Showing papers in "Nonlinearity in 2006"

Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays

Abstract: In this paper, the exponential stability problem is investigated for a class of Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays. By using the analysis method, inequality technique and the properties of an M-matrix, several novel sufficient conditions ensuring the existence, uniqueness and global exponential stability of the equilibrium point are derived. Moreover, the exponential convergence rate is estimated. The obtained results are less restrictive than those given in the earlier literature, and the boundedness and differentiability of the activation functions and differentiability of the time-varying delays are removed. Two examples with their simulations are given to show the effectiveness of the obtained results.

Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

Abstract: For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce the existence of classical solutions near equilibrium for the full nonlinear models associated with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof is based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.

Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems

Abstract: This paper is concerned with the existence of travelling wave solutions in a class of delayed reaction-diffusion systems without monotonicity, which concludes two-species diffusion-competition models with delays. Previous methods do not apply in solving these problems because the reaction terms do not satisfy either the so-called quasimonotonicity condition or non-quasimonotonicity condition. By using Schauder's fixed point theorem, a new cross-iteration scheme is given to establish the existence of travelling wave solutions. More precisely, by using such a new cross-iteration, we reduce the existence of travelling wave solutions to the existence of an admissible pair of upper and lower solutions which are easy to construct in practice. To illustrate our main results, we study the existence of travelling wave solutions in two delayed two-species diffusion-competition systems.

Escape rates and conditionally invariant measures

Abstract: We consider dynamical systems on domains that are not invariant under the dynamics—for example, a system with a hole in the phase space—and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, we are led quickly to conditionally invariant measures that are absolutely continuous with respect to Lebesgue. Comparisons with SRB measures are inevitable, yet there are important differences. Via informal discussions and examples, this paper seeks to clarify the ideas involved. It includes also a brief review of known results and possible directions of further work in this developing subject.

One component regularity for the Navier–Stokes equations

Abstract: We establish sufficient conditions for the regularity of solutions of the Navier–Stokes system based on conditions on one component of the velocity. The first result states that if , where and 54/23 ≤ r ≤ 18/5, then the solution is regular. The second result is that if , where and 24/5 ≤ r ≤ ∞, then the solution is regular. These statements improve earlier results on one component regularity.

Smooth attractors for strongly damped wave equations

Abstract: This paper is concerned with the semilinear strongly damped wave equation The existence of compact global attractors of optimal regularity is proved for nonlinearities of critical and supercritical growth.

A Modified-Leray-α subgrid scale model of turbulence

Abstract: Inspired by the remarkable performance of the Leray-α (and the Navier–Stokes alpha (NS-α), also known as the viscous Camassa–Holm) subgrid scale model of turbulence as a closure model to Reynolds averaged equations (RANS) for flows in turbulent channels and pipes, we introduce in this paper another subgrid scale model of turbulence, the modified Leray-α (ML-α) subgrid scale model of turbulence. The application of the ML-α to infinite channels and pipes gives, due to symmetry, similar reduced equations as Leray-α and NS-α. As a result the reduced ML-α model in infinite channels and pipes is equally impressive as a closure model to RANS equations as NS-α and all the other alpha subgrid scale models of turbulence (Leray-α and Clark-α). Motivated by this, we present an analytical study of the ML-α model in this paper. Specifically, we will show the global well-posedness of the ML-α equation and establish an upper bound for the dimension of its global attractor. Similarly to the analytical study of the NS-α and Leray-α subgrid scale models of turbulence we show that the ML-α model will follow the usual k−5/3 Kolmogorov power law for the energy spectrum for wavenumbers in the inertial range that are smaller than 1/α and then have a steeper power law for wavenumbers greater than 1/α (where α > 0 is the length scale associated with the width of the filter). This result essentially shows that there is some sort of parametrization of the large wavenumbers (larger than 1/α) in terms of the smaller wavenumbers. Therefore, the ML-α model can provide us another computationally sound analytical subgrid large eddy simulation model of turbulence.

Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions

Abstract: We consider the Navier–Stokes equations with Navier friction boundary conditions and prove two results. First, in the case of a bounded domain we prove that weak Leray solutions converge (locally in time in dimension ≥3 and globally in time in dimension 2) as the viscosity goes to 0 to a strong solution of the Euler equations, provided that the initial data converge in L2 to a sufficiently smooth limit. Second, we consider the case of a half-space and anisotropic viscosities: we fix the horizontal viscosity, send the vertical viscosity to 0 and prove convergence to the expected limit system under a weaker hypothesis on the initial data.

Chaos in low-dimensional Lotka Volterra models of competition

Abstract: The occurrence of chaos in basic Lotka–Volterra models of four competing species is studied. A brute-force numerical search conditioned on the largest Lyapunov exponent (LE) indicates that chaos occurs in a narrow region of parameter space but is robust to perturbations. The dynamics of the attractor for a maximally chaotic case are studied using symbolic dynamics, and the question of self-organized critical behaviour (scale-invariance) of the solution is considered.

Gap solitons in periodic discrete nonlinear Schrödinger equations

Abstract: It is shown that the periodic discrete nonlinear Schrodinger equation, with cubic nonlinearity, possesses gap solutions, i.e. standing waves, with the frequency in a spectral gap, that are exponentially localized in the spatial variable. The proof is based on the linking theorem in combination with periodic approximations.

The fractional Keller–Segel model

Abstract: The Keller–Segel model is a system of partial differential equations modelling chemotactic aggregation in cellular systems. This model has blowing-up solutions for large enough initial conditions in dimensions d ≥ 2, but all the solutions are regular in one dimension, a mathematical fact that crucially affects the patterns that can form in the biological system. One of the strongest assumptions of the Keller–Segel model is the diffusive character of the cellular motion, known to be false in many situations. We extend this model to such situations in which the cellular dispersal is better modelled by a fractional operator. We analyse this fractional Keller–Segel model and find that all solutions are again globally bounded in time in one dimension. This fact shows the robustness of the main biological conclusions obtained from the Keller–Segel model.

Generalized synchronization for delayed chaotic neural networks: a novel coupling scheme

Abstract: In this paper, a novel coupling scheme with different coupling delays is presented to achieve generalized synchronization (complete synchronization (CS), anticipating synchronization (AS) and lag synchronization (LS)). The Lyapunov–Krasovskii functional method is employed to investigate the global asymptotic stability of the error dynamical system. The theoretical analysis indicates that delayed chaotic neural networks coupled in this way can switch arbitrarily among CS, AS and LS under the newly proposed coupling configuration. Switching among different kinds of synchronization can be done by changing the transformation time of the coupling signal. Numerical simulations agree with the theoretical analysis.

Dynamic mechanisms of blood vessel growth.

Abstract: The formation of a polygonal configuration of proto-blood-vessels from initially dispersed cells is the first step in the development of the circulatory system in vertebrates. This initial vascular network later expands to form new blood vessels, primarily via a sprouting mechanism. We review a range of recent results obtained with a Monte Carlo model of chemotactically migrating cells which can explain both de novo blood vessel growth and aspects of blood vessel sprouting. We propose that the initial network forms via a percolation-like instability depending on cell shape, or through an alternative contact-inhibition of motility mechanism which also reproduces aspects of sprouting blood vessel growth.

A nonperturbative Eliasson's reducibility theorem

Abstract: This paper is concerned with discrete, one-dimensional Schrodinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasi-periodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated first-order system, a quasi-periodic skew-product, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schrodinger operators. Finally we prove that Cantor spectrum implies the existence of a Gδ-set of energies whose Schrodinger cocycle is not reducible to constant coefficients.

On multi-parametric bifurcations in a scalar piecewise-linear map

Abstract: In this work a one-dimensional piecewise-linear map is considered. The areas in the parameter space corresponding to specific periodic orbits are determined. Based on these results it is shown that the structure of the 2D and 3D parameter spaces can be simply described using the concept of multi-parametric bifurcations. It is demonstrated that an infinite number of two-parametric bifurcation lines starts at the origin of the 3D parameter space. Along each of these lines an infinite number of bifurcation planes starts, whereas the origin represents a three-parametric bifurcation.

Jump-defects in the nonlinear schrödinger model and other non-relativistic field theories

Abstract: Recent work on purely transmitting 'jump-defects' in the sine-Gordon model and other relativistic field theories is extended to non-relativistic models. In all the cases investigated the defect conditions are provided by 'frozen' B?cklund transformations and it is also shown via a Lax pair argument how integrability will be preserved in the presence of this type of defect. Explicit examples of the scattering of solitons by defects are given, and bound states associated with 'jump-defects' in the nonlinear Schr?dinger (NLS) model are described. Although the NLS model provides the principal example, some results are also presented for the Korteweg?de Vries and modified Korteweg?de Vries equations.

Multi-parametric bifurcations in a piecewise–linear discontinuous map

Abstract: In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. Additionally, the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained.

Chaotic attractors of relaxation oscillators

Abstract: We develop a general technique for proving the existence of chaotic attractors for three-dimensional vector fields with two time scales. Our results connect two important areas of dynamical systems: the theory of chaotic attractors for discrete two-dimensional Henon-like maps and geometric singular perturbation theory. Two-dimensional Henon-like maps are diffeomorphisms that limit on non-invertible one-dimensional maps. Wang and Young formulated hypotheses that suffice to prove the existence of chaotic attractors in these families. Three-dimensional singularly perturbed vector fields have return maps that are also two-dimensional diffeomorphisms limiting on one-dimensional maps. We describe a generic mechanism that produces folds in these return maps and demonstrate that the Wang–Young hypotheses are satisfied. Our analysis requires a careful study of the convergence of the return maps to their singular limits in the Ck topology for k ≥ 3. The theoretical results are illustrated with a numerical study of a variant of the forced van der Pol oscillator.

Quaternions and particle dynamics in the Euler fluid equations

Abstract: Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet-Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.

Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight

Abstract: We study polynomials that are orthogonal with respect to a varying quartic weight exp(− N(x2/2 + tx4/4)) for t < 0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its and Kitaev showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painleve I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann–Hilbert problem associated with the polynomials. Moreover, we extend the analysis to non-symmetric combinations of contours. Special features in the steepest descent analysis are a modified equililbrium problem and the use of Ψ-functions for the Painleve I equation in the construction of the local parametrix.

An algorithmic construction of entropies in higher-order nonlinear PDEs

Abstract: A new approach to the construction of entropies and entropy productions for a large class of nonlinear evolutionary PDEs of even order in one space dimension is presented. The task of proving entropy dissipation is reformulated as a decision problem for polynomial systems. The method is successfully applied to the porous medium equation, the thin film equation and the quantum drift–diffusion model. In all cases, an infinite number of entropy functionals together with the associated entropy productions is derived. Our technique can be extended to higher-order entropies, containing derivatives of the solution, and to several space dimensions. Furthermore, logarithmic Sobolev inequalities can be obtained.

Global bifurcations of the Lorenz manifold

Abstract: In this paper we consider the interaction of the Lorenz manifold—the two-dimensional stable manifold of the origin of the Lorenz equations—with the two-dimensional unstable manifolds of the secondary equilibria or bifurcating periodic orbits of saddle type. We compute these manifolds for varying values of the parameter in the Lorenz equations, which corresponds to the transition from simple to chaotic dynamics with the classic Lorenz butterfly attractor at = 28.Furthermore, we find and continue in the first 512 generic heteroclinic orbits that are given as the intersection curves of these two-dimensional manifolds. The branch of each heteroclinic orbit emerges from the well-known first codimension-one homoclinic explosion point at ≈ 13.9265, has a fold and then ends at another homoclinic explosion point with a specific -value. We describe the combinatorial structure of which heteroclinic orbit ends at which homoclinic explosion point. This is verified with our data for the 512 branches from which we automatically extract (by means of a small computer program) the relevant symbolic information.Our results on the manifold structure are complementary to previous work on the symbolic dynamics of periodic orbits in the Lorenz attractor. We point out the connections and discuss directions for future research.

Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids

Abstract: The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian mechanics on Lie groupoids. From a variational principle we derive the discrete Euler–Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we get include the classical discrete Euler–Lagrange equations, the discrete Euler–Poincare and discrete Lagrange–Poincare equations as particular cases. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems.

KAM tori for higher dimensional beam equations with constant potentials

Abstract: In this paper, we consider the higher dimensional nonlinear beam equations with periodic boundary conditions, where the nonlinearity f(u) is a real–analytic function near u = 0 with f(0) = f'(0) = 0 and σ is a real parameter in an interval . It is proved that for 'most' positive parameters σ lying in the finite interval , the above equations admit a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to a Cantor family of finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theorem, modified from (Geng and You 2006 Commun. Math. Phys. 262 343–72) and (Xu J et al 1996 Sci. China Ser. A 39 372–83, 383–94) with weaker non-degeneracy conditions.

Uncertainty estimates and L2 bounds for the Kuramoto-Sivashinsky equation

Abstract: We consider the Kuramoto–Sivashinsky (KS) equation in one dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko et al (1985 Physica D 16 155–83) and later improved by Collet et al (1993 Commun. Math. Phys. 152 203–14) and Goodman (1994 Commun. Pure Appl. Math. 47 293–306) to prove that This result is slightly weaker than a related result by Giacomelli and Otto (2005 Commun. Pure Appl. Math. 58 297–318), but applies in the presence of an additional linear destabilizing term. We further show that for a large class of functions x the exponent is the best possible from this line of argument. Finally, we mention several related results from the literature on equations related to the KS equation that can be improved using these ideas.

Partial preservation of frequencies in KAM theory

Abstract: We consider perturbations of moderately degenerate integrable or partially integrable Hamiltonian systems, so that unperturbed invariant n-tori with prescribed frequencies or frequency ratios do not persist, but there is preservation of, say, the first d < n frequencies or their ratios. Lagrangian and lower dimensional tori are treated in a unified way. The proofs are very simple and follow Herman's idea of 1990: we introduce external parameters to remove degeneracies and then eliminate these parameters making use of a suitable number-theoretical lemma concerning Diophantine approximations of dependent quantities. Parallel results for reversible, volume preserving and dissipative systems are also presented.

Stochastic models for selected slow variables in large deterministic systems

Abstract: A new stochastic mode-elimination procedure is introduced for a class of deterministic systems. Under assumptions of ergodicity and mixing, the procedure gives closed-form stochastic models for the slow variables in the limit of infinite separation of timescales. The procedure is applied to the truncated Burgers–Hopf (TBH) system as a test case where the separation of timescale is only approximate. It is shown that the stochastic models reproduce exactly the statistical behaviour of the slow modes in TBH when the fast modes are artificially accelerated to enforce the separation of timescales. It is shown that this operation of acceleration only has a moderate impact on the bulk statistical properties of the slow modes in TBH. As a result, the stochastic models are sound for the original TBH system.

Unfolding degenerate grazing dynamics in impact actuators

Abstract: In this paper, a selected analysis of the dynamics in an example impact microactuator is performed through a combination of numerical simulations and local analysis. Here, emphasis is placed on investigating the system response in the vicinity of the so-called grazing trajectories, i.e. motions that include zero-relative-velocity contact of the actuator parts, using the concept of discontinuity mappings that account for the effects of low-relative-velocity impacts and brief episodes of stick–slip motion. The analysis highlights the existence of isolated co-dimension-two grazing bifurcation points and the way in which these organize the behaviour of the impacting dynamics. In particular, it is shown how higher-order truncations of local maps of the near-grazing dynamics predict and enable the computation of global bifurcation curves emanating from such degenerate bifurcation points, thereby unfolding the near-grazing dynamics. Although the numerical results presented here are specific for the chosen model of an electrically driven and previously experimentally realized impact microactuator, the methodology generalizes naturally to arbitrary systems with impacts. Moreover, the qualitative nature of the near-grazing dynamics is expected to generalize to systems with similar nonlinearities.

Homogeneous three-cell networks

Abstract: A cell is a system of differential equations. Coupled cell systems are networks of cells. The architecture of a coupled cell network is a graph indicating which cells are identical and which cells are coupled to which. In this paper we continue the work of Stewart, Golubitsky, Pivato and Torok by classifying all homogeneous three-cell networks (where each cell has at most two inputs) and classifying all generic codimension one steady-state and Hopf bifurcations from a synchronous equilibrium. We use combinatorial arguments to show that there are 34 distinct homogeneous three-cell networks as opposed to only three such two-cell networks.We show that codimension one bifurcations in homogeneous three-cell networks can exhibit interesting features that are due to network architecture. Indeed, network architecture determines, even at linear level, the kind of generic transitions from a synchronous equilibrium that can occur as we vary one parameter and plays a crucial role in establishing how the solutions on the bifurcating branches manifest themselves in each cell.

Scattering and small data completeness for the critical nonlinear Schrödinger equation

Abstract: We prove asymptotic completeness of one-dimensional NLS with long range nonlinearities. We also prove existence and expansion of asymptotic solutions with large data at infinity.