Spatiotemporal chaos and quasipatterns in coupled reaction–diffusion systems

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.

Existence of quasipatterns in the superposition of two hexagonal patterns

“…We have not discussed stability of these quasipatterns: that is an important and difficult problem. However, the reason for including a quadratic term in the Swift-Hohenberg equation (1.1) is that three-wave interactions generated by quadratic terms, particularly in problems in which patterns on two length scales are simultaneously unstable, are known to play a key role in stabilising quasipatterns in a variety of contexts [3,4,11,16,27,29,32,34,36,37,42,43,50]. Despite this, we do not expect any of the new solutions to be stable in the Swift-Hohenberg equation, but they (or related solutions) may be stable in other situations.…”

“…It turns out that the same stabilization mechanism operates in the Faraday wave and the polymer crystalization systems [26,34]. In both cases, and indeed in other systems [11,18], a common feature is that a second unstable or weakly damped length scale plays a key role in stabilizing the pattern. See [38] for a recent review.…”

“…in Figure 1(c), while the point nature of the power spectrum in Figure 1(d) indicates that the pattern has long-range order. These two features: the lack of periodicity (implicit in this case from twelve-fold rotational symmetry) and the presence of long-range order, are characteristics of quasicrystals in metallic alloys [39] and soft matter [20], and in quasipatterns in fluid dynamics [16], reaction-diffusion systems [11] and optical systems [5].…”

A 3-D Multi-Stable System With a Peanut-Shaped Equilibrium Curve: Circuit Design, FPGA Realization, and an Application to Image Encryption

“…C HAOTIC OSCILLATORS have been a hot topic for research during the last years due to its usefulness in the development of chaotic secure communication systems and other applications that have been implemented using either analog or digital electronics, as already shown in [1]. Chaotic systems have attracted great attention in several application areas in engineering such as secure communication ( [2], [3]), encryption ( [4], [5]), networks [6], cryptosystems ( [7], [8]), chemical systems ( [9], [10]), memristive systems ( [11], [12]), neural networks ( [13], [14]), biology ( [15], [16]), lasers ( [17]- [19]), circuits ( [20], [21]), etc.…”

Patterns and Quasipatterns from the Superposition of Two Hexagonal Lattices

“…that the pattern has long-range order. These two features, the lack of periodicity (implicit in this case from twelve-fold rotational symmetry) and the presence of long-range order, are characteristics of quasicrystals in metallic alloys [44] and soft matter [23], and in quasipatterns in fluid dynamics [18], reaction-diffusion systems [12] and optical systems [6].…”