On the Stability of Large Amplitude Semi-discrete Shock Profiles by Means of an Evans Function in Infinite Dimensions

2001 ◽  
pp. 149-157
Author(s):  
Sylvie Benzoni-Gavage
Keyword(s):  
Large Amplitude ◽  
Evans Function ◽  
Infinite Dimensions ◽  
Shock Profiles ◽  
The Stability

scholarly journals Corrigendum

1979 ◽  
Vol 22 (3) ◽  
pp. 571-572
Author(s):  
E. Infeld ◽  
G. Rowlands
Keyword(s):  
Large Amplitude ◽  
Plasma Waves ◽  
Relativistic Plasma ◽  
Small Perturbations ◽  
Transverse Waves ◽  
The Stability

In this note we investigate the stability of large-amplitude longitudinal relativistic plasma waves. We find that they are secularly stable, that is, small perturbations grow in time proportional to time but not exponentially with time. Similar results have recently been obtained for transverse waves by Romeiras (1978)

Finite-amplitude internal wavepacket dispersion and breaking

2001 ◽  
Vol 429 ◽  
pp. 343-380 ◽  
Author(s):  
BRUCE R. SUTHERLAND
Keyword(s):  
Numerical Simulations ◽  
Internal Waves ◽  
Large Amplitude ◽  
Finite Amplitude ◽  
Critical Amplitude ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Stability ◽  
The Waves ◽  
Horizontal Wavelength

The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic internal wavepackets are examined analytically and by numerical simulations. The weakly nonlinear dispersion relation for horizontally periodic, vertically compact internal waves is derived and the results are applied to assess the stability of weakly nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of constant phase make with the vertical, the wavepackets are predicted to be unstable if [mid ]Θ[mid ] < Θc, where Θc = cos−1 (2/3)1/2 ≃ 35.3° is the angle corresponding to internal waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] < Θc increases initially, but then decreases as the wavepacket subdivides into a wave train, following the well-known Fermi–Pasta–Ulam recurrence phenomenon.If the initial wavepacket is of sufficiently large amplitude, it becomes unstable in the sense that eventually it convectively overturns. Two new analytic conditions for the stability of quasi-plane large-amplitude internal waves are proposed. These are qualitatively and quantitatively different from the parametric instability of plane periodic internal waves. The ‘breaking condition’ requires not only that the wave is statically unstable but that the convective instability growth rate is greater than the frequency of the waves. The critical amplitude for breaking to occur is found to be ACV = cot Θ (1 + cos2 Θ)/2π, where ACV is the ratio of the maximum vertical displacement of the wave to its horizontal wavelength. A second instability condition proposes that a statically stable wavepacket may evolve so that it becomes convectively unstable due to resonant interactions between the waves and the wave-induced mean flow. This hypothesis is based on the assumption that the resonant long wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves linearly in time, continues to amplify the waves in the fully nonlinear regime. Using linear theory estimates, the critical amplitude for instability is ASA = sin 2Θ/(8π2)1/2. The results of numerical simulations of horizontally periodic, vertically compact wavepackets show excellent agreement with this latter stability condition. However, for wavepackets with horizontal extent comparable with the horizontal wavelength, the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It is proposed that these results may explain why internal waves generated by turbulence in laboratory experiments are often observed to be excited within a narrow frequency band corresponding to Θ less than approximately 45°.

Evans function analysis of the stability of non-adiabatic flames

2003 ◽  
Vol 7 (3) ◽  
pp. 545-561 ◽  
Author(s):  
Peter L Simon ◽  
Serafim Kalliadasis ◽  
John H Merkin ◽  
Stephen K Scott
Keyword(s):  
Function Analysis ◽  
Evans Function ◽  
The Stability

STABILITY OF SHOCK PROFILES FOR NONCONVEX SCALAR VISCOUS CONSERVATION LAWS

1995 ◽  
Vol 05 (03) ◽  
pp. 279-296 ◽  
Author(s):  
MING MEI
Keyword(s):  
Conservation Laws ◽  
Decay Rates ◽  
Asymptotically Stable ◽  
Time Decay ◽  
Shock Condition ◽  
Viscous Conservation Laws ◽  
Time Decay Rates ◽  
Shock Profiles ◽  
Degenerate Shock ◽  
The Stability

This paper is to study the stability of shock profiles for nonconvex scalar viscous conservation laws with the nondegenerate and the degenerate shock conditions by means of an elementary energy method. In both cases, the shock profiles are proved to be asymptotically stable for suitably small initial disturbances. Moreover, in the case of nondegenerate shock condition, time decay rates of asymptotics are also obtained.

A stability index for travelling waves in activator-inhibitor systems

2019 ◽  
Vol 150 (1) ◽  
pp. 517-548
Author(s):  
Paul Cornwell ◽  
Christopher K. R. T. Jones
Keyword(s):  
Eigenvalue Problem ◽  
Symplectic Form ◽  
Travelling Waves ◽  
Stability Index ◽  
Maslov Index ◽  
Evans Function ◽  
Conjugate Points ◽  
Spatial Evolution ◽  
The Stability ◽  
Activator Inhibitor

AbstractWe consider the stability of nonlinear travelling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard symplectic form. This allows us to define a Maslov index for the wave corresponding to the spatial evolution of the unstable bundle. We formulate the Evans function for the eigenvalue problem and show that the parity of the Maslov index determines the sign of the derivative of the Evans function at the origin. The connection between the Evans function and the Maslov index is established by a ‘detection form,’ which identifies conjugate points for the curve of Lagrangian planes.

Multistability in the Centrifugal Governor System Under a Time-Delay Control Strategy

2019 ◽  
Vol 14 (11) ◽  
Author(s):  
Shuning Deng ◽  
Jinchen Ji ◽  
Shan Yin ◽  
Guilin Wen
Keyword(s):  
Time Delay ◽  
Control Strategy ◽  
Large Amplitude ◽  
Stability Boundary ◽  
Physical Parameters ◽  
Time Delay Control ◽  
Delay Control ◽  
Delayed Feedback Controller ◽  
Trivial Equilibrium ◽  
The Stability

Abstract The centrifugal governor system plays an indispensable role in maintaining the near-constant speed of engines. Although different arrangements have been developed, the governor systems are still applied in many machines for its simple mechanical structure. Therefore, the large-amplitude vibrations of the governor system which can lead to the function failure of the system should be attenuated to guarantee reliable operation. This paper adopts a time-delay control strategy to suppress the undesirable large-amplitude motions in the centrifugal governor system, which can be regarded as the practical application of the delayed feedback controller in this system. The stability region of the trivial equilibrium of the controlled system is determined by investigating the characteristic equation and generic Hopf bifurcations. It is found that the dynamic behavior of multistability can be induced by the Bautin bifurcation, arising on the stability boundary of the trivial equilibrium with a constant delay. More specifically, a coexistence of two desirable stable motions, i.e., an equilibrium or a small-amplitude periodic motion, can be observed in the controlled centrifugal governor system without changing the physical parameters. This is a new feature of the motion control in the centrifugal governor systems, which has not yet been reported in the existing studies. Finally, the results of theoretical analyses are verified by numerical simulations.

scholarly journals After Core Collapse. What?

1988 ◽  
Vol 126 ◽  
pp. 379-392
Author(s):  
Haldan Cohn
Keyword(s):  
Computer Simulations ◽  
Globular Cluster ◽  
Large Amplitude ◽  
Globular Clusters ◽  
Dynamical Evolution ◽  
Core Collapse ◽  
Expansion Phase ◽  
Cluster Evolution ◽  
Detailed Computer ◽  
The Stability

As our understanding of core collapse in globular clusters has improved through detailed computer simulations, attention has naturally turned to dynamical evolution of globular clusters after core collapse. The results of recent simulations of post-collapse cluster evolution are reviewed. An assessment is given of progress towards the goal of developing astrophysically realistic models that cover all phases of globular cluster evolution. A focus of this review is the stability of the post-collapse expansion phase to the large amplitude core oscillations first observed in the simulations of Sugimoto and Bettwieser and now confirmed by several other studies. The implications of core oscillations for the observation of post-collapse clusters are discussed.

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