scholarly journals Evans function computation for the stability of travelling waves

2018 ◽  
Vol 376 (2117) ◽  
pp. 20170184 ◽  
Author(s):  
B. Barker ◽  
J. Humpherys ◽  
G. Lyng ◽  
J. Lytle
Keyword(s):  
Travelling Waves ◽  
Evans Function ◽  
Detonation Waves ◽  
Model Parameters ◽  
Theme Issue ◽  
Function Computation ◽  
Computational Aspects ◽  
The Stability ◽  
Linearized Operator

In recent years, the Evans function has become an important tool for the determination of stability of travelling waves. This function, a Wronskian of decaying solutions of the eigenvalue equation, is useful both analytically and computationally for the spectral analysis of the linearized operator about the wave. In particular, Evans-function computation allows one to locate any unstable eigenvalues of the linear operator (if they exist); this allows one to establish spectral stability of a given wave and identify bifurcation points (loss of stability) as model parameters vary. In this paper, we review computational aspects of the Evans function and apply it to multidimensional detonation waves. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

Modeling the Vibratory Drilling Process

1999 ◽  
Author(s):  
Stephen A. Batzer ◽  
Alexander M. Gouskov ◽  
Sergey A. Voronov
Keyword(s):  
Rotational Velocity ◽  
Time Lag ◽  
Chip Thickness ◽  
Cutting Conditions ◽  
Model Parameters ◽  
Drilling Process ◽  
Workpiece Material ◽  
Velocity Amplitude ◽  
The Stability

Abstract The dynamic behavior of deep-hole vibratory drilling is analyzed. The mathematical model presented allows the determination of axial tool and workpiece displacements and cutting forces for significant dynamic system behavior such as the entrance of the cutting tool into workpiece material and exit. Model parameters include the actual rigidity of the tool and workpiece, time-varying chip thickness, time lag for chip formation due to tool rotation and possible disengagement of drill cutting edges from the workpiece due to tool and/or workpiece axial vibrations. The main features of this model are its nonlinearity and inclusion of time lag differential equations which require numeric solutions. The specific cutting conditions (feed, tool rotational velocity, amplitude and frequency of forced vibrations) necessary to obtain discontinuous chips and reliable removal are determined. The stability conditions of excited vibrations are also investigated. Calculated bifurcation diagrams make it possible to derive the domain of system parameters along with the determination of optimal cutting conditions.

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.

scholarly journals ANALYTICAL DETERMINATION OF INITIAL CONDITIONS LEADING TO FIRING IN NERVE FIBERS

2007 ◽  
Vol 17 (10) ◽  
pp. 3697-3701
Author(s):  
SABIR JACQUIR ◽  
STEPHANE BINCZAK ◽  
JEAN-MARIE BILBAULT
Keyword(s):  
Initial Conditions ◽  
Travelling Waves ◽  
Stationary Wave ◽  
Nerve Fibers ◽  
Analytical Determination ◽  
Action Potential Firing ◽  
Perturbative Method ◽  
Potential Firing ◽  
The Stability

An analytical solution characterizing initial conditions leading to action potential firing in smooth nerve fibers is determined, using the bistable equation. In the first place, we present a nontrivial stationary solution wave, then, using the perturbative method, we analyze the stability of this stationary wave. We show that it corresponds to a frontier between the initiation of the travelling waves and a decay to the resting state. Eventually, this analytical approach is extended to FitzHugh–Nagumo model.

PROBABILISTIC DETERMINATION OF STABILITY FOR A DELAY-DIFFERENTIAL MODEL OF THE GLUCOSE-INSULIN DYNAMICAL SYSTEM

1999 ◽  
Vol 07 (02) ◽  
pp. 131-144 ◽  
Author(s):  
A. DE GAETANO ◽  
O. ARINO
Keyword(s):  
Dynamical System ◽  
Equilibrium Point ◽  
Stability Criterion ◽  
Model Parameters ◽  
Qualitative Behavior ◽  
System Modelling ◽  
Differential Model ◽  
The Stability ◽  
Delay Differential

Some of the questions involved in the formulation of a new model for a physiological phenomenon, when the model represents a dynamical system, concern its qualitative behavior. The determination of the stability of a particular dynamical system is usually made analytically, from a linearization of the system around an equilibrium point. This analytic proof may often be complicated or impossible, leading to the imposition of conditions on the relative magnitude of the model structural parameters or to other partial results. We discuss a general technique whereby a probabilistic judgement is made on the stability of a dynamical system, and we apply it to the study of a particular delay differential system modelling the relationship between insulin secretion and glucose uptake. This technique is applicable in case experimental material is available: a stability criterion is obtained via the usual linearization around an equilibrium point and its distribution is approximated from the estimated dispersion of the model parameters. The probability that the stability criterion is such as to make the model stable can then be computed directly. While the conclusion is probabilistic in nature, it is generally applicable to a vast class of models.

STABILITY OF STEADY-STATE SOLUTIONS IN MULTIDIMENSIONAL COUPLED MAP LATTICES

2004 ◽  
Vol 14 (08) ◽  
pp. 2689-2699
Author(s):  
MIROSLAVA DUBCOVÁ ◽  
DANIEL TURZÍK ◽  
ALOIS KLÍČ
Keyword(s):  
Steady State ◽  
Coupled Map Lattices ◽  
Multidimensional Lattice ◽  
Spatially Periodic ◽  
Spatially Homogeneous ◽  
Periodic Steady State ◽  
The Stability ◽  
Linearized Operator ◽  
Steady State Solutions

Coupled map lattices with multidimensional lattice are considered. A method for the determination of the stability of spatially homogeneous and spatially periodic steady-state solutions is derived. This method is based on the determination of the spectrum of the linearized operator by means of Gelfand transformation of some appropriate Banach algebra. The results are applied to several examples.

scholarly journals An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices

2018 ◽  
Vol 376 (2117) ◽  
pp. 20170192 ◽  
Author(s):  
Haitao Xu ◽  
Jesús Cuevas-Maraver ◽  
Panayotis G. Kevrekidis ◽  
Anna Vainchtein
Keyword(s):  
Nonlinear Waves ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Point Of View ◽  
Floquet Multiplier ◽  
Theme Issue ◽  
Future Directions ◽  
Infinite Dimensional ◽  
The Stability ◽  
Delay Differential

In this work, we revisit a criterion, originally proposed in Friesecke & Pego (Friesecke & Pego 2004 Nonlinearity 17 , 207–227. ( doi:10.1088/0951715/17/1/013 )), for the stability of solitary travelling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-travelling frame, as well as at that of the Floquet problem arising when considering the travelling wave as a periodic orbit modulo shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the travelling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

scholarly journals Stability of Traveling Waves Based upon the Evans Function and Legendre Polynomials

2020 ◽  
Vol 10 (3) ◽  
pp. 846 ◽  
Author(s):  
H. M. Srivastava ◽  
H. I. Abdel-Gawad ◽  
Khaled M. Saad
Keyword(s):  
Stability Analysis ◽  
Fiber Optics ◽  
Traveling Waves ◽  
Legendre Polynomials ◽  
Reaction Diffusion ◽  
Evans Function ◽  
Reaction Diffusion Equations ◽  
Engineering Sciences ◽  
The Stability ◽  
Linearized Operator

One of the tools and techniques concerned with the stability of nonlinear waves is the Evans function which is an analytic function whose zeros give the eigenvalues of the linearized operator. Here, in this paper, we propose a direct approach, which is based essentially upon constructing the eigenfunction solution of the perturbed equation based upon the topological invariance in conjunction with usage of the Legendre polynomials, which have presumably not considered in the literature thus far. The associated Legendre eigenvalue problem arising from the stability analysis of traveling waves solutions is systematically studied here. The present work is of considerable interest in the engineering sciences as well as the mathematical and physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product. This can be controlled by depicting the initial concentration of the reactant, which is determined by its value at the bifurcation point. This analysis leads to the point separating stable and unstable solutions. As far as chemical reactions are described by reaction-diffusion equations, this specific concentration can be found mathematically. On the other hand, the study of stability analysis of solutions may depict whether or not a soliton pulse is well-propagated in fiber optics. This can, and should, be carried out by finding the solutions of the coupled nonlinear Schrödinger equations and by analyzing the stability of these solutions.

Determining the Norton’s Equivalent Model of Distribution System with Distributed Generation (DG) for Stability Analysis

2019 ◽  
Vol 12 (2) ◽  
pp. 190-198
Author(s):  
Sunny Katyara ◽  
Lukasz Staszewski ◽  
Faheem Akhtar Chachar
Keyword(s):  
Distributed Generation ◽  
Normal Condition ◽  
Distribution System ◽  
Distribution Networks ◽  
Equivalent Model ◽  
Stability Factor ◽  
Matlab Simulation ◽  
The Stability ◽  
Stability Issues

Background: Since the distribution networks are passive until Distributed Generation (DG) is not being installed into them, the stability issues occur in the distribution system after the integration of DG. Methods: In order to assure the simplicity during the calculations, many approximations have been proposed for finding the system’s parameters i.e. Voltage, active and reactive powers and load angle, more efficiently and accurately. This research presents an algorithm for finding the Norton’s equivalent model of distribution system with DG, considering from receiving end. Norton’s model of distribution system can be determined either from its complete configuration or through an algorithm using system’s voltage and current profiles. The algorithm involves the determination of derivative of apparent power against the current (dS/dIL) of the system. Results: This work also verifies the accuracy of proposed algorithm according to the relative variations in the phase angle of system’s impedance. This research also considers the varying states of distribution system due to switching in and out of DG and therefore Norton’s model needs to be updated accordingly. Conclusion: The efficacy of the proposed algorithm is verified through MATLAB simulation results under two scenarios, (i) normal condition and (ii) faulty condition. During normal condition, the stability factor near to 1 and change in dS/dIL was near to 0 while during fault condition, the stability factor was higher than 1 and the value of dS/dIL was away from 0.

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