Nonlinear Periodic Systems: Bands and Localization

2009 ◽  
Author(s):  
Alexander Vakakis
Keyword(s):  
Traveling Waves ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Near Field ◽  
Standing Waves ◽  
Nonlinear Resonance ◽  
Nonlinear Effects ◽  
Periodic Media ◽  
Spatially Extended ◽  
Nonlinear Standing Waves

We consider the dynamics of nonlinear mono-coupled periodic media. When coupling dominates over nonlinearity near-field standing waves and spatially extended traveling waves exist, inside stop and pass bands, respectively, of the nonlinear system. Nonlinear standing waves are analytically studied using a nonlinear normal mode formulation, whereas nonlinear traveling waves are analyzed by the method of multiple scales. When the nonlinear effects are of the same order with the coupling ones a completely different picture emerges, since nonlinear resonance interactions are unavoidable. As a result, infinite families of strongly and weakly localized nonlinear standing waves appear with frequencies lying in pass or stop bands of the corresponding linear periodic medium. Moreover, in the limit of weak coupling these solutions develop sensitive dependence on initial conditions, and the possibility of spatial chaos in the system exists. Some additional results on chaotic dynamics in linear periodic media with strongly nonlinear disorders are reviewed.

Dynamic Behavior of a Large Ring of Coupled Self-Excited Oscillators

2012 ◽  
Vol 8 (3) ◽  
Author(s):  
Miguel A. Barron ◽  
Mihir Sen
Keyword(s):  
Traveling Waves ◽  
Infinite Number ◽  
Initial Conditions ◽  
Standing Waves ◽  
Emergent Behavior ◽  
Van Der Pol ◽  
Coupling Matrix ◽  
Information Signal ◽  
Ring Dynamics ◽  
Van Der Pol Oscillators

Interconnected, self-excited oscillators are often found in nature and in engineered devices. In this work, a ring of van der Pol oscillators, each of which is connected to its immediate neighbors, is considered. The focus is on the emergent behavior of a large number of oscillators. Conditions are determined under which time-independent solutions are obtained, and the linear stability of these solutions is investigated. The effect of the singularity of the coupling matrix on the ring dynamics is explored. When this becomes singular, an infinite number of steady states is present, and the phenomenon of oscillation death arises. It is also possible to have, depending on initial conditions, all oscillators with in-phase synchrony, metachronal traveling waves with different wavelengths going around the ring, or standing waves. Interconnected oscillators can propagate information at a group velocity, and the information signal is present as an amplitude modulation.

Stability of slowly diverging jet flow

1976 ◽  
Vol 77 (2) ◽  
pp. 397-413 ◽  
Author(s):  
D. G. Crighton ◽  
M. Gaster
Keyword(s):  
Multiple Scales ◽  
Near Field ◽  
Mixing Layer ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Expansion Method ◽  
Inviscid Flow ◽  
Linear Instability ◽  
Mean Velocity ◽  
Instability Modes

Coherent axisymmetric structures in a turbulent jet are modelled as linear instability modes of the mean velocity profile, regarded as the profile of a, fictitious laminar inviscid flow. The usual multiple-scales expansion method is used in conjunction with a family of profiles consistent with similarity laws for the initial mixing region and approximating the profiles measured by Crow & Champagne (1971), Moore (1977) and other investigators, to deal with the effects of flow divergence. The downstream growth and approach to peak amplitude of axisymmetric wave modes with prescribed real frequency is calculated numerically, and comparisons are made with various sets of experimental data. Excellent agreement is found with the wavelength measurements of Crow & Champagne. Quantities such as the amplitude gain which depend on cumulative effects are less well predicted, though the agreement is still quite tolerable in view of the facts that this simple linear model of slowly diverging flow is being applied far outside its range of strict validity and that many of the published measurements are significantly contaminated by nonlinear effects. The predictions show that substantial variations are to be expected in such quantities as the phase speed and growth rate, according to the flow signal (velocity, pressure, etc.) measured, and that these variations depend not only on the axial measurement location but also on the cross-stream position. Trends of this kind help to explain differences in, for example, the preferred Strouhal number found by investigators using hot wires or pressure probes on the centre-line, in the mixing layer or in the near field.

Nonlinear Standing Waves in Cavities

2007 ◽  
pp. 237-247 ◽  
Author(s):  
Alan B. Coppens ◽  
Anthony A. Atchley
Keyword(s):  
Standing Waves ◽  
Nonlinear Standing Waves

scholarly journals Traveling and standing waves mediate pattern formation in cellular protrusions

2020 ◽  
Vol 6 (32) ◽  
pp. eaay7682
Author(s):  
Sayak Bhattacharya ◽  
Tatsat Banerjee ◽  
Yuchuan Miao ◽  
Huiwang Zhan ◽  
Peter N. Devreotes ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Traveling Waves ◽  
Standing Waves ◽  
Cell Cortex ◽  
Theoretical Studies ◽  
Direct Transformation ◽  
Diffusion Parameters ◽  
Excitable Systems ◽  
Mutant Strains ◽  
The Stability

The mechanisms regulating protrusions during amoeboid migration exhibit excitability. Theoretical studies have suggested the possible coexistence of traveling and standing waves in excitable systems. Here, we demonstrate the direct transformation of a traveling into a standing wave and establish conditions for the stability of this conversion. This theory combines excitable wave stopping and the emergence of a family of standing waves at zero velocity, without altering diffusion parameters. Experimentally, we show the existence of this phenomenon on the cell cortex of some Dictyostelium and mammalian mutant strains. We further predict a template that encompasses a spectrum of protrusive phenotypes, including pseudopodia and filopodia, through transitions between traveling and standing waves, allowing the cell to switch between excitability and bistability. Overall, this suggests that a previously-unidentified method of pattern formation, in which traveling waves spread, stop, and turn into standing waves that rearrange to form stable patterns, governs cell motility.

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