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Highlights 2010

Nonlinearityiopscience.org/non

Nonlinearity

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Cover image: Courtesy of U Ebert, W Hundsdorfer, V Ratushnaya CWI Amsterdam, PO Box 94079, 1090GB Amsterdam, the Netherlands; C Li, S Nijdam, Faculty of Applied Physics, Technische Universiteit Eindhoven, the Netherlands; F Brau, Groupe de Physique Nucleaire Théorique, Université de Mons-Hainaut, Belgium; G Derks, Department of Mathematics, University of Surrey, UK; C Y Kao, Department of Mathematics, Ohio State University, US; A Luque, Institito de Astrofisica de Andalucia, CSIC, Granada, Spain; B Meulenbroek, Institute of Applied Mathematics, Technische Universiteit Delft, the Netherlands; L Schäfer, Fachbereich Physik, Universität Duisburg-Essen, Germany; from their article ‘Multiple scales in streamer discharges, with an emphasis on moving boundary approximations’, which appears in issue 1 of volume 24 (image originally published in Journal of Physics D: Applied Physics).

Dear colleague,

Welcome to this collection of abstracts from some of the high-quality articles published in Nonlinearity during 2010. This brochure is designed to give you a taste of the exciting and wide-ranging science published in Nonlinearity. Over the next few pages you can browse the abstracts of articles covering everything from the nonlinear modelling of cancer and work on flocking models to the nonlinear Schrödinger equation and the invisible parts of attractors.

Nonlinearity is published jointly by the London Mathematical Society and IOP Publishing. Over the last 23 years, it has built a reputation for not only publishing high-quality and cutting-edge research but also for bringing together mathematicians and physicists from many fields, ranging from theoretical physics to pure mathematics and the biological sciences to engineering. The journal's coverage spans all aspects of nonlinear science, from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant interest.

We would like to take this opportunity to thank all of the authors and referees who have worked on Nonlinearity over the years. We hope that you enjoy perusing this collection. All of the articles presented in this brochure can be found online at iopscience.org/non and are free to access until the end of 2011.

Chris Wileman Publisher

Lara Finan, Joanna EvangelidesPublishing Editors

1.468*IMPACT FACTOR

*As listed in the 2010 ISI Journal Citation Reports®

Journal scopeAimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, Nonlinearity features nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance.

Subject coverage The journal's coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant mathematical and physical interest. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal. Because of the broad scope of the journal, authors are strongly encouraged to provide sufficient introductory material to appeal to a wide readership.

Nonlinearity

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Editorial Board

Editors-in-chiefJ P Keating University of Bristol, UK (for the IOP)Quantum mechanics of classically chaotic systems, semiclassical asymptotics and periodic orbit theoryA I Neishtadt Loughborough University, UK and Space Research Institute, Moscow, Russia (for the LMS)Perturbation theory, celestial mechanics, bifurcation theory

Editorial Board A L Bertozzi University of California Los Angeles, USANonlinear PDEs, fluid interfaces, swarming, image processingL Bunimovich Georgia Institute of Technology, Atlanta, USADynamical systemsA Chenciner Institut de Mecanique Celeste, Paris, FranceCelestial mechanics, N-body problemsC-Q Cheng Nanjing University, People’s Republic of ChinaHamiltonian dynamicsR de la Llave University of Texas at Austin, USAHamiltonian dynamics, hyperbolic systems, variational problemsD Dolgopyat University of Maryland, College Park, USADynamical systems and probability theoryB Dubrovin International School for Advanced Studies, Trieste, ItalyGeometrical, analytical and numerical methods in the theory of integrable systems and their perturbationsB Eckhardt Philipps-Universität Marburg, GermanyFluid mechanics, nonlinear dynamics, quantum chaosJ P Eckmann Université de Genève, SwitzerlandPDEs, non-equilibrium problemsM J Field University of Houston, TX, USABifurcation theory, equivariant dynamical systemsA S Fokas University of Cambridge, UKNonlinear integrable equations, symmetries, boundary value problems, inverse problems in medical imagingJ Glazier Indiana University, Bloomington, USABiological physics, biocomplexity, pattern formation, complex fluids, foams, rheology, hydrodynamics, turbulence, chaosA R Its Purdue University at Indianapolis, IN, USAIntegrable systems, Riemann--Hilbert method, asymptotic analysis, special functions, orthogonal polynomials, matrix modelsY G Kevrekidis Princeton University, NJ, USANumerical bifurcation theory, multiscale computationK Khanin University of Toronto, Ontario, CanadaDynamical systems, statistical mechanics, turbulenceC Le Bris CERMICS-ENPC, Marne-La-Vallée Cedex, FranceNonlinear PDEs, molecular simulation, materials scienceJ Lega University of Arizona, Tucson, USAModelling of nonlinear phenomena in physics and biology, pattern formation, stability theory

B Leimkuhler University of Edinburgh, UKStructure-preserving numerical algorithms (geometric integrators), methods for molecular dynamics, computational statistical mechanicsC Liverani Università di Roma 'Tor Vergata', ItalyClassical and quantum dynamical systemsD Lohse University of Twente, the NetherlandsHydrodynamics, turbulence, granular flow, acousticsL Mahadevan Harvard University, Cambridge, MA, USAApplied mathematics, macroscopic physics, materials science, quantitative biologyJ Marklof University of Bristol, UKDynamical systems and their applications, quantum chaosF Merle Université de Cergy-Pontoise, FranceQualitative behaviour and singularities in PDEsS Nonnenmacher CEA-Saclay, Gif-sur-Yvette, FranceQuantum chaos and maps, noisy classical and quantum dynamicsK Ohkitani University of Sheffield, UKFluid turbulence, vortex dynamicsL Ryzhik Stanford University, CA, USAWave propagation, random media, reaction-diffusion equationsC Schütte Freie Universität Berlin, GermanyComplex dynamical systemsJ A Sherratt Herriot-Watt University, UKMathematical biology, reaction-diffusion equations, travelling wavesE S Titi University of California, Irvine, USA and Weizmann Institute of Science, IsraelNonlinear PDEs, numerical analysis, fluid mechanics, infinite-dimensional dynamical systemsD Treschev M V Lomonosov Moscow State University, Leninskiye Gory, RussiaClassical mechanics, perturbation theory, integrability and nonintegrability, bifurcation theoryM Tsujii Kyushu University, Fukuoka, JapanErgodic theory of smooth dynamical systems, non-uniformly hyperbolic dynamical systemsL S Young New York University, USAGeometric and ergodic theories of dynamical systems and their applications

Our dedicated team at IOP Publishing is here to look after your work, ensuring that the whole process from submission to publication via the peer-review process and promotion is run in a satisfactory manner.

Journal team

Publisher Chris Wileman

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Contents

Nonlinear modelling of cancer: bridging the gap between cells and tumours 6J S Lowengrub, H B Frieboes, F Jin, Y-L Chuang, X Li, P Macklin, S M Wise and V Cristini

Dimension of non-conformal repellers: a survey 6Jianyu Chen and Yakov Pesin

Blow-up criteria for the 3D cubic nonlinear Schrödinger equation 6Justin Holmer, Rodrigo Platte and Svetlana Roudenko

A simple framework to justify linear response theory 7Martin Hairer and Andrew J Majda

Classification of solutions of the forced periodic nonlinear Schrödinger equation 7Eli Shlizerman and Vered Rom-Kedar

Some results on homoclinic and heteroclinic connections in planar systems 7Armengol Gasull, Hector Giacomini and Joan Torregrosa

Chaos in differential equations driven by a nonautonomous force 7Kening Lu and Qiudong Wang

Emergent behaviour of a generalized Viscek-type flocking model 8Seung-Yeal Ha, Eunhee Jeong and Moon-Jin Kang

Positive travelling fronts for reaction-diffusion systems with distributed delay 8Teresa Faria and Sergei Trofimchuk

A unified framework for mechanics: Hamilton-Jacobi equation and applications 8P Balseiro, J C Marrero, D Martín de Diego and E Padrón

Network periodic solutions: full oscillation and rigid synchrony 9Martin Golubitsky, David Romano and Yunjiao Wang

Excited states in the large density limit: a variational approach 9M P Coles, D E Pelinovsky and P G Kevrekidis

Pattern solutions of the Klausmeier Model for banded vegetation in semi-arid environments I 10Jonathan A Sherratt

Invisible parts of attractors 10Yu Ilyashenko and A Negut

2010 Highlights

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Nonlinear modelling of cancer: bridging the gap between cells and tumours

J S Lowengrub1,5, H B Frieboes2,3, F Jin2,3, Y L Chuang2, X Li3, P Macklin2, S M Wise4 and V Cristini2

1 Department of Biomedical Engineering, Center for Mathematical and Computational Biology, University of California at Irvine, Irvine, CA 92697, USA

2 School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, USA

3 Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA

4 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

5 Author to whom any correspondence should be addressed

2010 Nonlinearity 23 R1

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Dimension of non-conformal repellers: a survey

Jianyu Chen and Yakov Pesin

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

2010 Nonlinearity 23 R93

This paper is a survey of recent results on the dimension of repellers for expanding maps and limit sets for iterated function systems. While the case of conformal repellers is well understood, the study of non-conformal repellers is in its early stages though a number of interesting phenomena have been discovered, some remarkable results obtained and several interesting examples constructed. We will describe contemporary state of the art in the area with emphasis on some new emerging ideas and open problems.

Blow-up criteria for the 3D cubic nonlinear Schrödinger equation

Justin Holmer1, Rodrigo Platte2 and Svetlana Roudenko3

1 Mathematics Department, Brown University, 151 Thayer Street, Providence, RI 02912, USA

2 Computing Laboratory, Oxford University, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK

3 School of Mathematics and Statistics, Arizona State University, Arizona 85287–1804, USA

2010 Nonlinearity 23 977

We consider solutions u to the 3D nonlinear Schrödinger equation i∂

tu + Δu + |u|2u = 0. In particular, we are interested in finding criteria on the initial data u

0 that predict the asymptotic behaviour of u(t), e.g., whether u(t) blows up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for H1 data) (Duyckaerts–Holmer–Roudenko) if M[u]E[u] ≤ M[Q]E[Q], where M[u] and E[u] denote the mass and energy of u and Q denotes the ground state solution to −Q + ΔQ + |Q|2Q = 0. Here we consider the complementary case M[u]E[u] > M[Q]E[Q]. In the first (analytical) part of the paper, we present a result due to Lushnikov, based on the virial identity and the generalized uncertainty principle, giving a sufficient condition for blow-up. By replacing the uncertainty principle in his argument with an interpolation-type inequality, we obtain a new blow-up condition that in some cases improves upon Lushnikov's condition. Our approach also allows for an adaptation to radial infinite-variance initial data that has a conceptual interpretation: for real-valued initial data, if a certain fraction of the mass is contained within the ball of radius M[u], then blow up occurs. We also show analytically (if one takes the numerically computed value of ||Q||H1/2) that there exist Gaussian initial data u

0 with negative quadratic phase such that ||u

0||H1/2 < ||Q||H1/2 but the solution u(t) blows up. In the second (numerical) part of the paper, we examine several different classes of initial data—Gaussian, super Gaussian, off-centred Gaussian, and oscillatory Gaussian—and for each class give the theoretical predictions for scattering or blow-up provided by the above theorems as well as the results of

Hybrid tumour modelling: evolution of a tumour spheroid in the absence of the outer gel from the study by Othmer and co-workers. Necrosis is represented by the inner (white) region; it is enclosed by the continuum quiescent region and the outer cell-based region (space unit = 10 μm). Reprinted with permission from Kim et al Math. Models Methods Appl. Sci. 17 1790. Copyright © (2007) World Scientific.

2010 Highlightsq

A simple framework to justify linear response theory

Martin Hairer and Andrew J Majda

Courant Institute for Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA


The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced dissipative stochastic dynamical systems is developed. The main results are formulated in an abstract setting and apply to suitable systems, in finite and infinite dimensions, that are of interest in climate change science and other applications.

Some results on homoclinic and heteroclinic connections in planar systems

Armengol Gasull1, Hector Giacomini2 and Joan Torregrosa1,3

1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain

2 Laboratoire de Mathématiques et Physique Théorique, UMR 6083-CNRS- Faculté des Sciences et Techniques, Université de Tours, 37200 Tours, France

3 Author to whom any correspondence should be addressed


Consider a family of planar systems depending on two parameters (n, b) and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when Φ(n, b) = 0. We present a method that allows us to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set Φ(n, b) = 0. The method is applied to two quadratic families, one of them is the well-known Bogdanov–Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of n, given by

We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions.

Classification of solutions of the forced periodic nonlinear Schrödinger equation

Eli Shlizerman1 and Vered Rom-Kedar2

Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot, 76100, Israel1 From 1 September 2009 affiliated with the Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA

2 The Estrin family chair of Computer Science and Applied Mathematics

Nonlinearity 23 2183

The integrable structure of the periodic one-dimensional nonlinear Schrödinger equation is utilized to gain insights regarding the perturbed near-integrable dynamics. After recalling the known results regarding the structure and stability of the unperturbed standing and travelling waves solutions, two new stability results are presented: (1) it is shown numerically that the stability of the 'outer' (cnoidal) unperturbed solutions depends on their power (the L

2 norm): they undergo a finite sequence of Hamiltonian–Hopf bifurcations as their power is increased. (2) another proof that the 'inner' (dnoidal) unperturbed solutions with multiplicity ≥ 2 are linearly unstable is presented. Then, to study the global phase-space structure, an energy–momentum bifurcation diagram (PDE-EMBD) that consists of projections of the unperturbed standing and travelling waves solutions to the energy–power plane and includes information regarding their linear stability is constructed. The PDE-EMBD helps us to classify the behaviour near the plane wave solutions: the diagram demonstrates that below some known threshold amplitude, precisely three distinct observable chaotic mechanisms arise: homoclinic chaos, homoclinic resonance and, for some parameter values, parabolic-resonance. Moreover, it appears that the dynamics of the PDE chaotic solutions that exhibit the parabolic-resonance instability may be qualitatively predicted: these exhibit the

Phase portraits of system (1) for n > 0 and b < n.

( )b n n n n n75

240172

4529486530024

111083391669252352961656 0/ / /1 2 3 2 2 5 2

= + - - +

numerical simulation. We find that depending upon the form of the initial conditions, any of the three analytical criteria for blow-up can be optimal. We formulate several conjectures, among them that for real initial data, the quantity ||Q||H1/2 provides the threshold for scattering.

same dynamics as a recently derived parabolic-resonance low-dimensional normal form. In particular, these solutions undergo adiabatic chaos: they follow the level lines of an adiabatic invariant till they reach the separatrix set at which the adiabatic invariant undergoes essentially random jumps.

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Emergent behaviour of a generalized Viscek-type flocking model

Seung-Yeal Ha, Eunhee Jeong and Moon-Jin Kang

Department of Mathematical Sciences, Seoul National University, Seoul 151–747, Korea


We present a planar agent-based flocking model with a distance-dependent communication weight. We derive a sufficient condition for the asymptotic flocking in terms of the initial spatial and heading-angle diameters and a communication weight. For this, we employ differential inequalities for the spatial and phase diameters together with the Lyapunov functional approach. When the diameter of the agent's initial heading-angles is sufficiently small, we show that the diameter of the heading-angles converges to the average value of the initial heading-angles exponentially fast. As an application of flocking estimates, we also show that the Kuramoto model with a connected communication topology on the regular lattice Zd for identical oscillators exhibits a complete-phase-frequency synchronization, when coupled oscillators are initially distributed on the half circle.

Positive travelling fronts for reaction– diffusion systems with distributed delay

Teresa Faria1 and Sergei Trofimchuk2

1 Departamento de Matemática and CMAF, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749–016 Lisboa, Portugal

2 Instituto de Matemática y Fisica, Universidad de Talca, Casilla 747, Talca, Chile


We give sufficient conditions for the existence of positive travelling wave solutions for multi-dimensional autonomous reaction–diffusion systems with distributed delay. To prove the existence of travelling waves, we give an abstract formulation of the equation for the wave profiles in some suitable Banach spaces and apply known results about the index of some associated Fredholm operators. After a Lyapunov–Schmidt reduction, these waves are obtained via the Banach contraction principle, as perturbations of a positive heteroclinic solution for the associated system without diffusion, whose existence is proven under some requirements. By a careful analysis of the exponential decay of the travelling wave profiles at −∞, their positiveness is deduced. The existence of positive travelling waves is important in terms of applications to biological models. Our method applies to systems of delayed reaction–diffusion equations whose nonlinearities are not required to satisfy a quasi-monotonicity condition. Applications are given, and include the delayed Fisher–KPP equation.

Network periodic solutions: full oscillation and rigid synchrony

Martin Golubitsky1, David Romano2 and Yunjiao Wang1

1 Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43210, USA

2 Department of Mathematics and Statistics, Grinnell College, Grinnell, IA 50112, USA


We prove two results about hyperbolic periodic solutions in networks of systems of ODEs. First, we show that generically hyperbolic periodic solutions of network admissible systems of differential equations oscillate in each node if and only if the network is transitive. We can associate a polydiagonal Δ(Z(t)) with each hyperbolic periodic solution Z(t) as follows. The cell coordinates of a point in Δ(Z(t)) are equal if the corresponding cell coordinates of Z(t) are equal for all t; that is, the outputs from the two cells are synchronous. Second, we prove that Δ(Z(t)) is rigid (unchanged by small admissible perturbations) if and only if it is flow-invariant for all admissible vector fields.

A unified framework for mechanics: Hamilton–Jacobi equation and applications

P Balseiro1, J C Marrero2, D Martín de Diego3 and E Padrón4

1 Instituto de Matematica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil

2 Unidad Asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain

3 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain


In this paper, we construct Hamilton–Jacobi equations for a large variety of mechanical systems (nonholonomic systems subjected to linear or affine constraints, dissipative systems subjected to external forces, time-

dependent mechanical systems etc). We recover all these, in principle, different cases, using a unified framework based on skew-symmetric algebroids with a distinguished 1-cocycle. Several examples illustrate the theory.

Comparison of a free trajectory (without friction), on the left, and a trajectory with friction, on the right.

Excited states in the large density limit: a variational approach

M P Coles1, D E Pelinovsky1 and P G Kevrekidis2

1 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1 Canada

2 Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA


Excited states of Bose–Einstein condensates are considered in the large density limit of the Gross–Pitaevskii equation with repulsive inter-atomic interactions and a harmonic potential. The relative dynamics of dark solitons (density dips on the localized condensate) with respect to the harmonic potential and to each other is approximated using the averaged Lagrangian method. This permits a complete characterization of the equilibrium positions of the dark solitons as a function of the chemical potential. It also yields an analytical handle on the oscillation frequencies of dark solitons around such equilibria. The asymptotic predictions are generalized for an arbitrary number of dark solitons and are corroborated by numerical computations for 2- and 3-soliton configurations.

Left: the equilibrium position of the two dark solitons versus the chemical potential μ. The solid line shows the direct numerical result and the dashed–dotted line represents the asymptotic approximation. Right: the solid line shows the numerical solution υ(ξ) for μ = 17, while the dashed line represents the corresponding variational ansatz.

Chaos in differential equations driven by a nonautonomous force

Kening Lu1 and Qiudong Wang2

1 Department of Mathematics, Brigham Young University, Provo, UT 84602, USA

2 Department of Mathematics, University of Arizona, Tuscon, AZ 85721, USA


Nonautonomous forces appear in many applications. They could be periodic, quasiperiodic and almost periodic in time; or they could take the form of a sample path of a random forcing driven by a stochastic process, which is without any periodicity in time. In this paper, we study the chaotic behaviour of differential equations driven by a general nonautonomous forcing without assuming any periodicity in time, aiming at applications to systems driven by a bounded random force. As a direct application, we prove that, for the Duffing equation driven by a bounded stationary stochastic process induced by a Brownian motion, chaotic dynamics exist almost surely. We also obtain various chaotic behaviour that are exclusively associated with equations driven by nonautonomous forcing without any periodicity in time. It has turned out that, unlike the systems driven by a periodic or almost periodic forcing, the transversal intersections of the stable and unstable manifolds are neither necessary nor sufficient for chaotic dynamics to exist. Finally, we apply all our results to the Duffing equation.

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Nonlinearity

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Pattern solutions of the Klausmeier Model for banded vegetation in semi-arid environments I

Jonathan A Sherratt

Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK


In many semi-arid environments, vegetation cover is sparse, and is self-organized into large-scale spatial patterns. In particular, banded vegetation is typical on hillsides. Mathematical modelling is widely used to study these banded patterns, and many models are effectively extensions of a coupled reaction–diffusion–advection system proposed by Klausmeier (1999 Science 284 1826–8). However, there is currently very little mathematical theory on pattern solutions of these equations. This paper is the first in a series whose aim is a comprehensive understanding of these solutions, which can act as a springboard both for future simulation-based studies of the Klausmeier model, and for analysis of model extensions. The author focusses on a particular part of parameter space, and derives expressions for the boundaries of the parameter region in which patterns occur. The calculations are valid to leading order at large values of the 'slope parameter', which reflects a comparison of the rate of water flow downhill with the rate of vegetation dispersal. The form of the corresponding patterns is also studied, and the author shows that the leading order equations change close to one boundary of the parameter region in which there are patterns, leading to a homoclinic solution. Conclusions are drawn on the way in which changes in mean annual rainfall affect pattern properties, including overall biomass productivity.

Invisible parts of attractors

Yu Ilyashenko1,2,3,4 and A Negut3,5,6

1 Cornell University, Math Dept., Ithaca, NY 14853, USA2 Moscow State University, MechMath Dept., Leninckie Gory st., 117234 Moscow, Russia

3 Independent University of Moscow, 11 Bolshoi Vlasievski, 1190026 Moscow, Russia

4 Steklov Institute of Mathematics, 8 Dubkina st, 117966 Moscow, Russia 5 Princeton University, Princeton NJ 08544, USA6 Simion Stoilow Institute of Mathematics, Calea Grivitei 21, Bucharest, Romania


This paper deals with attractors of generic dynamical systems. We introduce the notion of ε-invisible set, which is an open set of the phase space in which almost all orbits spend on average a fraction of time no greater than ε. For extraordinarily small values of ε (say, smaller than 2−100), these are large neighbourhoods of some parts of the attractors in the phase space which an observer virtually never sees when following a generic orbit.For any n ≥ 100, we construct a set Qn in the space of skew products over a solenoid with the fibre a circle having the following properties. Any map from Qn is a structurally stable diffeomorphism; the Lipschitz constants of the map and its inverse are no greater than L (where L is a universal constant that does not depend on n, say L < 100). Moreover, any map from Qn has a 2−n-invisible part of its attractor, whose size is comparable to that of the whole attractor. The set Qn is a ball of radius O(n−2) in the space of skew products with the C1 metric. It consists of structurally stable skew products.Small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties. Thus for all such perturbations, a sizable portion of the attractor is almost never visited by generic orbits and is practically never seen by the observer.

Saddle-node bifurcation on a circle.

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