Resonance tongues and patterns in periodically forced reaction-diffusion systems

Lin, Anna L.; Hagberg, Aric; Meron, Ehud; Swinney, Harry L.

doi:10.1103/physreve.69.066217

Cited by 73 publications

(40 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There exist many results relating to spiral waves in reaction-diffusion systems; these include their response to temporally periodic forcing [24,25] or anisotropies of the domain [36], their motion near boundaries [38], the effects of differently shaped domains [43], and the interaction between two or more spirals [1]. A large open issue is to determine whether or not generic behavior of spiral waves in reaction-diffusion systems also occurs in systems with nonlocal spatial interactions.…”

Section: Discussionmentioning

confidence: 99%

Spiral Waves in Nonlocal Equations

Laing

2005

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

Abstract. We present a numerical study of rotating spiral waves in a partial integro-differential equation defined on a circular domain. This type of equation has been previously studied as a model for large scale pattern formation in the cortex and involves spatially nonlocal interactions through a convolution.The main results involve numerical continuation of spiral waves that are stationary in a rotating reference frame as various parameters are varied. We find that parameters controlling the strength of the nonlinear drive, the strength of local inhibitory feedback, and the steepness and threshold of the nonlinearity must all lie within particular intervals for stable spiral waves to exist. Beyond the ends of these intervals, either the whole domain becomes active or the whole domain becomes quiescent. An unexpected result is that the boundaries seem to play a much more significant role in determining stability and rotation speed of spirals, as compared with reaction-diffusion systems having only local interactions.

show abstract

Section: Discussionmentioning

confidence: 99%

Spiral Waves in Nonlocal Equations

Laing

2005

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

show abstract

“…Some of the contributions in the study of pattern formation induced by temporal forcing are Refs. [5][6][7][8][9][10][11][12][13][14][15][16][17]. We are not addressing cases in which the forcing is itself the origin of the pattern-forming instability, such as for instance in Faraday waves [1,18].…”

Section: Introductionmentioning

confidence: 99%

Theory of pattern forming systems under traveling-wave forcing

et al. 2007

View full text Add to dashboard Cite

The response of pattern forming systems to external forcing either spatial or temporal has received much attention for several decades. Combined spatio-temporal forcing has only been introduced recently, in particular in the form of a spatially resonant traveling-wave forcing [S. Rüdiger, D.G. Míguez, A.P. Muñuzuri, F. Sagués, J. Casademunt, Phys. Rev. Lett. 90 (2003) 128301]. Since then, both a series of experiments, in the context of Turing patterns in reaction-diffusion systems, and the development of the corresponding generic theory, have unveiled a wealth of new and unexpected phenomena. In this article we review these phenomena, we provide a unified and comprehensive description of them, and extend the theoretical analysis to new situations. We formulate the generic amplitude equations for different orders of spatial resonance for 1d and 2d patterns (stripes and hexagons). We identify and describe in detail the autonomous dynamical system which underlies the phenomenon of traveling-stripe resonance. For 1d we focus on localized solutions (kinks and pulses), their dynamics and their interaction for both 1:1 and 2:1 resonance. Specifically, we discuss the effect of wave-number mismatch (inexact resonance) combined with the non-gradient dynamics induced by the motion of the forcing, resulting in non-trivial interactions and complex spatio-temporal dynamics. Analytical results in the phase approximation are obtained, while numerical techniques are used to study the complete problem. We show that defect interaction is oscillatory with distance, allowing for the existence of locked chaotic wave trains. In 2d we discuss the modulational instabilities of striped patterns and focus mostly on the emergence of hexagons induced by traveling stripe forcing, for exact 1:1 resonance. In this case, we examine in detail the complex bifurcation scenario beyond primary instabilities. We finally discuss the problem of traveling-wave forcing in the context of a specific system, the photosensitive CDIMA reaction, where most experiments have been carried out. We compare experiments and theoretical predictions and propose other experimental systems where the study could be extended. Finally, we review related work by other authors and discuss possible further developments and open questions which hold the promise of new interesting findings.

show abstract

“…The calculations will illustrate how waves (front and back) interact during collisions, and how they are influenced by local distrurbances. There exist many interesting phenomena in periodically forced reaction-diffusion systems to be explored [7].…”

Section: Discussionmentioning

confidence: 99%

Stabilization of reaction-diffusion pulses by external forcing

2004

View full text Add to dashboard Cite

A pulse velocity equation under forcing is derived and the conditions for stationary waves (pinning conditions) are considered. It is found that there are two types of stationary pulses with symmetric and asymmetric parameter sets in the phase-amplitude diagram. The pulses with a symmetric set are always unstable, whereas the pulses with an asymmetric set may be stable. The stability criteria are presented.

show abstract

Resonance tongues and patterns in periodically forced reaction-diffusion systems

Cited by 73 publications

References 35 publications

Spiral Waves in Nonlocal Equations

Spiral Waves in Nonlocal Equations

Theory of pattern forming systems under traveling-wave forcing

Stabilization of reaction-diffusion pulses by external forcing

Contact Info

Product

Resources

About