ABUNDANCE OF TRAVELING WAVES IN COUPLED CIRCLE MAPS

2000 ◽  
Vol 10 (09) ◽  
pp. 2061-2073 ◽  
Author(s):  
WEN-XIN QIN
Keyword(s):  
Fixed Point ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Initial Conditions ◽  
Periodic Structures ◽  
Time Evolution Operator ◽  
Circle Maps ◽  
Temporal Complexity ◽  
Spatial Periodicity ◽  
The Stability

In this paper we study the existence of traveling waves with any rational velocity in coupled circle maps. We introduce an induced map, then a traveling wave with velocity p/q corresponds to a fixed point of the induced map. Moreover, the stability of a traveling wave is equivalent to that of the corresponding fixed point. Space translational system is chaotic on the set of traveling waves with rational velocities. In addition, time evolution operator exhibits sensitivity-like behavior with respect to initial conditions. By investigating the spatial periodicity of traveling waves, we obtain infinitely many space-time periodic structures. We also consider spatial asymptoticity of these traveling waves, which leads to the existence of fronts, defect solutions and soliton-like solutions. The abundance of traveling waves may be regarded as a signature of the spatial-temporal complexity in extended systems.

STABILITY OF STEADY STATE AND TRAVELING WAVES SOLUTIONS IN COUPLED MAP LATTICES

2008 ◽  
Vol 18 (01) ◽  
pp. 219-225 ◽  
Author(s):  
DANIEL TURZÍK ◽  
MIROSLAVA DUBCOVÁ
Keyword(s):  
Steady State ◽  
Essential Spectrum ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Linear Operators ◽  
Coupled Map Lattices ◽  
Basic Tool ◽  
Wave Solutions ◽  
The Stability

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

scholarly journals Synaptic propagation in neuronal networks with finite-support space dependent coupling

2020 ◽  
Author(s):  
Ricardo Erazo Toscano ◽  
Remus Osan
Keyword(s):  
Time Constant ◽  
Sensory Processing ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Neuronal Networks ◽  
Evolution Equations ◽  
Network Connectivity ◽  
Space Constant ◽  
The Stability ◽  
The Brain

1AbstractTraveling waves of electrical activity are ubiquitous in biological neuronal networks. Traveling waves in the brain are associated with sensory processing, phase coding, and sleep. The neuron and network parameters that determine traveling waves’ evolution are synaptic space constant, synaptic conductance, membrane time constant, and synaptic decay time constant. We used an abstract neuron model to investigate the propagation characteristics of traveling wave activity. We formulated a set of evolution equations based on the network connectivity parameters. We numerically investigated the stability of the traveling wave propagation with a series of perturbations with biological relevance.

scholarly journals Stability of Traveling Waves to a Burgers Equation with 2nd-Order Nonlinear Diffusion

2022 ◽  
Vol 4 (1) ◽  
pp. 77-85
Author(s):  
Mohammad Ghani
Keyword(s):  
Asymptotic Stability ◽  
Small Perturbation ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Nonlinear Diffusion ◽  
Wave Amplitude ◽  
Energy Estimate ◽  
Burgers Equation ◽  
Original Equation ◽  
The Stability

We are interested in the study of asymptotic stability for Burgers equation with second-order nonlinear diffusion. We first transform the original equation by the ansatz transformation to establish the existence of traveling wave. We further employ the energy estimate under small perturbation and arbitrary wave amplitude. This energy estimate is then used to establish the stability.

TRAVELING WAVES OF AN EPIDEMIC MODEL WITH VACCINATION

2013 ◽  
Vol 06 (05) ◽  
pp. 1350033 ◽  
Author(s):  
ZHIPING WANG ◽  
RUI XU
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Epidemic Model ◽  
Wave Solution ◽  
Local Stability ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solution ◽  
Spatial Diffusion

An epidemic model with vaccination and spatial diffusion is studied. By analyzing the corresponding characteristic equations, the local stability of each of feasible steady states to this model is discussed. The existence of a traveling wave solution is established by using the technique of upper and lower solutions and Schauder's fixed point theorem. Numerical simulations are carried out to illustrate the main results.

scholarly journals Vanishing Waves on Closed Intervals and Propagating Short-Range Phenomena

2008 ◽  
Vol 2008 ◽  
pp. 1-14 ◽  
Author(s):  
Ghiocel Toma ◽  
Flavia Doboga
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Initial Conditions ◽  
Propagation Direction ◽  
Closed Space ◽  
Middle Point ◽  
Closed Intervals ◽  
Null Values ◽  
Derivatives Of ◽  
Space Interval

This study presents mathematical aspects of wave equation considered on closed space intervals. It is shown that a solution of this equation can be represented by a certain superposition of traveling waves with null values for the amplitude and for the time derivatives of the resulting wave in the endpoints of this interval. Supplementary aspects connected with the possible existence of initial conditions for a secondorder differential system describing the amplitude of these localized oscillations are also studied, and requirements necessary for establishing a certain propagation direction for the wave (rejecting the possibility of reverse radiation) are also presented. Then it is shown that these aspects can be extended to a set of adjacent closed space intervals, by considering that a certain traveling wave propagating from an endpoint to the other can be defined on each space interval and a specific mathematical law (which can be approximated by a differential equation) describes the amplitude of these localized traveling waves as related to the space coordinates corresponding to the middle point of the interval. Using specific differential equations, it is shown that the existence of such propagating law for the amplitude of localized oscillations can generate periodical patterns and can explain fracture phenomena inside materials as well.

scholarly journals Nonlinear Stability of Periodic Traveling Wave Solutions for(n +1)-Dimensional Coupled Nonlinear Klein-Gordon Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Cong Sun ◽  
Bo Jiang
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Orbital Stability ◽  
Traveling Wave Solutions ◽  
Optical Fiber Communication ◽  
Fiber Communication ◽  
Periodic Traveling Waves ◽  
Klein Gordon ◽  
Periodic Traveling Wave ◽  
The Stability

We study the existence and orbital stability of smooth periodic traveling waves solutions of the(n +1)-dimensional coupled nonlinear Klein-Gordon equations. Such a system occurs in quantum mechanics, fluid mechanics, and optical fiber communication. Inspired by Angulo Pava’s results (2007), and by applying the stability theory established by Grillakis et al. (1987), we prove the existence of periodic traveling waves solutions and obtain the orbital stability of the solutions to this system.

STABILITY OF STEADY STATES AND EXISTENCE OF TRAVELING WAVES FOR A HOST-VECTOR EPIDEMIC

2011 ◽  
Vol 21 (06) ◽  
pp. 1667-1687 ◽  
Author(s):  
CHUFEN WU ◽  
PEIXUAN WENG
Keyword(s):  
Fixed Point ◽  
Singular Perturbation ◽  
Traveling Waves ◽  
Fixed Point Theorems ◽  
Convex Sets ◽  
Steady States ◽  
The Stability

We study the stability of steady states and establish the existence of traveling waves for a diffusive host-vector epidemic with a nonlocal spatiotemporal interaction. We develop the techniques of contracting-convex-sets, limit argument, singular perturbation and fixed point theorems.

scholarly journals Stability of Traveling Waves to the Lotka-Volterra Competition Model

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Ahmad Alhasanat ◽  
Chunhua Ou
Keyword(s):  
Global Stability ◽  
Comparison Principle ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Functional Space ◽  
Cooperative System ◽  
Wave Solutions ◽  
Diffusive Model ◽  
The Stability

In this paper, the stability of traveling wave solutions to the Lotka-Volterra diffusive model is investigated. First, we convert the model into a cooperative system by a special transformation. The local and the global stability of the traveling wavefronts are studied in a weighted functional space. For the global stability, comparison principle together with the squeezing technique is applied to derive the main results.

scholarly journals Traveling Wave Solutions for Some Three-Species Predator-Prey Systems

2021 ◽  
Vol 52 (1) ◽  
pp. 25-36
Author(s):  
Jong-Shenq Guo
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Predator Prey ◽  
Wave Solutions ◽  
Recent Developments ◽  
Wave Profiles ◽  
Predator Prey Models

In this paper, we present some recent developments on the application of Schauder’s fixed point theorem to the existence of traveling waves for some three-species predator-prey systems. The existence of traveling waves of predator-prey systems is closely related to the invasion phenomenon of some alien species to the habitat of aboriginal species. Three different three-species predator-prey models with different invaded and invading states are presented. In this paper, we focus on the methodology of deriving the convergence of stale tail of wave profiles.

NONLINEAR STABILITY OF LARGE AMPLITUDE VISCOUS SHOCK WAVES OF A GENERALIZED HYPERBOLIC–PARABOLIC SYSTEM ARISING IN CHEMOTAXIS

2010 ◽  
Vol 20 (11) ◽  
pp. 1967-1998 ◽  
Author(s):  
TONG LI ◽  
ZHI-AN WANG
Keyword(s):  
Nonlinear Stability ◽  
Parabolic System ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Band ◽  
Nonlinear Kinetics ◽  
Diffusive Limit ◽  
Viscous Shock Waves ◽  
The Stability

Traveling wave (band) behavior driven by chemotaxis was observed experimentally by Adler1,2 and was modeled by Keller and Segel.15 For a quasilinear hyperbolic–parabolic system that arises as a non-diffusive limit of the Keller–Segel model with nonlinear kinetics, we establish the existence and nonlinear stability of traveling wave solutions with large amplitudes. The numerical simulations are performed to show the stability of the traveling waves under various perturbations.

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