Showing papers in "Nonlinearity in 2004"

Extracting macroscopic dynamics: model problems and algorithms

Abstract: In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of coarse-grained, or macroscopic, quantities. The purpose of this review is twofold: first, to describe a number of simple model systems where the coarse-grained or macroscopic behaviour of a system can be explicitly determined from the full, or microscopic, description; and second, to overview some of the emerging algorithmic approaches that have been introduced to extract effective, lower-dimensional, macroscopic dynamics. The model problems we describe may be either stochastic or deterministic in both their microscopic and macroscopic behaviour, leading to four possibilities in the transition from microscopic to macroscopic descriptions. Model problems are given which illustrate all four situations, and mathematical tools for their study are introduced. These model problems are useful in the evaluation of algorithms. We use specific instances of the model problems to illustrate these algorithms. As the subject of algorithm development and analysis is, in many cases, in its infancy, the primary purpose here is to attempt to unify some of the emerging ideas so that individuals new to the field have a structured access to the literature. Furthermore, by discussing the algorithms in the context of the model problems, a platform for understanding existing algorithms and developing new ones is built.

Nonlinear dynamics and statistical physics of DNA

Abstract: DNA is not only an essential object of study for biologists—it also raises very interesting questions for physicists. This paper discuss its nonlinear dynamics, its statistical mechanics, and one of the experiments that one can now perform at the level of a single molecule and which leads to a non-equilibrium transition at the molecular scale.After a review of experimental facts about DNA, we introduce simple models of the molecule and show how they lead to nonlinear localization phenomena that could describe some of the experimental observations. In a second step we analyse the thermal denaturation of DNA, i.e. the separation of the two strands using standard statistical physics tools as well as an analysis based on the properties of a single nonlinear excitation of the model. The last part discusses the mechanical opening of the DNA double helix, performed in single molecule experiments. We show how transition state theory combined with the knowledge of the equilibrium statistical physics of the system can be used to analyse the results.

Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory

Abstract: In part I of this work (Bona J L, Chen M and Saut J-C 2002 Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory J. Nonlinear Sci. 12 283–318), a four-parameter family of Boussinesq systems was derived to describe the propagation of surface water waves. Similar systems are expected to arise in other physical settings where the dominant aspects of propagation are a balance between the nonlinear effects of convection and the linear effects of frequency dispersion. In addition to deriving these systems, we determined in part I exactly which of them are linearly well posed in various natural function classes. It was argued that linear well-posedness is a natural necessary requirement for the possible physical relevance of the model in question. In this paper, it is shown that the first-order correct models that are linearly well posed are in fact locally nonlinearly well posed. Moreover, in certain specific cases, global well-posedness is established for physically relevant initial data. In part I, higher-order correct models were also derived. A preliminary analysis of a promising subclass of these models shows them to be well posed.

Connecting orbits and invariant manifolds in the spatial restricted three-body problem

Abstract: The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a ‘Petit Grand Tour’ of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. This work extends the results to the three-dimensional case. Besides providing a full description of different kinds of libration motions in a large vicinity of these points, this paper numerically demonstrates the existence of heteroclinic connections between pairs of libration orbits, one around the libration point L_1 and the other around L_2. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from one libration point orbit to the other. A knowledge of these orbits can be very useful in the design of missions such as the Genesis Discovery Mission, and may provide the backbone for other interesting orbits in the future.

A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators

Abstract: A non-linear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. This model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the non-linear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of n degrees of freedom. Finally, the relation of this non-linear system to the harmonic oscillator on spaces of constant curvature, the two-dimensional sphere S2 and hyperbolic plane H2, is discussed.

Evolution of slow variables in a priori unstable Hamiltonian systems

Abstract: We study diffusion phenomena in a priori unstable (initially hyperbolic) Hamiltonian systems These systems are perturbations of integrable ones, which have a family of hyperbolic tori We prove that in the case of two and a half degrees of freedom the action variable generically drifts (ie changes on a trajectory by a quantity of order one) Moreover, there exists a trajectory such that the velocity of this drift is e/loge, where e is the parameter of the perturbation

Ruelle's linear response formula, ensemble adjoint schemes and Lévy flights

Abstract: A traditional subject in statistical physics is the linear response of a molecular dynamical system to changes in an external forcing agency, e.g. the Ohmic response of an electrical conductor to an applied electric field. For molecular systems the linear response matrices, such as the electrical conductivity, can be represented by Green–Kubo formulae as improper time-integrals of 2-time correlation functions in the system. Recently, Ruelle has extended the Green–Kubo formalism to describe the statistical, steady-state response of a 'sufficiently chaotic' nonlinear dynamical system to changes in its parameters. This formalism potentially has a number of important applications. For instance, in studies of global warming one wants to calculate the response of climate-mean temperature to a change in the atmospheric concentration of greenhouse gases. In general, a climate sensitivity is defined as the linear response of a long-time average to changes in external forces. We show that Ruelle's linear response formula can be computed by an ensemble adjoint technique and that this algorithm is equivalent to a more standard ensemble adjoint method proposed by Lea, Allen and Haine to calculate climate sensitivities.In a numerical implementation for the 3-variable, chaotic Lorenz model it is shown that the two methods perform very similarly. However, because of a power-law tail in the histogram of adjoint gradients their sum over ensemble members becomes a Levy flight, and the central limit theorem breaks down. The law of large numbers still holds and the ensemble-average converges to the desired sensitivity, but only very slowly, as the number of samples is increased. We discuss the implications of this example more generally for ensemble adjoint techniques and for the important practical issue of calculating climate sensitivities.

Solitary waves on Fermi-Pasta-Ulam lattices: IV. Proof of stability at low energy

Abstract: We establish the long-time stability of low-energy solitary waves in one-dimensional nonintegrable lattices with Hamiltonian with a general nearest-neighbour potential V. As a corollary we obtain a recurrence theorem related to numerical observations by Fermi, Pasta and Ulam.

On blowup for semilinear wave equations with a focusing nonlinearity

Abstract: In this paper we report on numerical studies of the formation of singularities for the semilinear wave equations with a focusing power nonlinearity utt − Δu = up in three space dimensions. We show that for generic large initial data that lead to singularities, the spatial pattern of blowup can be described in terms of linearized perturbations about the fundamental self-similar (homogeneous in space) solution. We consider also non-generic initial data which are fine-tuned to the threshold for blowup and identify critical solutions that separate blowup from dispersal for some values of the exponent p.

Solitary waves on Fermi Pasta Ulam lattices: III. Howland-type Floquet theory

Abstract: Parts II, III and IV of this series are devoted to proving long time stability of solitary waves in one-dimensional nonintegrable lattices with Hamiltonian with a general nearest-neighbour potential V. Here in part III we analyse the evolution equation obtained by linearizing the dynamics at a solitary wave. This equation is nonautonomous, because discrete solitary waves are not time-independent modulo a spatial shift (like their continuous counterparts), but time-periodic modulo a spatial shift.We develop a Floquet theory modulo shifts on the lattice that naturally characterizes the time-t evolution on the lattice in terms of a strongly continuous group of operators on the real line, in a manner reminiscent of Howland's treatment of quantum scattering with time-periodic potentials. This allows us to reduce the main hypothesis of our nonlinear stability theorem in part II (namely, exponential decay in the linearized dynamics on the symplectic complement to the solitary-wave manifold) to an eigenvalue condition on the generator of the group, which is a differential-difference operator on the real line. Physically, the eigenvalue condition means that no spatially localized modes of constant shape exist which travel at the solitary wave speed and have exponentially growing or neutral amplitude.

Infection in prey population may act as a biological control in ratio-dependent predator?prey models

Abstract: A ratio-dependent predator-prey model with infection in prey population is proposed and analysed. The behaviour of the system near the biological feasible equilibria is observed. The conditions for which no trajectory can reach the origin following any fixed direction or spirally are worked out. We investigate the criteria for which the system will persist. It is observed that the introduction of an infected population in the classical ratio-dependent predator-prey model may act as a biological control to save the population from extinction.

Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity

Abstract: We investigate a delay differential equation that models a pendulum stabilized in the upright position by a delayed linear horizontal control force Linear stability analysis reveals that the region of stability of the origin (the upright position of the pendulum) is bounded for positive delay We find that a codimension-three triple-zero eigenvalue bifurcation acts as an organizing centre of the dynamics It is studied by computing and then analysing a reduced three-dimensional vector field on the centre manifold The validity of this analysis is checked in the full delay model with the continuation software DDE-BIFTOOL Among other things, we find stable small-amplitude solutions outside the region of linear stability of the pendulum, which can be interpreted as a relaxed form of successful control

Strong statistical stability of non-uniformly expanding maps

Abstract: We consider families of transformations in multidimensional Riemannian manifolds with non-uniformly expanding behaviour. We give sufficient conditions for the continuous variation (in the L1-norm) of the densities of absolutely continuous (with respect to the Lebesgue measure) invariant probability measures for those transformations.

Dynamics of a strongly nonlocal reaction–diffusion population model

Abstract: We study the development of travelling waves in a population that competes with itself for resources in a spatially nonlocal manner. We model this situation as an initial value problem for the integro-differential reaction?diffusion equation with g an even function that satisfies g(y) ? 0 as y ? ? ?, , ? > 0, 0 0. We concentrate on the limit of highly nonlocal interactions, ? 1, focusing on the particular case g(y) = ? e?|y|, which is equivalent to the reaction?diffusion system Using numerical and asymptotic methods, we show that in different, well-defined regions of parameter space, steady travelling waves, unsteady travelling waves and periodic travelling waves develop from localized initial conditions. A key feature of the system for ? 1 is the local existence of travelling wave solutions that propagate with speed c < 2, and which, although they cannot exist globally, attract the solution of the initial value problem for an asymptotically long time. By using a Cole?Hopf transformation, we derive a first order hyperbolic equation for the gradient of log u ahead of the wavefront, where u is exponentially small. An analysis of this equation in terms of its characteristics, allowing for the formation of shocks where necessary, explains the dynamics of each of the different types of travelling wave. Moreover, we are able to show that the techniques that we develop for this particular case can be used for more general kernels g(y) and that we expect the same range of different types of travelling wave to be solutions of the initial value problem for appropriate parameter values. As another example, we briefly consider the case , for which the system cannot be simplified to a pair of partial differential equations.

The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities

Abstract: We consider the nonlinear model of an infinite oscillator chain embedded in a background field. We start from an appropriate modulation ansatz of the space–time periodic solutions to the linearized (microscopic) model and derive formally the associated (macroscopic) modulation equation, which turns out to be the nonlinear Schrodinger equation. Then we justify this necessary condition rigorously for the case of nonlinearities with cubic leading terms; i.e. we show that solutions that have the form of the assumed ansatz for t = 0 preserve this form over time-intervals with a positive macroscopic length. Finally, we transfer this result to the analogous case of a finite but large periodic chain and illustrate it by a numerical example.

Lower bounds for the energy in a crumpled elastic sheet: a minimal ridge

Abstract: We study the linearized Foppl–von Karman theory of a long, thin rectangular elastic membrane that is bent through an angle 2 α. We prove rigorous bounds for the minimum energy of this configuration in terms of the plate thickness, σ, and the bending angle. We show that the minimum energy scales as σ5/3 α7/3. This scaling is in sharp contrast with previously obtained results for the linearized theory of thin sheets with isotropic compression boundary conditions, where the energy scales as σ.

Action minimizing orbits in the n-body problem with simple choreography constraint

Abstract: In 1999 Chenciner and Montgomery found a remarkably simple choreographic motion for the planar three-body problem (see [11]). In this solution, three equal masses travel on an figure-of-eight shaped planar curve; this orbit is obtained by minimizing the action integral on the set of simple planar choreographies with some special symmetry constraints. In this work our aim is to study the problem of n masses moving in under an attractive force generated by a potential of the kind 1/rα, α > 0, with the only constraint to be a simple choreography: if q1(t),...,qn(t) are the n orbits then we impose the existence of such that where τ = 2π/n. In this setting, we first prove that for every and α > 0, the Lagrangian action attains its absolute minimum on the planar regular n-gon relative equilibrium. Next, we deal with the problem in a rotating frame and show a richer phenomenology: indeed, while for some values of the angular velocity, the minimizers are still relative equilibria, for others, the minima of the action are no longer rigid motions.

Effects of noise on elliptic bursters

Abstract: Elliptic bursting arises from fast–slow systems and involves recurrent alternation between active phases of large amplitude oscillations and silent phases of small amplitude oscillations. This paper is a geometric analysis of elliptic bursting with and without noise. We first prove the existence of elliptic bursting solutions for a class of fast–slow systems without noise by establishing an invariant region for the return map of the solutions. For noisy elliptic bursters, the bursting patterns depend on random variations associated with delayed bifurcations. We provide an exact formulation of the duration of delay and analyse its distribution. The duration of the delay, and consequently the durations of active and silent phases, is shown to be closely related to the logarithm of the amplitude of the noise. The treatment of noisy delayed bifurcation here is a general theory of delayed bifurcation valid for other systems involving delayed bifurcation as well and is a continuation of the rigorous Shishkova–Neishtadt theory on delayed bifurcation or delay of stability loss.

Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: a brief survey of recent results*

Abstract: Twelve open problems and one conjecture are posed in this exposition. Among them, eight open problems and the conjecture are on calculation of the exact value of the Hausdorff measure of self-similar sets, and four open problems are on topological entropy. We give a brief survey of the recent research results related to these open problems and conjecture.

Recursive tiling and geometry of piecewise rotations by π/7

Abstract: We study two piecewise affine maps on convex polygons, locally conjugate to a rotation by a multiple of ?/7. We obtain a finite-order recursive tiling of the phase space by return map sub-domains of triangles and periodic heptagonal domains (cells), with scaling factors given by algebraic units. This tiling allows one to construct efficiently periodic orbits of arbitrary period, and to obtain a convergent sequence of coverings of the closure of the discontinuity set ?. For every map for which such finite-order recursive tiling exists, we derive sufficient conditions for the equality of Hausdorff and box-counting dimensions, and for the existence of a finite, non-zero Hausdorff measure of . We then verify that these conditions apply to our models; we obtain an irreducible transcendental equation for the Hausdorff dimension involving fundamental units, and establish the existence of infinitely many disjoint invariant components of the residual set . We calculate numerically the asymptotic power law growth of the number of cells as a function of maximum return time, as well as the number of cells of diameter larger than a specified . In the latter case, the exponent is shown to coincide with the Hausdorff dimension.

An analysis of finite-dimensional approximations for the ground state solution of Bose?Einstein condensates

Abstract: In this paper, some finite-dimensional approximations for the ground state solution of Bose–Einstein condensates are investigated. The convergence of the finite-dimensional approximations obtained by a Galerkin discretization of the Gross–Pitaevskii equation is proved. Several upper bounds of the approximation errors are also established.

Golden gaskets: variations on the Sierpiński sieve

Abstract: We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, with a common contraction factor λ in (0, 1). As is well known, for λ = 1/2 the attractor, S_λ, is a fractal called the Sierpinski sieve and for λ < 1/2 it is also a fractal. Our goal is to study S_λ for this IFS for 1/2 < λ < 2/3 , i.e. when there are ‘overlaps’ in S_λ as well as ‘holes’. In this introductory paper we show that despite the overlaps (i.e. the breaking down of the open set condition (OSC)), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic λ (so-called multinacci numbers). We evaluate the ausdorff dimension of S_λ for these special values by showing that S_λ is essentially the attractor for an infinite IFS that does satisfy the OSC. We also show that the set of points in the attractor with a unique ‘address’ is self-similar and compute its dimension. For non-multinacci values of λ we show that if λ is close to 2/3 , then S_λ has a non-empty interior. Finally we discuss higher-dimensional analogues of the model in question.

The variational principle for products of non-negative matrices

Abstract: Let (ΣA, σ) be a subshift of finite type and let M(x) be a continuous function on ΣA taking values in the set of non-negative matrices. We set up the variational principle between the pressure function, entropy and Lyapunov exponent for M on ΣA. We also present some properties of equilibrium states.

On very singular similarity solutions of a higher-order semilinear parabolic equation

Abstract: We study the large-time behaviour of solutions of a semilinear 2mth-order parabolic equation 1,\end{equation*} \] SRC=http://ej.iop.org/images/0951-7715/17/3/017/non172874ude001.gif/> with bounded integrable initial data u0 decaying exponentially at infinity. For the semilinear heat equation (m = 1), the asymptotic behaviour was established in detail in the 1980s. Our main goal is to justify that, for any m ≥ 1 in the subcritical range 1 < p < p0 = 1 + (2m/N), there exists a finite number, M ~ N(p0 − p)/2(p − 1) → ∞ as p → 1+, of different very singular self-similar solutions of the form where each V is a radial, exponentially decaying solution of the elliptic equation By a perturbation technique, we establish the existence of radially symmetric very singular solution profiles Vl for p close to critical bifurcation exponents pl = 1 + (2m/(l + N)), l = 0, 2, ..., where the first one, V0, is shown to be stable. Discrete and countable subsets of other self-similar and approximately self-similar patterns are introduced.

Probabilistic cellular automata with conserved quantities

Abstract: We demonstrate that the concept of a conservation law can be naturally extended from deterministic to probabilistic cellular automata (PCA) rules. The local function for conservative PCA must satisfy conditions analogous to conservation conditions for deterministic cellular automata (CA). Conservation conditions for PCA can also be written in the form of a current conservation law. For deterministic nearest-neighbour CA the current can be computed exactly. Local structure approximation can partially predict the equilibrium current for non-deterministic cases. For linear segments of the fundamental diagram it actually produces exact results.

Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits

Abstract: For reversible two-dimensional diffeomorphisms we establish a new type of Newhouse regions (regions of structural instability density). We prove that in these regions there exists a dense set of diffeomorphisms having, simultaneously, infinitely many stable, infinitely many unstable, and infinitely many elliptic type periodic orbits.

Rotations by π/7

Abstract: Piecewise rotations are natural generalizations of interval exchange maps. They appear naturally in the theory of digital filters, Hamiltonian systems and polygonal dual billiards. We construct a rational piecewise rotation system with three atoms for which the return time to one of the atoms is unbounded. We show that the return map gives rise to a self-similar structure of induced atoms. The constructions are based on the angle of rotation π/7. Moreover, we construct a continuous class of examples with an infinite number of periodic cells. These periodic cells alternate between two atoms and they form a self-similar structure. Our investigation here may be viewed as generalizations of results obtained by Boshernitzan and Caroll, as well as Adler, Kitchens and Tresser, Kahng, Lowenstein and others. The main tools in the investigation are algebraic computations in a cyclotomic field determined by fourteenth roots of unity.

Feedback control of travelling wave solutions of the complex Ginzburg–Landau equation

Abstract: Through a linear stability analysis, we investigate the effectiveness of a noninvasive feedback control scheme in stabilizing travelling wave solutions, R eiKx+iωt, of the one-dimensional complex Ginzburg–Landau equation (CGLE) in the Benjamin–Feir unstable regime. The feedback control, a generalization of the time-delay method of Pyragas (1992 Phys. Lett. A 170 421), was proposed by Lu et al (1996 Phys. Rev. Lett. 76 3316) in the setting of nonlinear optics. It involves both spatial shifts, by the wavelength of the targeted travelling wave, and a time delay that coincides with the temporal period of the travelling wave. We derive a single necessary and sufficient stability criterion that determines whether a travelling wave is stable to all perturbation wavenumbers. This criterion has the benefit that it determines an optimal value for the time-delay feedback parameter. For various coefficients in the CGLE we use this algebraic stability criterion to numerically determine stable regions in the (K, ρ)-parameter plane, where ρ is the feedback parameter associated with the spatial translation. We find that the combination of the two feedbacks greatly enlarges the parameter regime where stabilization is possible, and that the stable regions take the form of stability tongues in the (K, ρ)-plane. We discuss possible resonance mechanisms that could account for the spacing with K of the stability tongues.

Chaotic dynamics of a nonlinear density dependent population model

Abstract: We study the dynamics of an overcompensatory Leslie population model where the fertility rates decay exponentially with population size. We find a plethora of complicated dynamical behaviour, some of which has not been previously observed in population models and which may give rise to new paradigms in population biology and demography.We study the two- and three-dimensional models and find a large variety of complicated behaviour: all codimension 1 local bifurcations, period doubling cascades, attracting closed curves that bifurcate into strange attractors, multiple coexisting strange attractors with large basins (which cause an intrinsic lack of 'ergodicity'), crises that can cause a discontinuous large population swing, merging of attractors, phase locking and transient chaos. We find (and explain) two different bifurcation cascades transforming an attracting invariant closed curve into a strange attractor. We also find one-parameter families that exhibit most of these phenomena. We show that some of the more exotic phenomena arise from homoclinic tangencies.

Recurrence and algorithmic information

Abstract: In this paper, we initiate a somewhat detailed investigation of the relationships between quantitative recurrence indicators and algorithmic complexity of orbits in weakly chaotic dynamical systems. We mainly focus on examples.