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Discrete and Continuous Dynamical Systems - S

September 2022 , Volume 15 , Issue 9

Issue on advances in the mathematical study of pattern formation

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Preface: Special issue on advances in the mathematical study of pattern formation
Paul Carter, Ryan Goh, Björn de Rijk and Qiliang Wu
2022, 15(9): i-ii doi: 10.3934/dcdss.2022132 +[Abstract](215) +[HTML](49) +[PDF](71.84KB)
Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci
Paul Carter and Alan R. Champneys
2022, 15(9): 2433-2466 doi: 10.3934/dcdss.2022036 +[Abstract](565) +[HTML](206) +[PDF](1631.07KB)

A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a canard explosion. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.

Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity
Grégory Faye, Thomas Giletti and Matt Holzer
2022, 15(9): 2467-2496 doi: 10.3934/dcdss.2021146 +[Abstract](723) +[HTML](223) +[PDF](1016.97KB)

We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at \begin{document}$ x = \pm \infty $\end{document}. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.

Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability
Christopher Logan Hambric, Chi-Kwong Li, Diane Christine Pelejo and Junping Shi
2022, 15(9): 2497-2511 doi: 10.3934/dcdss.2021128 +[Abstract](784) +[HTML](282) +[PDF](367.44KB)

It is shown that for any positive integer \begin{document}$ n \ge 3 $\end{document}, there is a stable irreducible \begin{document}$ n\times n $\end{document} matrix \begin{document}$ A $\end{document} with \begin{document}$ 2n+1-\lfloor\frac{n}{3}\rfloor $\end{document} nonzero entries exhibiting Turing instability. Moreover, when \begin{document}$ n = 3 $\end{document}, the result is best possible, i.e., every \begin{document}$ 3\times 3 $\end{document} stable matrix with five or fewer nonzero entries will not exhibit Turing instability. Furthermore, we determine all possible \begin{document}$ 3\times 3 $\end{document} irreducible sign pattern matrices with 6 nonzero entries which can be realized by a matrix \begin{document}$ A $\end{document} that exhibits Turing instability.

Rotating spirals in oscillatory media with nonlocal interactions and their normal form
Gabriela Jaramillo
2022, 15(9): 2513-2551 doi: 10.3934/dcdss.2022085 +[Abstract](368) +[HTML](143) +[PDF](794.15KB)

Biological and physical systems that can be classified as oscillatory media give rise to interesting phenomena like target patterns and spiral waves. The existence of these structures has been proven in the case of systems with local diffusive interactions. In this paper the more general case of oscillatory media with nonlocal coupling is considered. We model these systems using evolution equations where the nonlocal interactions are expressed via a diffusive convolution kernel, and prove the existence of rotating wave solutions for these systems. Since the nonlocal nature of the equations precludes the use of standard techniques from spatial dynamics, the method we use relies instead on a combination of a multiple-scales analysis and a construction similar to Lyapunov-Schmidt. This approach then allows us to derive a normal form, or reduced equation, that captures the leading order behavior of these solutions.

Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems
Juraj Kováč and Václav Klika
2022, 15(9): 2553-2579 doi: 10.3934/dcdss.2022133 +[Abstract](241) +[HTML](44) +[PDF](452.56KB)

Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations of the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The Liouville-Green approximation (also known as WKBJ method), one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar Liouville-Green approximation to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs without turning points. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence showing that Liouville-Green approximation is plausible for an arbitrary number of coupled species outside of turning points and generally valid for fast growing modes as long as the diffusivities are distinct. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy-like functions.

Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients
Joshua A. McGinnis and J. Douglas Wright
2022, 15(9): 2581-2607 doi: 10.3934/dcdss.2021100 +[Abstract](920) +[HTML](371) +[PDF](616.37KB)

We consider a linear Fermi-Pasta-Ulam-Tsingou lattice with random spatially varying material coefficients. Using the methods of stochastic homogenization we show that solutions with long wave initial data converge in an appropriate sense to solutions of a wave equation. The convergence is strong and both almost sure and in expectation, but the rate is quite slow. The technique combines energy estimates with powerful classical results about random walks, specifically the law of the iterated logarithm.

Large-amplitude modulation of periodic traveling waves
Guy Métivier and Kevin Zumbrun
2022, 15(9): 2609-2632 doi: 10.3934/dcdss.2022070 +[Abstract](348) +[HTML](108) +[PDF](472.21KB)

We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.

Undulated bilayer interfaces in the planar functionalized Cahn-Hilliard equation
Keith Promislow and Qiliang Wu
2022, 15(9): 2633-2656 doi: 10.3934/dcdss.2022035 +[Abstract](385) +[HTML](143) +[PDF](686.07KB)

Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [8]. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter \begin{document}$ \varepsilon\ll1 $\end{document}, we show that this induces undulated bilayer solutions whose width perturbations decay on an \begin{document}$ O\!\left( \varepsilon^{-1/2}\right) $\end{document} inner length scale that is long in comparison to the \begin{document}$ O(1) $\end{document} scale that characterizes the bilayer width.

The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability
Guido Schneider and Matthias Winter
2022, 15(9): 2657-2672 doi: 10.3934/dcdss.2021119 +[Abstract](789) +[HTML](330) +[PDF](383.25KB)

We consider reaction-diffusion systems for which the trivial solution simultaneously becomes unstable via a short-wave Turing and a long-wave Hopf instability. The Brusseletor, Gierer-Meinhardt system and Schnakenberg model are prototype biological pattern forming systems which show this kind of behavior for certain parameter regimes. In this paper we prove the validity of the amplitude system associated to this kind of instability. Our analytical approach is based on the use of mode filters and normal form transformations. The amplitude system allows us an efficient numerical simulation of the original multiple scaling problems close to the instability.

Hopf bifurcations in the full SKT model and where to find them
Cinzia Soresina
2022, 15(9): 2673-2693 doi: 10.3934/dcdss.2022120 +[Abstract](364) +[HTML](70) +[PDF](1375.06KB)

In this paper, we consider the Shigesada–Kawasaki–Teramoto (SKT) model, which presents cross-diffusion terms describing competition pressure effects. Even though the reaction part does not present the activator–inhibitor structure, cross-diffusion can destabilise the homogeneous equilibrium. However, in the full cross-diffusion system and weak competition regime, the cross-diffusion terms have an opposite effect and the bifurcation structure of the system modifies as the interspecific competition pressure increases. The major changes in the bifurcation structure, the type of pitchfork bifurcations on the homogeneous branch, as well as the presence of Hopf bifurcation points are here investigated. Through weakly nonlinear analysis, we can predict the type of pitchfork bifurcation. Increasing the additional cross-diffusion coefficients, the first two pitchfork bifurcation points from super-critical become sub-critical, leading to the appearance of a multi-stability region. The interspecific competition pressure also influences the possible appearance of stable time-period spatial patterns appearing through a Hopf bifurcation point.

Optimal spatial patterns in feeding, fishing, and pollution
Hannes Uecker
2022, 15(9): 2695-2717 doi: 10.3934/dcdss.2021099 +[Abstract](876) +[HTML](430) +[PDF](1059.66KB)

Infinite time horizon spatially distributed optimal control problems may show so–called optimal diffusion induced instabilities, which may lead to patterned optimal steady states, although the problem itself is completely homogeneous. Here we show that this can be considered as a generic phenomenon, in problems with scalar distributed states, by computing optimal spatial patterns and their canonical paths in three examples: optimal feeding, optimal fishing, and optimal pollution. The (numerical) analysis uses the continuation and bifurcation package \begin{document}$\mathtt{pde2path} $\end{document} to first compute bifurcation diagrams of canonical steady states, and then time–dependent optimal controls to control the systems from some initial states to a target steady state as \begin{document}$ t\to\infty $\end{document}. We consider two setups: The case of discrete patches in space, which allows to gain intuition and to compute domains of attraction of canonical steady states, and the spatially continuous (PDE) case.

An SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects
Shankar C. Venkataramani
2022, 15(9): 2719-2746 doi: 10.3934/dcdss.2022101 +[Abstract](364) +[HTML](71) +[PDF](3635.18KB)

We investigate stripe patterns formation far from threshold using a combination of topological, analytic, and numerical methods. We first give a definition of the mathematical structure of 'multi-valued' phase functions that are needed for describing layered structures or stripe patterns containing defects. This definition yields insight into the appropriate 'gauge symmetries' of patterns, and leads to the formulation of variational problems, in the class of special functions with bounded variation, to model patterns with defects. We then discuss approaches to discretize and numerically solve these variational problems. These energy minimizing solutions support defects having the same character as seen in experiments.

Dynamics of patchy vegetation patterns in the two-dimensional generalized Klausmeier model
Tony Wong and Michael J. Ward
2022, 15(9): 2747-2793 doi: 10.3934/dcdss.2022043 +[Abstract](422) +[HTML](155) +[PDF](5800.87KB)

We study the dynamical and steady-state behavior of self-organized spatially localized patches or "spots" of vegetation for the Klausmeier reaction-diffusion (RD) system of spatial ecology that models the interaction between surface water and vegetation biomass on a 2-D spatial landscape with a spatially uniform terrain slope gradient. In this context, we develop and implement a hybrid asymptotic-numerical theory to analyze the existence, linear stability, and slow dynamics of multi-spot quasi-equilibrium spot patterns for the Klausmeier model in the singularly perturbed limit where the biomass diffusivity is much smaller than that of the water resource. From the resulting differential-algebraic (DAE) system of ODEs for the spot locations, one primary focus is to analyze how the constant slope gradient influences the steady-state spot configuration. Our second primary focus is to examine bifurcations in quasi-equilibrium multi-spot patterns that are triggered by a slowly varying time-dependent rainfall rate. Many full numerical simulations of the Klausmeier RD system are performed both to illustrate the effect of the terrain slope and rainfall rate on localized spot patterns, as well as to validate the predictions from our hybrid asymptotic-numerical theory.

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6

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