Abstract
In coupled reaction–diffusion systems, modes with two different length scales can interact to produce a wide variety of spatiotemporal patterns. Three-wave interactions between these modes can explain the occurrence of spatially complex steady patterns and time-varying states including spatiotemporal chaos. The interactions can take the form of two short waves with different orientations interacting with one long wave, or vice verse. We investigate the role of such three-wave interactions in a coupled Brusselator system. As well as finding simple steady patterns when the waves reinforce each other, we can also find spatially complex but steady patterns, including quasipatterns. When the waves compete with each other, time varying states such as spatiotemporal chaos are also possible. The signs of the quadratic coefficients in three-wave interaction equations distinguish between these two cases. By manipulating parameters of the chemical model, the formation of these various states can be encouraged, as we confirm through extensive numerical simulation. Our arguments allow us to predict when spatiotemporal chaos might be found: standard nonlinear methods fail in this case. The arguments are quite general and apply to a wide class of pattern-forming systems, including the Faraday wave experiment.
Highlights
Two substances that react and diffuse can form patterns, an insight first highlighted in the work of Alan Turing [1,2]
We review what is known about the role of three-wave interactions in the formation of complex spatiotemporal patterns and outline our hypotheses regarding how quadratic coefficients would influence observed patterns
With quadratic coefficients Qzw and Qzz of opposite sign, the preliminary 8 × 8 calculations often yield time-dependent patterns, with relatively simple oscillatory, chaotic or heteroclinic cycle dynamics. When we extend these into 30 × 30 domains, the simple time dependence is often replaced by spatiotemporal chaos, supporting the infinite set of wavevectors picture implied by Fig. 3(c)
Summary
Two substances that react and diffuse can form patterns, an insight first highlighted in the work of Alan Turing [1,2]. This work demonstrated that by changing the interlayer coupling strength, one can manipulate the ratio of the length scales associated with two resonantly interacting Turing instabilities and encourage the formation of certain complex patterns in the Brusselator model. Rather than slaving away one set of critical modes and studying cubic terms, as described above, we instead see how much understanding may be gleaned by restricting our attention to quadratic terms near the codimension-two point This approach, namely, studying the effect of three-wave interactions on spatiotemporal pattern formation in reaction–diffusion systems by looking at quadratic coefficients, has proven successful in the past [27,28]. Our present work develops a more exhaustive investigation in the context of layered Turing systems, though the ideas are applicable wherever a pattern-forming system can have two unstable length scales, including the Faraday wave experiment.
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