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Physica D: Nonlinear Phenomena
Journal Prestige (SJR): 0.861
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  Hybrid Journal Hybrid journal (It can contain Open Access articles)
ISSN (Print) 0167-2789 - ISSN (Online) 0167-2789
Published by Elsevier Homepage  [3207 journals]
  • Reduction approach to the dynamics of interacting front solutions in a
           bistable reaction–diffusion system and its application to heterogeneous
    • Abstract: Publication date: Available online 4 April 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Kei Nishi, Yasumasa Nishiura, Takashi TeramotoAbstractThe dynamics of pulse solutions in a bistable reaction–diffusion system are studied analytically by reducing partial differential equations (PDEs) to finite-dimensional ordinary differential equations (ODEs). For the reduction, we apply the multiple-scales method to the mixed ODE-PDE system obtained by taking a singular limit of the PDEs. The reduced equations describe the interface motion of a pulse solution formed by two interacting front solutions. This motion is in qualitatively good agreement with that observed for the original PDE system. Furthermore, it is found that the reduction not only facilitates the analytical study of the pulse solution, especially the specification of the onset of local bifurcations, but also allows us to elucidate the global bifurcation structure behind the pulse behavior. As an application, the pulse dynamics in a heterogeneous bump-type medium are explored numerically and analytically. The reduced ODEs clarify the transition mechanisms between four pulse behaviors that occur at different parameter values.
  • Self-similarity of bubble size distributions in the aging metastable foams
    • Abstract: Publication date: Available online 2 April 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): D.A. Zimnyakov, A.I. Zemlyanukhin, S.A. Yuvchenko, A.V. Bochkarev, I.O. Slavnetskov, S.A. Gavrilov, D.D. TumachevAbstractThe quantitative approach to self-similar growth of gas bubbles in slowly evolving metastable liquid foams is provided using a solution to an evolution equation in the form of the non-linear Fokker–Planck equation. The obtained self-similar solution is compared to the experimental data obtained using the image analysis of liquid films at the container wall and the low-coherence reflectometry of foam samples. In addition, the modelled data are compared to the results of x-ray tomographic study of slowly aging foams, which were taken from the work of J. Lambert et al. The analyzed size distributions for surface film cells and bulk bubbles in the isolated samples of modeled liquid foams (Gillette shaving cream) allow for fitting with good accuracy using the self-similar solutions to the non-linear Fokker–Planck equation at the various aging times and various temperatures. The relationship between probability distributions for the normalized radii of the film cells and bulk bubbles is established. It is shown that the temperature-dependent rate factor of self-similar growth of the film cells and bulk bubbles can be considered in terms of the Arrhenius equation.
  • Chaotic attractor of the normal form map for grazing bifurcations of
           impact oscillators
    • Abstract: Publication date: Available online 29 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Pengcheng Miao, Denghui Li, Yuan Yue, Jianhua Xie, Celso GrebogiAbstractGrazing bifurcations can cause impact oscillators to exhibit chaotic motions. Such dynamical behavior can be described by a normal form map (called the Nordmark map). A main feature of the Nordmark map is that it has a square-root term. The purpose of this paper is to study the structure of the chaotic attractor of the Nordmark map from the topological point of view. First, the trapping region of the asymptotic dynamics of the map is constructed. It is then proven that, for some set of parameter values having positive Lebesgue measure, the ω-limit set of each point of the trapping region is contained in a invariant set which is just the closure of the unstable manifold of the hyperbolic fixed point of the map. Besides, the dynamics on the invariant set is topologically mixing. Accordingly the invariant set is a chaotic attractor of the map.
  • Supercritical and subcritical turing pattern formation in a hyperbolic
           vegetation model for flat arid environments
    • Abstract: Publication date: Available online 26 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Giancarlo Consolo, Carmela Currò, Giovanna ValentiAbstractPatterned vegetation dynamics in flat arid environments are investigated in the framework of a hyperbolic modified Klausmeier model. In particular, this study aims at elucidating how the properties exhibited by supercritical and subcritical patterns during the transient regime are affected by the inertial times.The present work encloses linear stability analysis to deduce the threshold condition for Turing pattern formation and weakly nonlinear analysis to describe the time evolution of the pattern amplitude close to the instability threshold.In our analysis, we consider the case in which the emerging patterns do not have any spatial structure, as it is typically assumed in small domains, as well as the scenario in which patterns form sequentially and propagate over a large domain in the form of a traveling wavefront.The hyperbolic model is also integrated numerically to validate the analytical predictions and to characterize more deeply the spatio-temporal evolution of vegetation patterns in both supercritical and subcritical regime.
  • Existence of rotating magnetic stars
    • Abstract: Publication date: Available online 25 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Juhi Jang, Walter A. Strauss, Yilun WuAbstractWe consider a star as a compressible fluid subject to gravitational and magnetic forces. This leads to an Euler-Poisson system coupled to a magnetic field, which may be regarded as an MHD model together with gravity. The star executes steadily rotating motion about a fixed axis. We prove, for the first time, the existence of such stars provided that the rotation speed and the magnetic field are sufficiently small.
  • On the modeling of equatorial shallow-water waves with the Coriolis effect
    • Abstract: Publication date: April 2019Source: Physica D: Nonlinear Phenomena, Volume 391Author(s): Tianqiao Hu, Yue LiuAbstractIn the present study a simplified phenomenological model of shallow-water wave propagating mainly in the equatorial ocean regions with the Coriolis effect caused by the Earth’s rotation is formally derived. The model equation which is analogous to the Green–Naghdi equations with the second-order approximation of the Camassa–Holm scaling captures stronger nonlinear effects than the classical dispersive integrable equations like the Korteweg–de Vries and two-component Camassa–Holm system. The local well-posedness of the Cauchy problem is then established by the linear transport theory and wave-breaking phenomenon is investigated based on the method of characteristics and the Riccati-type differential inequality. Finally, the condition of permanent waves is demonstrated by analyzing competition between the slope of average of horizontal velocity component and the free surface component.
  • Bellerophon state in the Kuramoto model with gravitation rules
    • Abstract: Publication date: April 2019Source: Physica D: Nonlinear Phenomena, Volume 391Author(s): Liuhua ZhuAbstractWe study a variant of the Kuramoto model in a two-dimensional grid, in which the coupling strength is governed by gravitation rules. Here the magnitude of gravitation is directly proportional to the product of natural frequencies of the two oscillators, but inversely proportional to the αth power of their distance. The numerical simulations indicate that for a large enough coupling strength, phase locking of oscillators is possible and it is more likely to occur in networks with small decay exponent α and large oscillator density ρ. When the value of α equals the dimensions of the system, the Bellerophon state appears, in which the oscillators form several different subgroups by recruiting their own peers. In each subgroup, the instantaneous frequencies of oscillators are different from each other, but they do correlate with each other in some form, so that their effective frequencies tend to be consistent. This result drops a hint that the Bellerophon state might be universal in improved Kuramoto models as long as the intrinsic properties of the oscillators are not exactly the same.
  • Elementary symmetrization of inviscid two-fluid flow equations giving a
           number of instant results
    • Abstract: Publication date: April 2019Source: Physica D: Nonlinear Phenomena, Volume 391Author(s): Lizhi Ruan, Yuri TrakhininAbstractWe consider two models of a compressible inviscid isentropic two-fluid flow. The first one describes the liquid–gas two-phase flow. The second one can describe the mixture of two fluids of different densities or the mixture of fluid and particles. Introducing an entropy-like function, we reduce the equations of both models to a symmetric form which looks like the compressible Euler equations written in the nonconservative form in terms of the pressure, the velocity and the entropy. Basing on existing results for the Euler equations, this gives a number of instant results for both models. In particular, we conclude that all compressive shock waves in these models exist locally in time. For the 2D case, we make the conclusion about the local-in-time existence of vortex sheets under a “supersonic” stability condition. In the sense of a much lower regularity requirement for the initial data, our result for 2D vortex sheets essentially improves a recent result for vortex sheets in the liquid–gas two-phase flow.
  • Explicit transversality conditions and local bifurcation diagrams for
           Bogdanov–Takens bifurcation on center manifolds
    • Abstract: Publication date: April 2019Source: Physica D: Nonlinear Phenomena, Volume 391Author(s): Yang Li, Hiroshi Kokubu, Kazuyuki AiharaAbstractThis application-oriented study is concerned with the derivation of parameter-dependent normal forms for the codimension-two Bogdanov–Takens bifurcation of n-dimensional, m-parameterized systems on the basis of the homological method. In the case of an enduring equilibrium, simple formulas are obtained for the transformation of parameters, enabling the formulation of explicit transversality conditions and bifurcation diagrams to at most the second order. Moreover, in Z2-symmetric systems, the calculation can be further limited within certain subspaces. In the general case, existing results are re-derived, and a revision necessary for determining the bifurcation diagrams to the second order is then provided. These results facilitate the derivation of normal forms, check of transversality and depiction of bifurcation diagrams for the Bogdanov–Takens bifurcation.
  • On the dynamical stability and instability of Parker problem
    • Abstract: Publication date: April 2019Source: Physica D: Nonlinear Phenomena, Volume 391Author(s): Fei Jiang, Song JiangAbstractWe investigate a Parker problem for the three-dimensional compressible isentropic viscous magnetohydrodynamic system with zero resistivity in the presence of a modified gravitational force in a vertical strip domain in which the velocity of the fluid is non-slip on the boundary, and focus on the stabilizing effect of the (equilibrium) magnetic field through the non-slip boundary condition. We show that there is a discriminant Ξ, depending on the known physical parameters, for the stability/instability of the Parker problem. More precisely, if Ξ>0, then the Parker problem is unstable, i.e., the Parker instability occurs, while if Ξ
  • Local representation and construction of Beltrami fields
    • Abstract: Publication date: April 2019Source: Physica D: Nonlinear Phenomena, Volume 391Author(s): N. Sato, M. YamadaAbstractA Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie–Darboux theorem of differential geometry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold–Beltrami–Childress flow with two of the parameters set to zero. Furthermore, a Beltrami flow admits two local invariants, a coordinate representing the physical plane of the flow, and an angular momentum-like quantity in the direction across the plane. As a consequence of the theorem, we derive a method to construct Beltrami fields with given proportionality factor. This method, based on the solution of the eikonal equation, guarantees the existence of Beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal. We construct several solenoidal and non-solenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors.
  • The second Painlevé equation, a related nonautonomous semidiscrete
           equation, and a limit to the first Painlevé equation: Scalar and matrix
    • Abstract: Publication date: April 2019Source: Physica D: Nonlinear Phenomena, Volume 391Author(s): Andrew Pickering, Pilar R. Gordoa, Jonathan A.D. WattisAbstractIn this paper we consider the matrix nonautonomous semidiscrete (or lattice) equation ddtUn=(2n−1)(Un+1−Un−1)−1, as well as the scalar case thereof. This equation was recently derived in the context of auto-Bäcklund transformations for a matrix partial differential equation. We use asymptotic techniques to reveal a connection between this equation and the matrix (or, as appropriate, scalar) first Painlevé equation. In the matrix case, we also discuss our asymptotic analysis more generally, as well as considering a component-wise approach. In addition, Hamiltonian formulations of the matrix first and second Painlevé equations are given, as well as a discussion of classes of solutions of the matrix second Painlevé equation.
  • On the synchronization phenomenon of a parallel array of spin torque
    • Abstract: Publication date: Available online 21 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Brian Sturgis-Jensen, Pietro-Luciano Buono, Antonio Palacios, James Turtle, Visarath In, Patrick LonghiniAbstractThe ability for a Spin Torque Nano Oscillator (STNO) to perform as a nano-scaled microwave voltage oscillator continues to be the focus of extensive research. Due to their small size (on the order of 10nm), low power consumption, and ultrawide frequency range STNOs demonstrate significant potential for practical applications in microwave generation. However, the low power output produced by a single STNO currently renders them inoperable for applications. In response, various groups have proposed the synchronization of a network of STNOs such that the coherent signal produces a strong enough microwave signal at the nanoscale. Achieving synchronization, however, has proven to be a challenging task. In this work we analyze the problem of synchronization for networks of identical STNOs connected in parallel. Bifurcation diagrams for small networks of STNOs are computed which depicts bistability between in-phase and out-of-phase limit cycle oscillations for much of the phase space. Numerical approximations of the relative sizes of the basins of attraction are generated and further suggest the inherent difficulty in achieving synchronization. To extend the analysis to large networks of STNOs, we exploit the SN symmetry exhibited by the all-to-all coupled array of parallel STNOs. We develop implicit analytic expressions for Hopf bifurcations which yield synchronized limit cycle oscillations, allowing for the computation of the Hopf loci and stability for an arbitrarily large network of oscillators.
  • A collective coordinate framework to study the dynamics of travelling
           waves in stochastic partial differential equations
    • Abstract: Publication date: Available online 19 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Madeleine Cartwright, Georg A. GottwaldAbstractWe propose a formal framework based on collective coordinates to reduce infinite-dimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finite-dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group. We study SPDEs arising in population dynamics with multiplicative noise and additive symmetry breaking noise. The collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour, which would otherwise only be available through costly computational experiments. We corroborate our analytical results with numerical simulations of the full SPDE.
  • An information-theoretic study of fish swimming in the wake of a pitching
    • Abstract: Publication date: Available online 19 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Peng Zhang, Elizabeth Krasner, Sean D. Peterson, Maurizio PorfiriAbstractSwimming in schools affords several advantages for fish, including enhanced ability to escape from predators, searching for food, and finding correct migratory routes. However, the role of hydrodynamics in coordinated swimming is still not fully understood due to a lack of data-driven approaches to disentangle causes from effects. In an effort to elucidate the mechanisms underlying fish schooling, we propose an empirical study that integrates information theory and experimental biology. We studied the interactions between an actively pitching airfoil and a fish swimming in a flow. The pitching frequency of the airfoil was varied randomly over time, eliciting an information-rich interaction between the airfoil and the fish. Within an information-theoretic framework, we examined the information content of fish tail beating and information transfer from the airfoil to the fish. The proposed framework may help improve our understanding of the role of hydrodynamics in fish swimming, thereby supporting hypothesis-driven studies on the hydrodynamic advantages of fish schooling.
  • Global search for localised modes in scalar and vector nonlinear
           Schrödinger-type equations
    • Abstract: Publication date: Available online 18 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): G.L. Alfimov, I.V. Barashenkov, A.P. Fedotov, V.V. Smirnov, D.A. ZezyulinAbstractWe present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schrödinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we set apart identify the bounded, nonsingular, solutions — and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances X± to the nearest forward and backward singularities; large X+ or X− indicate the proximity to a bounded solution. We illustrate our method with the Gross–Pitaevskii equation with a PT-symmetric complex potential, a system of coupled Gross–Pitaevskii equations with real potentials, and the Lugiato–Lefever equation with normal dispersion.
  • Model and experiments for resonant generation of second harmonic
           capillary–gravity waves
    • Abstract: Publication date: Available online 15 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Adriano Alippi, Andrea Bettucci, Massimo GermanoAbstractThe generation and propagation of a second harmonic water wave has been investigated in the frequency range between 7 Hz and 60 Hz where the velocity vs. frequency curve attains its minimum value. A model is proposed by assuming that the second harmonic is locally generated by point sources on the wavefront of the fundamental wave, and that at any point along the propagation direction the second harmonic be given by the cumulative contribution from all the sources up to the considered point. In the frequency range examined the combined effects of gravity and capillarity yield the so called resonance condition where the fundamental and the second harmonic waves share the very same phase velocity. In such a case, wave shape matching condition is maintained between the two waves along all the propagation direction, with the amplitude of the second harmonic only limited by the attenuation effect. Evidence is given experimentally of such effect through the wavenumbers mismatching produced by the model vs. frequency and the detection of the maximum distance of second harmonic amplitude from the wave source. Furthermore, it is found that the resonance condition is a threshold effect with respect to water depth.
  • Blow-up dynamics in the mass super-critical NLS equations
    • Abstract: Publication date: Available online 15 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Kai Yang, Svetlana Roudenko, Yanxiang ZhaoAbstractWe study stable blow-up dynamics in the L2-supercritical nonlinear Schrödinger equation with radial symmetry in various dimensions. We first investigate the profile equation and extend the result of Wang (1990) and Budd et al. (1999) on the existence and local uniqueness of solutions of the cubic profile equation to other L2-supercritical nonlinearities and dimensions d≥2. We then numerically observe the multi-bump structure of such solutions, and in particular, exhibit the Q1,0 solution, a candidate for the stable blow-up profile. Next, using the dynamic rescaling method, we investigate stable blow-up solutions in the L2-supercritical NLS and confirm the square root rate of the blow-up as well as the convergence of blow-up profiles to the Q1,0 profile.
  • Attractor bifurcation analysis for an electrically conducting fluid flow
           between two rotating cylinders
    • Abstract: Publication date: Available online 14 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Ruikuan Liu, Quan WangAbstractThis work formulates the simplified governing equations for an electrically conducting fluid flow between two concentric rotating cylinders. We consider the case where the flow is under the influence of an electric current (axial current) of suitable constant density passing axially through the fluid and a line current (center current) moving along the axis of the two concentric cylinders. Furthermore, we show that the simplified governing equations bifurcate to an S1 attractor (a 1-dimensional sphere) when the magnetic Taylor number crosses a critical value. Notably, the S1 attractor contains precisely eight singular points, four of which are stable nodes and the remaining are saddle points, with the impact of the magnetic field generated only by the axial electric current. If the magnetic field is mainly produced by the center electric current, the number of singular points of the S1 attractor can only be four, with two saddle points, and two stable nodes. In addition, our research shows that the axial current accelerates the S1 attractor bifurcation but the center current has the opposite effect, in comparison with the flow without the influence of a magnetic field.
  • A projector-based convergence proof of the Ginelli algorithm for covariant
           Lyapunov vectors
    • Abstract: Publication date: Available online 5 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Florian NoethenAbstractLinear perturbations of solutions of dynamical systems exhibit different asymptotic growth rates, which are naturally characterized by so-called covariant Lyapunov vectors (CLVs). Due to an increased interest of CLVs in applications, several algorithms were developed to compute them. The Ginelli algorithm is among the most commonly used. Although several properties of the algorithm have been analyzed, there exists no mathematically rigorous convergence proof yet.In this article we extend existing approaches in order to construct a projector-based convergence proof of Ginelli’s algorithm. One of the main ingredients will be an asymptotic characterization of CLVs via the Multiplicative Ergodic Theorem. In the proof, we keep a rather general setting allowing even for degenerate Lyapunov spectra.
  • Inter-layer synchronization of periodic solutions in two coupled rings
           with time delay
    • Abstract: Publication date: Available online 1 March 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Xu Xu, Dongyuan Yu, Zaihua WangAbstractA coupled two-ring-network with time delay is developed to avoid the weakness of a simple ring topology. The divide and conquer algorithm is used to analyze the distribution of the eigenvalues of the adjacent matrix. The stability of the equilibrium and inter-layer synchronization of the periodic dynamics are studied, and the dynamical interactions between coupled rings which lead to complicated mirror and reflection waves are further analyzed. Special attention is paid on the generalized synchronization between two rings when the adjacent matrix has repeated eigenvalues. Numerical simulations verify the theoretical results, and show that the appropriate settings of the time delay and transverse coupling can guarantee the “backup and recovery” procedure.
  • Asymptotic models for free boundary flow in porous media
    • Abstract: Publication date: Available online 28 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Rafael Granero-Belinchón, Stefano ScrobognaAbstractWe provide rigorous asymptotic models for the free boundary Darcy and Forchheimer problem under the assumption of weak nonlinear interaction, in a regime in which the steepness parameter of the interface is considered to be very small. The models we derive capture the nonlinear interaction of the original free boundary Darcy and Forchheimer problem up to quadratic terms. Furthermore, we provide models that consider both the two-dimensional and three-dimensional cases, with and without bottom topography.
  • On the possible time singularities for the 3D Navier–Stokes
    • Abstract: Publication date: Available online 23 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Xiaoyutao LuoAbstractWe prove a local-in-time regularity criterion for the 3D Navier–Stokes equations. In particular, it follows from the criterion that the Hausdorff dimension of possible singular times of Leray–Hopf weak solutions u∈LtrBs,∞α for some α>0, s>3 and r>2 is less than r2(3s+2r−α−1). The main contribution is that we do not assume the suitability of weak solutions.
  • Computation of domains of analyticity for the dissipative standard map in
           the limit of small dissipation
    • Abstract: Publication date: Available online 23 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Adrián P. Bustamante, Renato C. CallejaAbstractConformally symplectic systems include mechanical systems with a friction proportional to the velocity. Geometrically, these systems transform a symplectic form into a multiple of itself making the systems dissipative or expanding. In the present work we consider the limit of small dissipation. The example we study is a family of conformally symplectic standard maps of the cylinder for which the conformal factor, b(ε), is a function of a small complex parameter, ε.We assume that for ε=0 the map preserves the symplectic form and the dependence on ε is cubic, i.e., b(ε)=1−ε3. We compute perturbative expansions formally in ε and use them to estimate the shape of the domains of analyticity of invariant circles as functions of ε. We also give evidence that the functions might belong to a Gevrey class at ε=0. We also perform numerical continuation of the solutions as they pass through the boundary of the domain to illustrate that the monodromy of the solutions is trivial. The numerical computations we perform support conjectures on the shape of the domains of analyticity.
  • p -Schrödinger+equations+under+the+mixed+boundary+conditions+on+networks&rft.title=Physica+D:+Nonlinear+Phenomena&rft.issn=0167-2789&">The discrete p -Schrödinger equations under the mixed boundary conditions
           on networks
    • Abstract: Publication date: Available online 22 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Soon-Yeong Chung, Jaeho HwangAbstractIn this paper, we study the discrete eigenvalue problems for the p-Schrödinger equations under the mixed boundary conditions defined on networks as follows: −Δp,ωϕ(x)+q(x) ϕ(x) p−2ϕ(x)=λ ϕ(x) p−2ϕ(x),x∈S,μ(z)∂ϕ∂pn(z)+σ(z) ϕ(z) p−2ϕ(z)=0,z∈∂S,where p>1, q is a real-valued function on a network S, and μ,σ are nonnegative functions on the boundary ∂S of S, with μ(z)+σ(z)>0, z∈∂S. Next, we use the above result to provide the existence of a positive solution to the discrete Poisson equation −Δp,ωu(x)+q(x) u(x) p−2u(x)=f(x),x∈S,μ(z)∂u∂pn(z)+σ(z) u(z) p−2u(...
  • Interactions of solitary pulses of E. coli in a one-dimensional
           nutrient gradient
    • Abstract: Publication date: Available online 19 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Glenn Young, Mahmut Demir, Hanna Salman, G. Bard Ermentrout, Jonathan E. RubinAbstractWe study the interaction of two bacterial pulses in a one-dimensional nutrient gradient. Simulations of the Keller–Segel chemotaxis model reveal two qualitatively distinct behaviors. As the two pulses approach one another, they either combine and move as a single pulse or, surprisingly, change direction and begin moving away from each other in the direction from which they originated. To study this phenomenon, we introduce a heuristic approximation to the spatial profiles of the pulses in the Keller–Segel model and derive a system of ordinary differential equations approximating the dynamics of the pulse centers of mass and widths. This approximation simplifies analysis of the global dynamics of the bacterial system and allows us to efficiently explore qualitative behavior changes under a range of parameter variations. We end by presenting experimental data showing that populations of E. coli display behavior that qualitatively agrees with our theoretical results.
  • Time-dependent saddle–node bifurcation: Breaking time and the point of
           no return in a non-autonomous model of critical transitions
    • Abstract: Publication date: Available online 19 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Jeremiah H. Li, Felix X.-F. Ye, Hong Qian, Sui HuangAbstractThere is a growing awareness that catastrophic phenomena in biology and medicine can be mathematically represented in terms of saddle–node bifurcations. In particular, the term “tipping”, or critical transition has in recent years entered the discourse of the general public in relation to ecology, medicine, and public health. The saddle–node bifurcation and its associated theory of catastrophe as put forth by Thom and Zeeman has seen applications in a wide range of fields including molecular biophysics, mesoscopic physics, and climate science. In this paper, we investigate a simple model of a non-autonomous system with a time-dependent parameter p(τ) and its corresponding “dynamic” (time-dependent) saddle–node bifurcation by the modern theory of non-autonomous dynamical systems. We show that the actual point of no return for a system undergoing tipping can be significantly delayed in comparison to the breaking time τˆ at which the corresponding autonomous system with a time-independent parameter pa=p(τˆ) undergoes a bifurcation. A dimensionless parameter α=λp03V−2 is introduced, in which λ is the curvature of the autonomous saddle–node bifurcation according to parameter p(τ), which has an initial value of p0 and a constant rate of change V. We find that the breaking time τˆ is always less than the actual point of no return τ∗ after which the critical transition is irreversible; specifically, the relation τ∗−τˆ≃2.338(λV)−13 is analytically obtained. For a system with a small λV, there exists a significant window of opportunity (τˆ,τ∗) during which rapid reversal of the environment can save the system from catastrophe.
  • Moran model of spatial alignment in microbial colonies
    • Abstract: Publication date: Available online 18 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): B.R. Karamched, W. Ott, I. Timofeyev, R.N. Alnahhas, M.R. Bennett, K. JosićAbstractWe describe a spatial Moran model that captures mechanical interactions and directional growth in spatially extended populations. The model is analytically tractable and completely solvable under a mean-field approximation and can elucidate the mechanisms that drive the formation of population-level patterns. As an example we model a population of E. coli growing in a rectangular microfluidic trap. We show that spatial patterns can arise as a result of a tug-of-war between boundary effects and growth rate modulations due to cell–cell interactions: Cells align parallel to the long side of the trap when boundary effects dominate. However, when cell–cell interactions exceed a critical value, cells align orthogonally to the trap’s long side. This modeling approach and analysis can be extended to directionally-growing cells in a variety of domains to provide insight into how local and global interactions shape collective behavior.
  • Existence and stability of steady compressible Navier–Stokes solutions
           on a finite interval with noncharacteristic boundary conditions
    • Abstract: Publication date: Available online 15 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Benjamin Melinand, Kevin ZumbrunAbstractWe study existence and stability of steady solutions of the isentropic compressible Navier–Stokes equations on a finite interval with noncharacteristic boundary conditions, for general not necessarily small-amplitude data. We show that there exists a unique solution, about which the linearized spatial operator possesses (i) a spectral gap between neutral and growing/decaying modes, and (ii) an even number of nonstable eigenvalues λ (with a nonnegative real part). In the case that there are no nonstable eigenvalues, i.e., of spectral stability, we show this solution to be nonlinearly exponentially stable in H2×H3. Using “Goodman-type” weighted energy estimates, we establish spectral stability for small-amplitude data. For large-amplitude data, we obtain high-frequency stability, reducing stability investigations to a bounded frequency regime. On this remaining, bounded-frequency regime, we carry out a numerical Evans function study, with results again indicating universal stability of solutions.
  • Mathematical modeling of cyclic population dynamics
    • Abstract: Publication date: Available online 12 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): A. Bayliss, A.A. Nepomnyashchy, V.A. VolpertAbstractA rock–paper–scissors three species cyclic ecosystem is considered. Deterministic mathematical models based on delayed ODEs and nonlocal PDEs are proposed and studied both analytically and numerically. Transitions between the coexistence state that is associated with biodiversity, limit cycles and the heteroclinic cycle are discussed for the ODE model. Traveling waves between the coexistence state and single species states are studied for the PDE model. We show that delay promotes oscillatory instabilities of the coexistence state while nonlocality promotes stationary cellular instabilities.
  • Polynomial mixing under a certain stationary Euler flow
    • Abstract: Publication date: Available online 6 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Gianluca Crippa, Renato Lucà, Christian SchulzeAbstractWe study the mixing properties of a scalar ρ on the unit disk advected by a certain incompressible velocity field u, which is a stationary radial solution of the Euler equation. The scalar ρ solves the continuity equation with the velocity field u and we can measure the degree of “mixedness” of ρ with two different scales commonly used in this setting, namely the geometric and the functional mixing scale. We develop a physical space approach well adapted to the quantitative analysis of the decay in time of the geometric mixing scale, which turns out to be polynomial for a large class of initial data. This extends previous results for the functional mixing scale, based on the explicit expression for the solution in Fourier variable, results that are also partially recovered by our approach.
  • Volume scavenging of networked droplets
    • Abstract: Publication date: Available online 5 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Thomas C. Hagen, Paul H. SteenAbstractA system of N spherical-cap fluid droplets protruding from circular openings on a plane is connected through channels. This system is governed by surface tension acting on the droplets and viscous stresses inside the fluid channels. The fluid rheology is given by the Ostwald–de Waele power law, thus permitting shear thinning. The pressure acting on each droplet is caused by capillarity and given in terms of the droplet volume via the Young–Laplace law. Liquid is exchanged along the network of fluid conduits due to an imbalance of the Laplace pressures between the droplets. In this way some droplets gain volume at the expense of others. This mechanism, christened “volume scavenging,” leads to interesting dynamics.Numerical experiments show that an initial droplet configuration is driven to a stable equilibrium exhibiting 1 super-hemispherical droplet and N−1 sub-hemispherical ones when the initial droplet volumes are large. The selection of this “winning” droplet depends not only on the channel network and the fluid volume, but also notably on the fluid rheology. The rheology is also observed to drastically change the transition to equilibrium. For smaller droplet volumes the long-term behavior is seen to be more complicated since the types of equilibria differ from those arising for larger volumes. These observations motivate our analytical study of equilibria and their stability for the corresponding nonlinear dynamical system. The identification of equilibria is accomplished by locating the zeros of a mass polynomial, defined through the constant volume/mass constraint. The key tool in our stability analysis is a pressure–volume work functional, related to the total surface area, which serves as a Lyapunov function for the dynamical system. This functional is useful since equilibria are typically not hyperbolic and linearization techniques not available. Equilibria will be shown to be hierarchically organized in terms of size of the pressure–volume work functional. For larger droplet volumes this ordering exhibits one hierarchy of equilibria. Two hierarchies exist when the volumes are smaller. The minimizing equilibria in either case are asymptotically stable.
  • Mutual transitions between stationary and moving dissipative solitons
    • Abstract: Publication date: Available online 4 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Bogdan A. KochetovAbstractApplication of stable localized dissipative solitons as basic carriers of information promises the significant progress in the development of new optical communication networks. The success development of such systems requires getting the full control over soliton waveforms. In this paper we use the fundamental model of dissipative solitons in the form of the complex Ginzburg–Landau equation with a potential term to demonstrate controllable transitions between different types of coexisted waveforms of stationary and moving dissipative solitons. Namely, we consider mutual transitions between so-called plain (fundamental soliton), composite, and moving pulses. We found necessary features of transverse spatial distributions of locally applied (along the propagation distance) attractive potentials to perform those waveform transitions. We revealed that a one-peaked symmetric potential transits the input pulses to the plain pulse, while a two-peaked symmetric (asymmetric) potential performs the transitions of the input pulses to the composite (moving) pulse.
  • A kinematic evolution equation for the dynamic contact angle and some
    • Abstract: Publication date: Available online 1 February 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Mathis Fricke, Matthias Köhne, Dieter BotheAbstractWe investigate the moving contact line problem for two-phase incompressible flows with a kinematic approach. The key idea is to derive an evolution equation for the contact angle in terms of the transporting velocity field. It turns out that the resulting equation has a simple structure and expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. Thus, the kinematic evolution equation is a tool to analyze the evolution of the contact angle. Since the transporting velocity field is required only on the moving interface, the kinematic evolution equation also applies when the interface moves with its own velocity independent of the fluid velocity. We apply the developed tool to a class of moving contact line models which employ the Navier slip boundary condition. We derive an explicit form of the contact angle evolution for sufficiently regular solutions, showing that such solutions are unphysical. Within the simplest model, this rigorously shows that the contact angle can only relax to equilibrium if some kind of singularity is present at the contact line. Moreover, we analyze more general models including surface tension gradients at the contact line, slip at the fluid–fluid interface and mass transfer across the fluid–fluid interface.
  • Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model
    • Abstract: Publication date: Available online 31 January 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Jian-Guo Liu, Jianfeng Lu, Dionisios Margetis, Jeremy L. MarzuolaAbstractIn the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study analytically an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a continuum evolution law expressed by a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the nonlinear gradient flow of a convex, singular surface free energy with a certain exponential mobility in homoepitaxy. This evolution law is motivated both by an atomistic broken-bond model and a mesoscale model for crystal steps. By constructing an explicit solution to this PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the continuum law via a regularized surface free energy.
  • The effect of integral control in oscillatory and chaotic reaction kinetic
    • Abstract: Publication date: Available online 30 January 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Kristian Thorsen, Tormod Drengstig, Peter RuoffAbstractIntegral control is ubiquitously used in industrial processes to keep variables robustly regulated at a given setpoint. Integral control is also present in many biological systems where it, implemented through reaction kinetic networks of genes, proteins and molecules, protects the organism against external variations. One difference between industrial control systems and organisms is that oscillatory behavior seems to be more common in biology. This is probably because engineers can choose to design systems that avoid oscillations. Looking at regulation from the viewpoint of biological systems, the prevalence of oscillations leads to a question which is not often asked in traditional control engineering: how can regulatory and adaptive mechanisms function and coexist with oscillations' And furthermore: does integral control provide some kind of robust regulation in oscillatory systems' Here we present an analysis of the effect of integral control in oscillatory systems. We study nonlinear reaction kinetic networks where integral control is internally present and how these systems behave for parameter values that produce periodic and chaotic oscillations. In addition, we also study how the behavior of an oscillatory reaction kinetic network, the Brusselator, changes when integral control is added to it. Our results show that integral control, when internally present, in an oscillatory system robustly defends the average level of a controlled variable. This is true for both periodic and chaotic oscillations. Although we use reaction kinetic networks in our study, the properties we find are applicable to all systems that contain integral control.
  • Travelling-wave spatially periodic forcing of asymmetric binary mixtures
    • Abstract: Publication date: Available online 22 January 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Lennon Ó NáraighAbstractWe study travelling-wave spatially periodic solutions of a forced Cahn–Hilliard equation. This is a model for phase separation of a binary mixture, subject to external forcing. We look at arbitrary values of the mean mixture concentration, corresponding to asymmetric mixtures (previous studies have only considered the symmetric case). We characterize in depth one particular solution which consists of an oscillation around the mean concentration level, using a range of techniques, both numerical and analytical. We determine the stability of this solution to small-amplitude perturbations. Next, we use methods developed elsewhere in the context of shallow-water waves to uncover a (possibly infinite) family of multiple-spike solutions for the concentration profile, which linear stability analysis demonstrates to be unstable. Throughout the work, we perform thorough parametric studies to outline for which parameter values the different solution types occur.
  • A study of pattern forming systems with a fully nonlocal interaction
           kernel of possibly changing sign
    • Abstract: Publication date: Available online 8 January 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Xiaofeng Ren, Jeremy TrageserAbstractA nonlocal model with a possibly sign changing kernel is proposed to study pattern formation problems in physical and biological systems with self-organization properties. One defines pre-solutions of a related linear problem and gives sufficient conditions for pre-solutions to be jump discontinuous solutions of the nonlocal model. Among a multitude of solutions a selection criterion is used to single out more significant critical solutions. These solutions are further classified into minimizing solutions, maximizing solutions and saddle solutions. Minimizing solutions are the most relevant for patterned states and they exist if the kernel changes sign. A non-negative kernel on the other hand behaves in some ways like the gradient term in the standard local model.
  • Perturbation solutions to the nonlinear motion of displaced orbits
    • Abstract: Publication date: Available online 6 January 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Xiao Pan, Ming Xu, Yuechen Ma, Yanchao HeAbstractThis paper analytically describes the nonlinear motion of displaced orbits obtained by low-thrust propulsion, and focuses on the first-order analytical derivation of perturbation solutions of osculating Keplerian element (OKE) time histories. A virtual Earth (VE) model is developed to transform non-Keplerian orbits above the Earth into Keplerian orbits around the VE by treating the thrust and coordinate transformation effects as perturbations. The numerical OKEs computed from the Cartesian position and velocity exhibit secular and periodic variations. The short-period perturbations of the orbital elements are derived by the conventional quasi-mean element method. The differential equations of the secular and long-period perturbations are linearized for an analytical solution. A comparison between the analytical OKEs and numerical OKEs of a specific displaced orbit validates the correctness and good accuracy of the analytical derivation. One contribution of this paper is that the analytical OKEs can be applied for a fast calculation of a displaced orbit in the preliminary mission design phase, which has more significance than accurate but complicated time-consuming calculations by complete variational equations. Another contribution is revealing the connections between the amplitude as well as frequency of perturbation solutions and displaced orbits from a physical perspective, from which the topology and geometrical size of a displaced orbit can be easily inferred from the time history of OKEs.
  • Kinetic equation for nonlinear wave-particle interaction: Solution
           properties and asymptotic dynamics
    • Abstract: Publication date: Available online 2 January 2019Source: Physica D: Nonlinear PhenomenaAuthor(s): Anton Artemyev, Anatoly Neishtadt, Alexei VasilievAbstractWe consider a kinetic equation describing evolution of the particle distribution function in a system with nonlinear wave-particle interactions (trappings into resonance and nonlinear scatterings). We study properties of its solutions and show that the only stationary solution is a constant, and that all solutions with smooth initial conditions tend to a constant as time grows. The resulting flattening of the distribution function in the domain of nonlinear interactions is similar to one described by the quasi-linear plasma theory, but the distribution evolves much faster. The results are confirmed numerically for a model problem.
  • Stability, well-posedness and blow-up criterion for the Incompressible
           Slice Model
    • Abstract: Publication date: Available online 31 December 2018Source: Physica D: Nonlinear PhenomenaAuthor(s): Diego Alonso-Orán, Aythami Bethencourt de LeónAbstractIn atmospheric science, slice models are frequently used to study the behaviour of weather, and specifically the formation of atmospheric fronts, whose prediction is fundamental in meteorology. In 2013, Cotter and Holm introduced a new slice model, which they formulated using Hamilton’s variational principle, modified for this purpose. In this paper, we show the local existence and uniqueness of strong solutions of the related ISM (Incompressible Slice Model). The ISM is a modified version of the Cotter–Holm Slice Model (CHSM) that we have obtained by adapting the Lagrangian function in Hamilton’s principle for CHSM to the Euler–Boussinesq Eady incompressible case. Besides proving local existence and uniqueness, in this paper we also construct a blow-up criterion for the ISM, and study Arnold’s stability around a restricted class of equilibrium solutions. These results establish the potential applicability of the ISM equations in physically meaningful situations.
  • Non blowup of a generalized Boussinesq–Burgers system with nonlinear
           dispersion relation and large data
    • Abstract: Publication date: Available online 28 December 2018Source: Physica D: Nonlinear PhenomenaAuthor(s): Neng Zhu, Zhengrong Liu, Kun ZhaoAbstractWe study the qualitative behavior of classical solutions to the Cauchy problem of a generalized Boussinesq–Burgers system in one space dimension. Assuming initial data belong to H2(R) and utilizing energy methods, we show that there exist unique global-in-time classical solutions to the Cauchy problem of the model, and the solutions converge to constant equilibrium states as time goes to infinity, regardless of the magnitude of the initial data. Moreover, it is shown that the viscous and inviscid models are consistent in the process of vanishing viscosity limit.
  • Micropolar meets Newtonian. The Rayleigh–Bénard problem
    • Abstract: Publication date: Available online 21 December 2018Source: Physica D: Nonlinear PhenomenaAuthor(s): Piotr Kalita, José A. Langa, Grzegorz ŁukaszewiczAbstractWe consider the Rayleigh–Bénard problem for the two-dimensional Boussinesq system for the micropolar fluid. Our main goal is to compare the value of the critical Rayleigh number, and estimates of the Nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical Navier–Stokes system. Our estimates reveal the stabilizing effects of micropolarity in comparison with the homogeneous Navier–Stokes fluid. In particular, the critical Rayleigh number for the micropolar model is larger than that for the Navier–Stokes one, and large micropolar viscosity may even prevent the averaged heat transport in the upward vertical direction. The estimate of the fractal dimension of the global attractor for the considered problem is better than that for the same problem for the Navier–Stokes system. Besides, we obtain also other results which relate the heat convection in the micropolar fluids to the Newtonian ones, among them, relations between the Nusselt number and the energy dissipation rate and upper semicontinuous convergence of global attractors to the Rayleigh–Bénard attractor for the considered model.
  • Using statistical functionals for effective control of inhomogeneous
           complex turbulent dynamical systems
    • Abstract: Publication date: Available online 14 December 2018Source: Physica D: Nonlinear PhenomenaAuthor(s): Andrew J. Majda, Di QiAbstractEfficient statistical control strategies are developed for general complex turbulent systems with energy conserving nonlinearity. Instead of direct control on the high-dimensional turbulent equations concerning a large number of instabilities, a statistical functional that characterizes the total statistical structure of the complex system is adopted here as the control object. First the statistical energy equation reduces the control of the complex nonlinear system to a linear statistical control problem; then the explicit form of the forcing control is recovered through nonlocal inversion of the optimal control functional using approximate statistical linear response theory for attribution of the feedback. Through this control strategy with statistical energy conservation, the explicit form of the control forcing is determined offline only requiring the initial configuration of total statistical energy change and the autocorrelation functions in the most sensitive modes of the target statistical equilibrium, with no need of knowing the explicit forcing history and running the complex system. The general framework of the statistical control method can be applied directly on various scenarios with both homogeneous and inhomogeneous perturbations. The effectiveness of the statistical control strategy is demonstrated using the Lorenz ’96 system and a turbulent barotropic system with topography and a large number of instabilities.
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Heriot-Watt University
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