Reduction Principle ◽  
Critical Stability ◽  
Center Manifold Reduction ◽  
Infinite Dimensional ◽  
Linearized Stability ◽  
Dichotomy Spectrum ◽  
Nonautonomous Case ◽  
Necessary And Sufficient ◽  
Manifold Reduction

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

scholarly journals Existence and linearized stability of solitary waves for a quasilinear Benney system

2016 ◽  
Vol 146 (3) ◽  
pp. 547-564 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira ◽  
Filipe Oliveira
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Compact Support ◽  
Travelling Waves ◽  
Standing Waves ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Linearized Stability ◽  
Stability Of Solitary Waves ◽  
Image Position

We prove the existence of solitary wave solutions to the quasilinear Benney systemwhere , –1 < p < +∞ and a, γ > 0. We establish, in particular, the existence of travelling waves with speed arbitrarily large if p < 0 and arbitrarily close to 0 if . We also show the existence of standing waves in the case , with compact support if – 1 < p < 0. Finally, we obtain, under certain conditions, the linearized stability of such solutions.

Dynamics of a vortex filament in a shear flow

1984 ◽  
Vol 148 ◽  
pp. 477-497 ◽  
Author(s):  
Hassan Aref ◽  
Edward P. Flinchem
Keyword(s):  
Mixing Layer ◽  
Background Field ◽  
Nucleation Site ◽  
Finite Amplitude ◽  
Vortex Filament ◽  
Attractive Alternative ◽  
Morphological Similarity ◽  
Single Vortex ◽  
Linearized Stability ◽  
Model Equations

Motions of a single vortex filament in a background flow are studied by numerical simulation of a set of model equations. The model, which in essence is due to Hama, treats the self-interaction of the filament through the so-called ‘localized-induction approximation’ (LIA). Interaction with the prescribed background field is treated by simply advecting the filament appropriately. We are particularly interested in elucidating the evolution of sinuous vortices such as the ‘wiggle’ seen by Breidenthal in the transition to three-dimensionality in the mixing layer. The model studied embodies two of the simplest ingredients that must enter into any dynamical explanation: induction and advection. For finite-amplitude phenomena we make contact with the theory of solitons on strong vortices developed by Betchov and Hasimoto. In a shear, solitons cannot exist, but solitary waves can, and their interactions with the shear are found to be key ingredients for an understanding of the behaviour of the vortex filament. When sheared, a soliton seems to act as a ‘nucleation site’ for the generation of a family of waves. Computed sequences are shown that display a remarkable morphological similarity to flow-visualization studies. The present application of fully nonlinear dynamics to a model presents an attractive alternative to the extrapolations from linearized stability theory applied to the full equations that have so far constituted the theoretical basis for understanding the experimental results.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Stability of charged rotating (2 + 1)-dimensional wormholes

2020 ◽  
Vol 29 (01) ◽  
pp. 2050007 ◽  
Author(s):  
M. Sharif ◽  
Faisal Javed
Keyword(s):  
Thin Shell ◽  
Geometrical Structure ◽  
Energy Tensor ◽  
Exterior Region ◽  
Barotropic Fluid ◽  
Linearized Stability ◽  
Stress Energy ◽  
Thin Shell Wormholes ◽  
The Stability ◽  
Fast Rotating

This paper investigates the effects of charge on linearized stability of rotating thin-shell wormholes (WHs) filled with a barotropic fluid. We use Visser cut and paste technique to construct thin-shell from charged rotating Bañados–Teitelboim–Zanelli (BTZ) black holes (BHs). The components of stress-energy tensor are evaluated through Israel thin-shell formalism. The angular momentum for both interior as well as exterior region at the WH throat remains the same but opposite in direction, i.e. thin-shell WH at the throat is counter-rotated. It is found that the geometrical structure of WHs is more stable for highly charged and fast rotating thin-shell. We conclude that the stability regions of charged rotating WHs are larger than the uncharged rotating WHs.

scholarly journals Thermosolutal Instability in Compressible Viscoelastic Dusty Fluid through Porous Medium

2013 ◽  
Vol 18 (1) ◽  
pp. 99-112 ◽  
Author(s):  
P. Kumar ◽  
H. Mohan
Keyword(s):  
Porous Medium ◽  
Viscoelastic Fluid ◽  
Normal Mode Analysis ◽  
Suspended Particles ◽  
Mode Analysis ◽  
Stationary Convection ◽  
Linearized Stability ◽  
Medium Permeability ◽  
The Stability ◽  
Solute Gradient

Thermosolutal instability in a compressible Walters B’ viscoelastic fluid with suspended particles through a porous medium is considered. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For stationary convection, the Walters B’ viscoelastic fluid behaves like a Newtonian fluid and it is found that suspended particles and medium permeability have a destabilizing effect whereas the stable solute gradient and compressibility have a stabilizing effect on the system. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. The stable solute gradient and viscoelasticity are found to introduce oscillatory modes in the system which are non-existent in their absence.

Surface-tension-driven breakup of viscoelastic liquid threads

1982 ◽  
Vol 120 ◽  
pp. 245-266 ◽  
Author(s):  
Simon L. Goren ◽  
Moshe Gottlieb
Keyword(s):  
Relaxation Time ◽  
Stress Relaxation ◽  
Polymer Solutions ◽  
Small Diameter ◽  
Shear Thinning ◽  
Prior History ◽  
Axial Tension ◽  
Dilute Polymer Solutions ◽  
Linearized Stability ◽  
Newtonian Liquids

A linearized stability analysis is carried out for the breakup of small-diameter liquid filaments of dilute polymer solutions into droplets. Oldroyd's 8-constant model expressed in a corotational reference frame is used as the rheological equation of state. The crucial idea in this theory is the recognition that the liquid may be subject to an unrelaxed axial tension due to its prior history. If the tension is zero, the present analysis predicts that jets of shear-thinning liquids are less stable than comparable jets of Newtonian liquids; this is in agreement with previous analyses. However, when the axial tension is not zero, and provided the stress relaxation time constant is sufficiently large, the new theory predicts that the axial elastic tension can be a significant stabilizing influence. With reasonable values for the tension and stress relaxation time the theory explains the great stability observed for jets of some shear- thinning, dilute polymer solutions. The theory explains why drops produced from jets of such liquids are larger than drops from corresponding Newtonian liquids. The theory also appears capable of explaining the sudden appearance of irregularly spaced bulges on jets after long distances of t,ravel with little amplification of disturbances.

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