Nonlinear patterns shaping the domain on which they live

Ruppert, Mirko; Ziebert, Falko; Zimmermann, Walter

doi:10.1088/1367-2630/ab7f92

Cited by 8 publications

(6 citation statements)

References 51 publications

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“…In this work, we observed that the orientation effect of the gradient prevailed in many, but not all, of the explored configurations, either starting from a preexisting structure of stripes parallel or perpendicular to the gradient, or with a pseudorandom initial condition. The orientation effect is relevant for many physical systems presenting periodic patterns, such as in developmental biology [20,39], smectic mesophases [8], and localized sand patterns [40], whose dynamics have been studied by Swift-Hohenberg type equations, but present mechanisms of stripe orientation that are not well understood. The monotonic decay of the finite differences version is enforced, provided that the internal iterations converge [24].…”

Section: Discussionmentioning

confidence: 99%

Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Coelho,

Vitral,

Pontes

et al. 2020

Preprint

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Spatio-temporal pattern formation in natural systems presents rich nonlinear dynamics which lead to the emergence of periodic nonequilibrium structures. One of the most successful equations currently available for theoretically investigating the behavior of these structures is the Swift-Hohenberg (SH), which contains a bifurcation parameter (forcing) that controls the dynamics by changing the energy landscape of the system. Though a large part of the literature on pattern formation addresses uniformly forced systems, nonuniform forcings are also observed in several natural systems, for instance, in developmental biology and in soft matter applications. In these cases, an orientation effect due to a forcing gradient is a new factor playing a role in the development of patterns, particularly in the class of stripe patterns, which we investigate through the nonuniformly forced SH dynamics. The present work addresses the stability of stripes, and the competition between the orientation effect of the gradient and other bulk, boundary, and geometric effects taking part in the selection of the emerging patterns. A weakly nonlinear analysis shows that stripes tend to align with the gradient, and become unstable when perpendicular to the preferred direction. This analysis is complemented by a numerical work that accounts for other effects, confirming that stripes align, or even reorient from preexisting conditions. However, we observe that the orientation effect does not always prevail in face of competing effects. Other than the cubic SH equation (SH3), the quadratic-cubic (SH23) and cubic-quintic (SH35) equations are also studied. In the SH23 case, not only do forcing gradients lead to stripe orientation, but also interfere in the transition from hexagonal patterns to stripes.

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Section: Discussionmentioning

confidence: 99%

Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Coelho,

Vitral,

Pontes

et al. 2020

Preprint

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“…where ν is a new parameter. Together with (14), this defines a differential-algebraic system. For two spikes, it has up to four real solutions for each configuration (x 1 , x 2 ).…”

Section: Comparison With the Turing Systemmentioning

confidence: 99%

“…So although we can determine which modes are initially excited, we can not predict, in general, which pattern is finally obtained. Furthermore, many studies have examined the effect of external constraints such as fixed boundary conditions, parameter ramps, template patterns, geometry and deformable boundaries [12][13][14][15], especially near onset (see [10,16] for reviews). Here, however, we study unforced systems away from onset with reflexive or periodic boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Pattern selection in reaction diffusion systems

Subramanian,

Murray

2020

Preprint

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Turing's theory of pattern formation has been used to describe self-organisation in many biological, chemical and physical systems. However, while conditions sufficient for the existence of patterns are known, the general nonlinear mechanisms responsible for pattern selection are not. Here, we show that the movement and final position of peaks within a Turing pattern are determined by the flow of mass through the system and the minimisation of the total mass of the fast species. The apparent success, for particular parameters choice, of linear stability analysis in predicting the final pattern is found to be a coincidence. Furthermore, mass minimisation also underlies a competition instability that results in peak-coarsening of quasi-steady state patterns and correctly predicts the number of peaks at the true steady-state.

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“…In contrast, periodic and no-flux boundary conditions, for example, impose no restriction on the stable wavenumber band, except that the wavenumbers can take only discrete values. In rectangular systems with amplitude-suppressing boundary conditions, stripe patterns prefer an orientation perpendicular to these edges 1, 44,45 .…”

Section: Introductionmentioning

confidence: 99%

“…In such examples, orientations are often perpendicular to non-resonant control parameter drops 45,51 . In the case of steep control parameter decays, stationary strips may orient parallel to the boundaries due to resonance effects 51 .…”

Section: Introductionmentioning

confidence: 99%

On the band-width of stable nonlinear stripe patterns in finite size systems

Ruppert,

Zimmermann

2021

Preprint

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Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share the universal property of being stable at different wavenumbers q, i.e., they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus-and zigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for Turing pattern in cells. In addition, we explain why below a certain system width stripe pattern behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in mediumsized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.Nonlinear stripe patterns are ubiquitous in nature, and their driving mechanisms are as diverse as the systems themselves in which they occur [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] . Stripe patterns have the universal property of being stable at different values of the wavenumber, and these stable wavenumber regions are the so-called Busse balloons after their pioneer 1,3,11,16,17 . To the instabilities bounding the stable wavenumber range of stripe patterns count the generic Eckhaus instability, a longwavelength longitudinal (compressional) instability, and the zigzag instability, a long-wavelength transverse instability 1,18 .Stable wavenumber ranges are restricted even in large systems by pattern suppressing boundary conditions at the domain sides 19,20 or are even selected by spatial inhomogeneities, e.g. via so-called ramps [21][22][23] . In contrast, in short systems the stable wavenumber range can be enlarged, as in Ref. 24 for quasione-dimensional systems predicted and experimentally confirmed in Ref. 25,26 . Such a range extension depends on the boundary conditions and the second spatial dimension as we explain in this work analytically and numerically by investigating the generic Swift-Hohenberg model and the universal Newell-Whitehead-Segel equation in rectangular domains. Finite size effects on patterns are also highly relevant for Turing patterns in small systems, as for instance in cells [27][28][29][30] .

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Nonlinear patterns shaping the domain on which they live

Cited by 8 publications

References 51 publications

Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Pattern selection in reaction diffusion systems

On the band-width of stable nonlinear stripe patterns in finite size systems

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