Showing papers in "Nonlinearity in 2008"

Energy conservation and Onsager's conjecture for the Euler equations

Abstract: Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space . We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood?Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques

Abstract: The aim of this review is to introduce the reader to some of the physical notions and the mathematical methods that are relevant to the study of nonlinear waves in Bose–Einstein condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyse some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g. the linear or the nonlinear limit or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.

Finite-time blow-up in a quasilinear system of chemotaxis

Abstract: We consider an elliptic–parabolic system of the Keller–Segel type which involves nonlinear diffusion. We find a critical exponent of the nonlinearity in the diffusion, measuring the strength of diffusion at points of high (population) densities, which distinguishes between finite-time blow-up and global-in-time existence of uniformly bounded solutions. This critical exponent depends on the space dimension n ≥ 1, where apart from the physically relevant cases n = 2 and n = 3 also the result obtained in the one-dimensional setting might be of mathematical interest: here, namely, finite-time explosion of solutions occurs although the Lyapunov functional associated with the system is bounded from below. Additionally this one-dimensional case is an example to show that L∞ estimates of solutions to non-uniformly parabolic drift–diffusion equations cannot be expected even when boundedness of the gradient of the drift term is presupposed.

Wigner's dynamical transition state theory in phase space: classical and quantum

Abstract: We develop Wigner's approach to a dynamical transition state theory in phase space in both the classical and quantum mechanical settings. The key to our development is the construction of a normal form for describing the dynamics in the neighbourhood of a specific type of saddle point that governs the evolution from reactants to products in high dimensional systems. In the classical case this is the standard Poincare–Birkhoff normal form. In the quantum case we develop a normal form based on the Weyl calculus and an explicit algorithm for computing this quantum normal form. The classical normal form allows us to discover and compute the phase space structures that govern classical reaction dynamics. From this knowledge we are able to provide a direct construction of an energy dependent dividing surface in phase space having the properties that trajectories do not locally 're-cross' the surface and the directional flux across the surface is minimal. Using this, we are able to give a formula for the directional flux through the dividing surface that goes beyond the harmonic approximation. We relate this construction to the flux–flux autocorrelation function which is a standard ingredient in the expression for the reaction rate in the chemistry community. We also give a classical mechanical interpretation of the activated complex as a normally hyperbolic invariant manifold (NHIM), and further describe the structure of the NHIM. The quantum normal form provides us with an efficient algorithm to compute quantum reaction rates and we relate this algorithm to the quantum version of the flux–flux autocorrelation function formalism. The significance of the classical phase space structures for the quantum mechanics of reactions is elucidated by studying the phase space distribution of scattering states. The quantum normal form also provides an efficient way of computing Gamov–Siegert resonances. We relate these resonances to the lifetimes of the quantum activated complex. We consider several one, two and three degree-of-freedom systems and show explicitly how calculations of the above quantities can be carried out. Our theoretical framework is valid for Hamiltonian systems with an arbitrary number of degrees of freedom and we demonstrate that in several situations it gives rise to algorithms that are computationally more efficient than existing methods.

Spatially localized structures in dissipative systems: open problems

Abstract: Stationary spatially localized structures, sometimes called dissipative solitons, arise in many interesting and important applications, including buckling of slender structures under compression, nonlinear optics, fluid flow, surface catalysis, neurobiology and many more. The recent resurgence in interest in these structures has led to significant advances in our understanding of the origin and properties of these states, and these in turn suggest new questions, both general and system-specific. This paper surveys these results focusing on open problems, both mathematical and computational, as well as on new applications.

Complexities of splashing

Abstract: We study the impact of a drop of liquid onto a thin layer of the same liquid. We give an overview of the sequence of events that occur as the two most important dimensionless control parameters are varied. In particular, multiple cohorts of droplets can be ejected at different stages after impact due to different mechanisms. Edgerton's famous Milkdrop Coronet is only observed for a narrow range of parameters. Outside this range, the splash is either qualitatively different, or suffers from a much lower level of regularity.

Asymptotic behaviour of solutions of planar elliptic systems with strong competition

Abstract: We study a class of planar nonlinear elliptic systems with competition which includes the Hartree–Fock type approximation for a system of Bose–Einstein condensates in multiple hyperfine states as derived by Esry et al (1997 Phys. Rev. Lett. 78 3594–7). We study the limit behaviour of solutions in the case where the repulsive interaction tends to infinity and phase separation is expected. In particular, we prove the continuity of the limit shape and derive limit equations satisfied within its nodal sets. By this we complement recent work of Chang et al (2004 Physique D 196 341–61) where additional assumptions had to be made.

Linear response formula for piecewise expanding unimodal maps

Abstract: The average of a smooth function with respect to the SRB measure ?t of a smooth one-parameter family ft of piecewise expanding interval maps is not always Lipschitz (Baladi 2007 Commun. Math. Phys. 275 839?59, Mazzolena 2007 Master's Thesis Rome 2, Tor Vergata). We prove that if ft is tangent to the topological class of f, and if ?t ft|t = 0 = X f, then is differentiable at zero, and coincides with the resummation proposed (Baladi 2007) of the (a priori divergent) series given by Ruelle's conjecture. In fact, we show that t ?t is differentiable within Radon measures. Linear response is violated if and only if ft is transversal to the topological class of f.

On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law

Abstract: Understanding of spatial and temporal behaviour of interacting species or reactants in ecological or chemical systems has become a central issue, and rigorously determining the formation of patterns in models from various mechanisms is of particular interest to applied mathematicians. In this paper, we study a bimolecular autocatalytic reaction–diffusion model with saturation law and are mainly concerned with the corresponding steady-state problem subject to the homogeneous Neumann boundary condition. In particular, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain reactant is large or small. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system. Our theoretical analysis shows that the diffusion rate of this reactant and the size of the reactor play decisive roles in leading to the formation of stationary patterns.

On a generalization of the Constantin?Lax?Majda equation

Abstract: We present evidence on the global existence of solutions of De Gregorio's equation, based on numerical computation and a mathematical criterion analogous to the Beale–Kato–Majda theorem. Its meaning in the context of a generalized Constantin–Lax–Majda equation will be discussed. We then argue that a convection term, if set in a proper form and in a proper magnitude, can deplete solutions of blow-up.

Orbits with historic behaviour, or non-existence of averages

Abstract: We say that an orbit {xn} of a dynamical system has historic behaviour if for some continuous function f on the state space the average does not exist (or converge). The problem is under which conditions such orbits can occur in a persistent way.

Global solution for a chemotactic-haptotactic model of cancer invasion

Abstract: This paper deals with a mathematical model of cancer invasion of tissue recently proposed by Chaplain and Lolas. The model consists of a reaction–diffusion-taxis partial differential equation (PDE) describing the evolution of tumour cell density, a reaction–diffusion PDE governing the evolution of the proteolytic enzyme concentration and an ordinary differential equation modelling the proteolysis of the extracellular matrix (ECM). In addition to random motion, the tumour cells are directed not only by haptotaxis (cellular locomotion directed in response to a concentration gradient of adhesive molecules along the ECM) but also by chemotaxis (cellular locomotion directed in response to a concentration gradient of the diffusible proteolytic enzyme). In one space dimension, the global existence and uniqueness of a classical solution to this combined chemotactic–haptotactic model is proved for any chemotactic coefficient χ > 0. In two and three space dimensions, the global existence is proved for small χ/μ (where μ is the logistic growth rate of the tumour cells). The fundamental point of proof is to raise the regularity of a solution from L1 to Lp (p > 1). Furthermore, the existence of blow-up solutions to a sub-model in two space dimensions for large χ shows, to some extent, that the condition that χ/μ is small is necessary for the global existence of a solution to the full model.

Long-time convergence of an adaptive biasing force method

Abstract: We propose a proof of convergence of an adaptive method used in molecular dynamics to compute free energy profiles (see Darve and Porohille 2001 J. Chem. Phys. 115 9169–83, Henin and Chipot 2004 J. Chem. Phys. 121 2904–14, Lelievre et al J. Chem. Phys. 126 134111). Mathematically, it amounts to studying the long-time behaviour of a stochastic process which satisfies a nonlinear stochastic differential equation, where the drift depends on conditional expectations of some functionals of the process. We use entropy techniques to prove exponential convergence to the stationary state.

A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits

Abstract: We present a numerical method for finding and continuing heteroclinic connections of vector fields that involve periodic orbits. Specifically, we concentrate on the case of a codimension-d heteroclinic connection from a saddle equilibrium to a saddle periodic orbit, denoted EtoP connection for short. By employing a Lin's method approach we construct a boundary value problem that has as its solution two orbit segments, one from the equilibrium to a suitable section Σ and the other from Σ to the periodic orbit. The difference between their two end points in Σ can be chosen in a d-dimensional subspace, and this gives rise to d well-defined test functions that are called the Lin gaps. A connecting orbit can be found in a systematic way by closing the Lin gaps one by one in d consecutive continuation runs. Indeed, any common zero of the Lin gaps corresponds to an EtoP connection, which can then be continued in system parameters.The performance of our method is demonstrated with a number of examples. Specifically, we computate different types of EtoP orbits in the Lorenz system, in a vector-field model of a saddle-node Hopf bifurcation with global reinjection and in a four-dimensional Duffing-type system. Finally, we demonstrate the versatility of our geometric approach by finding a codimension-zero heteroclinic connection between two saddle periodic orbits in a four-dimensional vector field.

On the Gross–Pitaevskii equation for trapped dipolar quantum gases

Abstract: We study the time-dependent Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases. Existence and uniqueness as well as the possible blow-up of solutions are studied. Moreover, we discuss the problem of dimension reduction for this nonlinear and nonlocal Schrodinger equation.

A quasineutral type limit for the Navier–Stokes–Poisson system with large data

Abstract: In this paper we investigate a quasineutral type limit for the Navier–Stokes–Poisson system. We prove that the projection of the approximating velocity fields on the divergence-free vector field is relatively compact and converges to a Leray weak solution of the incompressible Navier–Stokes equation. By exploiting the wave equation structure of the density fluctuation we achieve the convergence of the approximating sequences by means of a dispersive estimate of the Strichartz type.

On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I

Abstract: The phenomenon of the generic coexistence of infinitely many periodic orbits with different numbers of positive Lyapunov exponents is analysed. Bifurcations of periodic orbits near a homoclinic tangency are studied. Criteria for the coexistence of infinitely many stable periodic orbits and for the coexistence of infinitely many stable invariant tori are given.

Partial differential equations for self-organization in cellular and developmental biology

Abstract: Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field.

Zero singularities of codimension two and three in delay differential equations

Abstract: We give conditions under which a general class of delay differential equations has a point of Bogdanov?Takens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to two- or three-dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depend on the original parameters in the model. Finally we apply our results to two neural models and compare the predictions of the theory with numerical bifurcation analysis of the full equations. One model involves a transcritical bifurcation, hence we derive and analyse the appropriate unfoldings for this case.

The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations

Abstract: In this work, we give a complete description of the asymptotic behaviour of quasi-geostrophic equations in the subcritical range . We first show that its solutions simplify asymptotically as t → ∞. More precisely, solutions behave as a particular self-similar solution normalized by the mass as t → ∞ and when the initial data belong to . On the other hand, we show that solutions with initial data in decay towards zero as t → ∞ in this space. All results are obtained regardless of the size of the initial condition.

Open questions leading to a global perspective in dynamics

Abstract: In this paper, I address one of the most challenging and central problems in dynamical systems, which here means flows, diffeomophisms or, more generally, transformations, defined on a closed manifold (compact, without boundary or an interval on the real line): can we describe the behaviour in the long run of typical trajectories for the 'majority' of systems? Poincare was probably the first to point in this direction and stress its importance. Certainly this is so in the context of modern dynamics, whose foundation we attribute to him. Alongside Poincare, this sense of dynamical globality can be seen in the work of Birkhoff, Morse, Andronov and Pontyagin, Peixoto, among others, and most notably Smale in the early 1960s. Smale conjectured that the limit set of all trajectories of a typical dynamical system should display a hyperbolic behaviour: along trajectories, distances increase and decrease exponentially in complementary dimensions, or in complementary dimensions transversally to flow trajectories. By the end of the decade, a number of counter examples to the conjecture were provided. Quite a few years afterwards, in 1995, see Palis (2000 Asterisque 261 339–51), based on my previous work with Takens, Newhouse, Viana and Yoccoz and several other colleagues' work, I was able to set up a program of interrelated conjectures aimed at describing the behaviour in the long run of a typical trajectory of a typical system in finite-dimensional parameter families of dynamics, the families also being typical. In brief, for a typical dynamical system, almost all trajectories have only finitely many choices, of (transitive) attractors, where to accumulate upon in the future and such attractors should be stochastically stable. Different to the more common topological viewpoint in the 1960s, the approach here is a probabilistic one, as publicized by the Russian school many years ago, and typicality is taken in terms of Lebesgue probability both in parameter and phase spaces. There is also a flavour of randomness in the behaviour at large of a typical trajectory, and of sensitivity with respect to initial conditions, unless the associated limit set attractor is just a fixed point or periodic attracting trajectory. Hence, we are also proposing a description of most chaotic systems. A strategy to verify the validity of such a global scenario is to show that this is indeed the case in the absence, in a robust way, of homoclinic tangencies, introduced by Poincare, or heterodimensional cycles, introduced by Newhouse and myself, and much developed by Bonatti, Diaz, Rocha and others. Some of the considerable progress made along the main conjecture and related ones are commented on in this paper.

Turbulence transition in pipe flow: some open questions

Abstract: The transition to turbulence in pipe flow is a longstanding problem in fluid dynamics. In contrast to many other transitions it is not connected with linear instabilities of the laminar profile and hence follows a different route. Experimental and numerical studies within the last few years have revealed many unexpected connections to the nonlinear dynamics of strange saddles and have considerably improved our understanding of this transition. The text summarizes some of these insights and points to some outstanding problems in areas where valuable contributions from nonlinear dynamics can be expected.

On sensitive sets in topological dynamics

Abstract: In this paper notions of sensitive sets (S-sets) and regionally proximal sets (Q-sets) are introduced. It is shown that a transitive system is sensitive if and only if there is an S-set with Card(S) ≥ 2, and for a transitive system each S-set is a Q-set. Moreover, the converse holds when (X, T) is minimal. It turns out that each transitive (X, T) has a maximal almost equicontinuous factor.According to the cardinalities of the S-sets, transitive systems are divided into several classes. Characterizations and examples are given for this classification both in minimal and transitive non-minimal settings. It is proved that for a transitive system any entropy set is an S-set, and consequently, a transitive system which has no uncountable S-sets has zero topological entropy. Moreover, it is shown that a transitive, non-minimal system with dense set of minimal points has an infinite S-set, and there exists a Devaney chaotic system which has no uncountable S-set. Finally, a non-minimal sensitive E-system is constructed such that each of its S-set has cardinality at most 4.

On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

Abstract: In this paper we study the non-existence and the uniqueness of limit cycles for the Lienard differential equation of the form x'' − f(x)x' + g(x) = 0 where the functions f and g satisfy xf(x) > 0 and xg(x) > 0 for x ≠ 0 but can be discontinuous at x = 0.In particular, our results allow us to prove the non-existence of limit cycles under suitable assumptions, and also prove the existence and uniqueness of a limit cycle in a class of discontinuous Lienard systems which are relevant in engineering applications.

An averaging principle for a completely integrable stochastic Hamiltonian system

Abstract: We investigate the effective behaviour of a small transversal perturbation of order to a completely integrable stochastic Hamiltonian system, by which we mean a stochastic differential equation whose diffusion vector fields are formed from a completely integrable family of Hamiltonian functions Hi, i = 1, ..., n. An averaging principle is shown to hold and the action component of the solution converges, as → 0, to the solution of a deterministic system of differential equations when the time is rescaled at 1/. An estimate for the rate of the convergence is given. In the case when the perturbation is a Hamiltonian vector field, the limiting deterministic system is constant in which case we show that the action component of the solution scaled at 1/2 converges to that of a limiting stochastic differentiable equation.

Filtering nonlinear dynamical systems with linear stochastic models

Abstract: An important emerging scientific issue is the real time filtering through observations of noisy signals for nonlinear dynamical systems as well as the statistical accuracy of spatio-temporal discretizations for filtering such systems. From the practical standpoint, the demand for operationally practical filtering methods escalates as the model resolution is significantly increased. For example, in numerical weather forecasting the current generation of global circulation models with resolution of 35?km has a total of billions of state variables. Numerous ensemble based Kalman filters (Evensen 2003 Ocean Dyn. 53 343?67; Bishop et al 2001 Mon. Weather Rev. 129 420?36; Anderson 2001 Mon. Weather Rev. 129 2884?903; Szunyogh et al 2005 Tellus A 57 528?45; Hunt et al 2007 Physica D 230 112?26) show promising results in addressing this issue; however, all these methods are very sensitive to model resolution, observation frequency, and the nature of the turbulent signals when a practical limited ensemble size (typically less than 100) is used. In this paper, we implement a radical filtering approach to a relatively low (40) dimensional toy model, the L-96 model (Lorenz 1996 Proc. on Predictability (ECMWF, 4?8 September 1995) pp 1?18) in various chaotic regimes in order to address the 'curse of ensemble size' for complex nonlinear systems. Practically, our approach has several desirable features such as extremely high computational efficiency, filter robustness towards variations of ensemble size (we found that the filter is reasonably stable even with a single realization) which makes it feasible for high dimensional problems, and it is independent of any tunable parameters such as the variance inflation coefficient in an ensemble Kalman filter.This radical filtering strategy decouples the problem of filtering a spatially extended nonlinear deterministic system to filtering a Fourier diagonal system of parametrized linear stochastic differential equations (Majda and Grote 2007 Proc. Natl Acad. Sci. 104 1124?9; Castronovo et al 2008 J. Comput. Phys. 227 3678?714); for the linear stochastically forced partial differential equations with constant coefficients such as in (Castronovo et al 2008 J. Comput. Phys. 227 3678?714), this Fourier diagonal decoupling is a natural approach provided that the system noise is chosen to be independent in the Fourier space; for a nonlinear problem, however, there is a strong mixing and correlations between different Fourier modes. Our strategy is to radically assume for the purposes of filtering that different Fourier modes are uncorrelated. In particular, we introduce physical model errors by replacing the nonlinearity in the original model with a suitable Ornstein?Uhlenbeck process. We show that even with this 'poor-man's' stochastic model, when the appropriate parametrization strategy is guided by mathematical offline test criteria, it is able to produce reasonably skilful filtered solutions. In the highly turbulent regime with infrequent observation time, this approach is at least as good as trusting the observations while the ensemble Kalman filter implemented in a perfect model scenario diverges. Since these Fourier diagonal linear filters have large model error compared with the nonlinear dynamics, an essential part of the study below is the interplay between this error and the mathematical criteria for a given linear filter in order to produce skilful filtered solutions through the radical strategy.

The simplest problem in the collective dynamics of neural networks: is synchrony stable?

Abstract: For spiking neural networks we consider the stability problem of global synchrony, arguably the simplest non-trivial collective dynamics in such networks. We find that even this simplest dynamical problem—local stability of synchrony—is non-trivial to solve and requires novel methods for its solution. In particular, the discrete mode of pulsed communication together with the complicated connectivity of neural interaction networks requires a non-standard approach. The dynamics in the vicinity of the synchronous state is determined by a multitude of linear operators, in contrast to a single stability matrix in conventional linear stability theory. This unusual property qualitatively depends on network topology and may be neglected for globally coupled homogeneous networks. For generic networks, however, the number of operators increases exponentially with the size of the network. We present methods to treat this multi-operator problem exactly. First, based on the Gershgorin and Perron–Frobenius theorems, we derive bounds on the eigenvalues that provide important information about the synchronization process but are not sufficient to establish the asymptotic stability or instability of the synchronous state. We then present a complete analysis of asymptotic stability for topologically strongly connected networks using simple graphtheoretical considerations. For inhibitory interactions between dissipative (leaky) oscillatory neurons the synchronous state is stable, independent of the parameters and the network connectivity. These results indicate that pulse-like interactions play a profound role in network dynamical systems, and in particular in the dynamics of biological synchronization, unless the coupling is homogeneous and all-to-all. The concepts introduced here are expected to also facilitate the exact analysis of

Modelling DNA at the mesoscale : a challenge for nonlinear science?

Abstract: When it is viewed at the scale of a base pair, DNA appears as a nonlinear lattice. Modelling its properties is a fascinating goal. The detailed experiments that can be performed on this system impose constraints on the models and can be used as a guide to improve them. There are nevertheless many open problems, particularly to describe DNA at the scale of a few tens of base pairs, which is relevant for many biological phenomena.

Shear-Induced Chaos

Abstract: Guided by a geometric understanding developed in earlier works of Wang and Young, we carry out numerical studies of shear-induced chaos in several parallel but different situations. The settings considered include periodic kicking of limit cycles, random kicks at Poisson times and continuous-time driving by white noise. The forcing of a quasi-periodic model describing two coupled oscillators is also investigated. In all cases, positive Lyapunov exponents are found in suitable parameter ranges when the forcing is suitably directed.

Asymptotic stability of Toda lattice solitons

Abstract: We establish an asymptotic stability result for Toda lattice soliton solutions, by making use of a linearized Backlund transformation whose domain has codimension one Combining a linear stability result with a general theory of nonlinear stability by Friesecke and Pego for solitary waves in lattice equations, we conclude that all solitons in the Toda lattice are asymptotically stable in an exponentially weighted norm In addition, we determine the complete spectrum of an operator naturally associated with the Floquet theory for these lattice solitons