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’.

Numerical Nonlinear Analysis for Dynamic Stability of an Ankle-Hip Model of Balance on a Balance Board

2019 ◽  
Vol 14 (10) ◽  
Author(s):  
Erik Chumacero-Polanco ◽  
James Yang ◽  
James Chagdes
Keyword(s):  
Dynamical System ◽  
Dynamic Stability ◽  
Point Of View ◽  
Local Dynamic ◽  
Stability Properties ◽  
Dynamical Behaviors ◽  
Local Dynamic Stability ◽  
The Stability ◽  
Delay Differential ◽  
Balance Board

Abstract Study of human upright posture (UP) stability is of great relevance to fall prevention and rehabilitation, especially for those with balance deficits for whom a balance board (BB) is a widely used mechanism to improve balance. The stability of the human-BB system has been widely investigated from a dynamical system point of view. However, most studies assume small disturbances, which allow to linearize the nonlinear human-BB dynamical system, neglecting the effect of the nonlinear terms on the stability. Such assumption has been useful to simplify the system and use bifurcation analyses to determine local dynamic stability properties. However, dynamic stability analysis results through such linearization of the system have not been verified. Moreover, bifurcation analyses cannot provide insight on dynamical behaviors for different points within the stable and unstable regions. In this study, we numerically solve the nonlinear delay differential equation that describes the human-BB dynamics for a range of selected parameters (proprioceptive feedback and time-delays). The resulting solutions in time domain are used to verify the stability properties given by the bifurcation analyses and to compare different dynamical behaviors within the regions. Results show that the selected bifurcation parameters have significant impacts not only on UP stability but also on the amplitude, frequency, and increasing or decaying rate of the resulting trajectory solutions.

EXISTENCE OF TRAVELLING WAVES IN NONLINEAR SI EPIDEMIC MODELS

2009 ◽  
Vol 17 (04) ◽  
pp. 643-657 ◽  
Author(s):  
FEN-FEN ZHANG ◽  
GANG HUO ◽  
QUAN-XING LIU ◽  
GUI-QUAN SUN ◽  
ZHEN JIN
Keyword(s):  
Control Strategies ◽  
Travelling Waves ◽  
Equilibrium Points ◽  
Heteroclinic Orbit ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Point Of View ◽  
Spatially Extended ◽  
Theoretical Results ◽  
Disease Free

In this paper, we investigate a spatially extended SI epidemic system with a nonlinear incidence rate. Using mathematical analysis, we study the existence of a heteroclinic orbit connecting two equilibrium points in R3 which corresponds to a travelling wave solution connecting the disease-free and endemic equilibria for the reaction-diffusion system. In other words, the travelling wave solutions of the model are studied to determine the speed of disease dissemination, form the biological point of view. Moreover, this wave speed is obtained as a function of the model's parameters, in order to assess the control strategies. Also, our theoretical results are confirmed by numerical simulations. The obtained results confirm that travelling wave can enhance the spread of the disease, which can provide some insights into controlling the disease.

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’.

An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

2011 ◽  
Vol 141 (6) ◽  
pp. 1141-1173 ◽  
Author(s):  
Jared C. Bronski ◽  
Mathew A. Johnson ◽  
Todd Kapitula
Keyword(s):  
Blow Up ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Classical Mechanics ◽  
Index Theorem ◽  
Travelling Wave ◽  
Kdv Equations ◽  
The Kdv Equation ◽  
De Vries ◽  
The Stability

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

Convergence rates to travelling waves for a nonconvex relaxation model

1998 ◽  
Vol 128 (5) ◽  
pp. 1053-1068 ◽  
Author(s):  
Ming Mei ◽  
Tong Yang
Keyword(s):  
Convergence Rates ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Decay Rates ◽  
Relaxation Model ◽  
Weighted Energy Method ◽  
Time Decay Rates ◽  
Algebraic Forms ◽  
The Stability ◽  
Nonconvex Relaxation

In this paper we study the asymptotic behaviour of the solution for a nonconvex relaxation model. The time decay rates in both the exponential and algebraic forms of the travelling wave solutions are shown by the weighted energy method. Our results develop and improve the stability theory in [8,9].

scholarly journals Thermal solitons: travelling waves in combustion

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120587 ◽  
Author(s):  
Lawrence K. Forbes
Keyword(s):  
Travelling Waves ◽  
Reaction System ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Governing Equations ◽  
Competitive Reaction ◽  
Chemical Reagent ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
The Stability

A competitive reaction system is considered, under which some chemical reagent decays by means of two simultaneous chemical reactions to form two separate inert products. One reaction is exothermic, and the other is endothermic. The governing equations for the model are presented, and a weakly nonlinear theory is then generated using the method of strained coordinates. Travelling-wave solutions are possible in the model, and the temperature is found to have a classical sech-squared profile. The stability of these moderate-amplitude temperature solitons is confirmed both analytically and numerically.

AN ABSTRACT DIFFERENTIAL EQUATION ARISING FROM CELL POPULATION DYNAMICS

1995 ◽  
Vol 03 (02) ◽  
pp. 469-481
Author(s):  
OVIDE ARINO ◽  
EVA SÁNCHEZ
Keyword(s):  
Differential Equation ◽  
Population Dynamics ◽  
Cell Population ◽  
Abstract Differential Equation ◽  
Cell Population Dynamics ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Stability ◽  
The One ◽  
Delay Differential

We provide an analysis of the stability and bifurcation properties of the solutions of an abstract differential nonlinear equation arising from cell population dynamics. The work surveyed here stems from a remark we made with respect to these equations: that it is possible to associate to any of them a delay differential equation on an infinite dimensional vector space. Perturbation theory for nonlinear equations similar to the one known for delay differential equations on finite dimensional spaces could possibly yield the same results as for those equations.

scholarly journals Stability of nonlinear waves and patterns and related topics

2018 ◽  
Vol 376 (2117) ◽  
pp. 20180001
Author(s):  
Anna Ghazaryan ◽  
Stephane Lafortune ◽  
Vahagn Manukian
Keyword(s):  
Coherent Structures ◽  
Nonlinear Waves ◽  
Dynamic Properties ◽  
Travelling Waves ◽  
Complex Structure ◽  
Nonlinear Partial Differential Equations ◽  
Theme Issue ◽  
Hybrid Techniques ◽  
Wave Trains ◽  
The Waves

Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more complex structure often occur in natural and experimentally built systems. In mathematics, these objects are realized as solutions of nonlinear partial differential equations. The existence, dynamic properties and bifurcations of those solutions are of interest. In particular, their stability is important for applications, as the waves that are observable are usually stable. When the waves are unstable, further investigation is warranted of the way the instability is exhibited, i.e. the nature of the instability, and also coherent structures that appear as a result of an instability of travelling waves. A variety of analytical, numerical and hybrid techniques are used to study travelling waves and their properties. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

Stability of Up-milling and Down-milling Operations with Variable Spindle Speed

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1151-1168 ◽  
Author(s):  
Xinhua Long ◽  
B. Balachandran
Keyword(s):  
Spindle Speed ◽  
Depth Of Cut ◽  
Control Parameters ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Milling Operations ◽  
The Stability ◽  
Delay Differential ◽  
Milling Dynamics

In this article, a stability treatment is presented for up-milling and down-milling processes with a variable spindle speed (VSS). This speed variation is introduced by superimposing a sinusoidal modulation on a nominal spindle speed. The VSS milling dynamics is described by a set of delay differential equations with time varying periodic coefficients and a time delay. A semi-discretization scheme is used to discretize the system over one period, and the infinite-dimensional transition matrix is reduced to a finite-dimensional matrix over this period. The eigenvalues of this finite-dimensional matrix provide information on VSS milling stability with respect to control parameters, such as the axial depth of cut and the nominal spindle speed. The stability charts obtained for VSS milling operations are compared with those obtained for constant spindle speed milling operations, and the benefits of VSS milling operations are discussed.

Stability Analysis of a Variable Spindle Speed Milling Process

2005 ◽  
Author(s):  
X.-H. Long ◽  
B. Balachandran
Keyword(s):  
Milling Process ◽  
Spindle Speed ◽  
Depth Of Cut ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Milling Operations ◽  
Milling Operation ◽  
The Stability ◽  
Delay Differential

In this effort, a stability treatment is presented for a milling process with a variable spindle speed (VSS). This variation is caused by superimposing a sinusoidal modulation on a nominal spindle speed. The dynamics of the VSS milling process is described by a set of delay differential equations (DDEs) with time varying periodic coefficients and a time delay. A semi-discretization scheme is used to discretize the system over one period, and the infinite dimensional transition matrix is converted to a finite dimensional matrix over this period. The eigenvalues of this finite dimensional matrix are used to determine the stability of the VSS milling operation with respect to selected control parameters, such as the axis depth of cut and the nominal spindle speed. The benefits of VSS milling operations are discussed by comparing the stability charts obtained for VSS milling operations with those obtained for constant spindle speed (CSS) milling operations.

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