Showing papers in "Nonlinearity in 1998"
TL;DR: In this paper, the vanishing viscosity limit is considered for the incompressible 2D Navier-Stokes equations in a bounded domain, and the existence of the regular solutions for the Navier Stokes equations with smooth compatible data is proved.
Abstract: The vanishing viscosity limit is considered for the incompressible 2D Navier-Stokes equations in a bounded domain. Motivated by studies of turbulent flow we suppose Navier's friction condition in the tangential direction, i.e. the creation of a vorticity proportional to the tangential velocity. We prove the existence of the regular solutions for the Navier-Stokes equations with smooth compatible data and of the solutions with bounded vorticity for initial vorticity being only bounded. Finally, we establish a uniform -bound for the vorticity and convergence to the incompressible 2D Euler equations in the inviscid limit.
238 citations
TL;DR: In this article, an exact two-loop soliton solution to the Vakhnenko equation is found, and the key step in finding this solution is to transform the independent variables in the equation, leading to a transformed equation for which it is straightforward to find an exact explicit 2-soliton solution by use of Hirota's method.
Abstract: An exact two loop soliton solution to the Vakhnenko equation is found. The key step in finding this solution is to transform the independent variables in the equation. This leads to a transformed equation for which it is straightforward to find an exact explicit 2-soliton solution by use of Hirota's method. The exact two loop soliton solution to the Vakhnenko equation is then found in implicit form by means of a transformation back to the original independent variables. The nature of the interaction between the two loop solitons depends on the ratio of their amplitudes.
195 citations
TL;DR: A representative model of a return map near homoclinic bifurcation is studied in this paper, which is the so-called fattened Arnold map, a diffeomorphism of the annulus.
Abstract: A representative model of a return map near homoclinic bifurcation is studied This model is the so-called fattened Arnold map, a diffeomorphism of the annulus The dynamics is extremely rich, involving periodicity, quasiperiodicity and chaos The method of study is a mixture of analytic perturbation theory, numerical continuation, iteration to an attractor and experiments, in which the guesses are inspired by the theory In rum the results lead to fine-tuning of the theory This approach is a natural paradigm for the study of complicated dynamical systems By following generic bifurcations, both local and homoclinic, various routes to chaos and strange attractors are detected Here, particularly, the 'large' strange attractors which wind around the annulus are of interest Furthermore, a global phenomenon regarding Arnold tongues is important This concerns the accumulation of tongues on lines of homoclinic bifurcation This phenomenon sheds some new light on the occurrence of infinitely many sinks in certain cases, as predicted by the theory
163 citations
TL;DR: In this article, the authors show how to classify singularities of vector fields up to codimension 4 for the notion of equivalence and show that a semianalytic stratification is not possible even for weak equivalence.
Abstract: In his well known paper `Singularities of vector fields', Takens made a topological classification of vector fields up to codimension 2 and introduced a semialgebraic stratification to distinguish the different cases; from dimensions he had to use the notion of `weak--equivalence'. In this paper we show how to classify singularities of vector fields on up to codimension 4 for the notion of equivalence. To separate the different cases we use a semianalytic stratification and show that a semialgebraic one is not possible, even for the notion of weak--equivalence. Up to codimension 3 the stratification is semialgebraic. We will always suppose that the vector fields are , although it will be clear that the results are valid for , with r sufficiently big. We provide a complete, but short, survey of the different techniques to be used, referring to the existing literature for precise calculations and pictures. We put much emphasis on the new results.
159 citations
TL;DR: In this paper, the energy-equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented.
Abstract: The energy-equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. As examples, three specific and physically relevant problems are considered, namely the flows of a second-grade fluid, the flows of a Newtonian fluid in an infinite channel past an obstacle, and a weakly damped, forced Korteweg-de Vries equation on the whole line.
153 citations
TL;DR: In this paper, the buck converter model is reviewed and the most fascinating features of its dynamical behaviour are reviewed. But the authors focus on a local map which explains how grazing bifurcations cause sharp turning points in the bifurlcation diagram of periodic orbits and how these orbits accumulate onto a sliding trajectory through a ''spiralling' impact adding scenario.
Abstract: This paper provides an analytical insight into the observed nonlinear behaviour of a simple widely used power electronic circuit (the buck converter) and draws parallels with a wider class of piecewise-smooth systems. After introducing the buck converter model and background, the most fascinating features of its dynamical behaviour are reviewed. So-called grazing and sliding solutions are discussed and their role in determining many of the buck converter's dynamical oddities is demonstrated. In particular, a local map is studied which explains how grazing bifurcations cause sharp turning points in the bifurcation diagram of periodic orbits. Moreover, these orbits are shown to accumulate onto a sliding trajectory through a `spiralling' impact adding scenario. The structure of such a diagram is derived analytically and is shown to be closely related to the analysis of homoclinic bifurcations. The results are shown to match perfectly with numerical simulations. The sudden jump to large-scale chaos and the fingered structure of the resulting attractor are also explained.
140 citations
TL;DR: In this article, it was shown that in most conjugacy classes there is a collision-free reduced periodic solution to Newton's N-body equations, which are the classes that are 'tied' in the sense of Gordon.
Abstract: A reduced periodic orbit is one which is periodic modulo a rigid motion. If such an orbit for the planar N-body problem is collision free then it represents a conjugacy class in the projective coloured braid group. Under a `strong force' assumption which excludes the original Newtonian potential we prove that in most conjugacy classes there is a collision-free reduced periodic solution to Newton's N-body equations. These are the classes that are `tied' in the sense of Gordon. We give explicit homological conditions which ensure that a class is tied. The method of proof is the direct method of the calculus of variations. For the three-body problem we obtain qualitative information regarding the shape of our solutions which leads to a partial symbolic dynamics.
108 citations
TL;DR: Using Sinai - Ruelle - Bowen measures to describe nonequilibrium steady states, one can in principle compute the coefficients of expansions around equilibrium as discussed by the authors, and this can be done in practice, and how the results correspond to the zero noise limit when there is a stochastic perturbation.
Abstract: Using Sinai - Ruelle - Bowen measures to describe nonequilibrium steady states, one can in principle compute the coefficients of expansions around equilibrium We discuss how this can be done in practice, and how the results correspond to the zero noise limit when there is a stochastic perturbation The approach used is formal rather than rigorous
96 citations
TL;DR: An explicit upper bound for the number of zeros of the integral of degree n polynomials f, g, on the open interval for which the cubic curve contains an oval was derived in this article.
Abstract: An explicit upper bound is derived for the number of the zeros of the integral of degree n polynomials f, g, on the open interval for which the cubic curve contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals , i,j=0,1, generating the module of complete Abelian integrals I(h) (over the ring of polynomials in h).
92 citations
TL;DR: In this paper, the authors consider piecewise twice differentiable maps T on [0, 1] with indifferent fixed points giving rise to infinite invariant measures and prove that the interval decomposes into a finite number of ergodic cycles with exact powers plus a dissipative part.
Abstract: We consider piecewise twice differentiable maps T on [0,1] with indifferent fixed points giving rise to infinite invariant measures. Without assuming the existence of a Markov partition and only requiring that the first image of the fundamental partition is finite, we prove that the interval decomposes into a finite number of ergodic cycles with exact powers plus a dissipative part. T is shown to be exact on components containing indifferent fixed points. We also determine the order of the singularities of the invariant densities.
91 citations
TL;DR: In this article, the authors study the dynamics of Lyapunov vectors in various models of one-dimensional distributed systems with spacetime chaos and demonstrate that the vector corresponding to the maximum exponent is always localized and the localization region wanders irregularly.
Abstract: We study the dynamics of Lyapunov vectors in various models of one-dimensional distributed systems with spacetime chaos. We demonstrate that the vector corresponding to the maximum exponent is always localized and the localization region wanders irregularly. This localization is explained by interpreting the logarithm of the Lyapunov vector as a roughening interface. We show that for many systems, the `interface' belongs to the Kardar-Parisi-Zhang universality class. Accordingly, we discuss the scaling behaviour of finite-size effects and self-averaging properties of the Lyapunov exponents.
TL;DR: In this article, the global bifurcation diagram of a three-parameter family of cubic Lienard systems is derived, which is a universal character in that its bifuration diagram appears in many models from applications for which a combination of hysteretic and self-oscillatory behavior is essential.
Abstract: We derive the global bifurcation diagram of a three-parameter family of cubic Lienard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications for which a combination of hysteretic and self-oscillatory behaviour is essential. The family emerges as a partial unfolding of a doubly degenerate Bogdanov-Takens point, that is, of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form. We give a new presentation of a local four-parameter bifurcation diagram which is a candidate for the universal unfolding of this singularity.
TL;DR: In this article, it was shown that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in the ''nearby'' Hamiltonian flows.
Abstract: It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in the `nearby' Hamiltonian flows i.e. in a family of two-degrees-of-freedom smooth Hamiltonian flows which converge to the singular billiard flow smoothly where the billiard flow is smooth and continuously where it is continuous. Such Hamiltonians exist; indeed, sufficient conditions are supplied, and thus it is proved that a large class of smooth Hamiltonians converges to billiard flows in this manner. These results imply that ergodicity may be lost in the physical setting, where smooth Hamiltonians which are arbitrarily close to the ergodic billiards, arise.
TL;DR: In this paper, the authors prove criteria for stability and propagation of relative equilibria in symmetric Hamiltonian systems at singular points of the momentum map, and prove AMS classification scheme numbers: 58F05, 70H33.
Abstract: We prove criteria for stability and propagation of relative equilibria in symmetric Hamiltonian systems at singular points of the momentum map. AMS classification scheme numbers: 58F05, 70H33
TL;DR: In this paper, the authors consider motion in a periodic potential in a classical, quantum, and semiclassical context and prove various results on the distribution of asymptotic velocities.
Abstract: We consider motion in a periodic potential in a classical, quantum, and semiclassical context. Various results on the distribution of asymptotic velocities are proven.
TL;DR: Blank and Keller as discussed by the authors showed that the isolated eigenvalues of small perturbations of one-dimensional maps are close to isolated eigvalues of the unperturbed system.
Abstract: For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system (Here `eigenvalue' means eigenvalue of the corresponding Perron-Frobenius operator acting on the space of functions of bounded variation) This result applies eg to the approximation of the system by a finite state Markov chain and generalizes Ulam's conjecture about the approximation of the Sinai-Bowen-Ruelle invariant measure of such a map We provide several simple examples showing that for maps with periodic turning points and for general multidimensional smooth hyperbolic maps isolated eigenvalues are typically unstable under random perturbations Our main tool in the one-dimendional case is a special technique for `interchanging' the map and the perturbation, developed in our previous paper (Blank M L and Keller G 1997 Stochastic stability versus localization in chaotic dynamical systems Nonlinearity 10 81-107), combined with a compactness argument
TL;DR: In this paper, the authors study classical solutions of the vector O(3) sigma model in (2 + 1) dimensions, spontaneously broken to O(1) and show that different types of waves can propagate on the wall, including waves carrying a topological charge.
Abstract: We study classical solutions of the vector O(3) sigma model in (2 + 1) dimensions, spontaneously broken to . The model possesses Skyrmion-type solutions as well as stable domain walls which connect different vacua. We show that different types of waves can propagate on the wall, including waves carrying a topological charge. The domain wall can also absorb Skyrmions and, under appropriate initial conditions, it is possible to emit a Skyrmion from the wall.
TL;DR: In this article, a family of quadratic volume-preserving diffeomorphisms whose inverse is also quadratically convex is studied, which generalize the Henon areapreserving map and the family of symplectic quad ratic maps studied by Moser.
Abstract: We study quadratic, volume-preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Henon area-preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume-preserving maps in three-space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.
TL;DR: In this paper, complete proofs are given for some claims of Middleton and of Floria and Mazo about the asymptotic behaviour of chains of balls and springs in a tilted periodic potential and generalizations, under gradient dynamics.
Abstract: Complete proofs are given for some claims of Middleton and of Floria and Mazo about the asymptotic behaviour of chains of balls and springs in a tilted periodic potential and generalizations, under gradient dynamics. AMS classification number: 58F22
TL;DR: In this paper, the authors derived a result of stability over exponentially long times with respect to the inverse of the distance to an elliptic equilibrium point which has a definite torsion.
Abstract: Using a scheme given by Lochak, we derive a result of stability over exponentially long times with respect to the inverse of the distance to an elliptic equilibrium point which has a definite torsion. At the price of this assumption, our study is valid without arithmetical properties of the linearized system while the previous theorems of this kind rely on a Diophantine condition on the linear spectrum. Actually, under the latter condition and a definite torsion, a result of stability over superexponentially long times can be proved. Finally, the same kind of theorems are also valid for an elliptic lower-dimensional invariant torus.
TL;DR: In this paper, a functional framework for nonlinear dynamics is proposed, where the nonlinearities are (very) short-range perturbation of a given Hamiltonian, and the global in time existence and uniqueness of a solution for the time dependent problem and the existence of steady states of the form which can be specified are proved.
Abstract: We propose a functional framework for some nonlinear dynamics, where the nonlinearities are (very) short-range perturbation of a given Hamiltonian. In this framework one can prove the global in time existence and uniqueness of a solution for the time dependent problem and the existence of steady states of the form which can be specified. This problem arises from the modeling of semiconductors and covers different applications.
TL;DR: In this article, a general, coordinate-free theory for the reduction of volume-preserving flows with a volumepreserving symmetry on three-manifolds is developed, where the reduced flow is generated by a one-degree-of-freedom Hamiltonian which is the generalization of the Bernoulli invariant from hydrodynamics.
Abstract: We develop a general, coordinate-free theory for the reduction of volume-preserving flows with a volume-preserving symmetry on three-manifolds. The reduced flow is generated by a one-degree-of-freedom Hamiltonian which is the generalization of the Bernoulli invariant from hydrodynamics. The reduction procedure also provides global coordinates for the study of symmetry-breaking perturbations. Our theory gives a unified geometric treatment of the integrability of three-dimensional, steady Euler flows and two-dimensional, unsteady Euler flows, as well as quasigeostrophic and magnetohydrodynamic flows.
TL;DR: In this article, the dynamics of merons in a two-dimensional isotropic Heisenberg antiferromagnet immersed in a uniform magnetic field are studied numerically within the discrete spin model as well as analytically within a continuum approximation based on a suitable extension of the nonlinear model.
Abstract: Vortices or merons are the relevant topological solitons in a two-dimensional isotropic Heisenberg antiferromagnet immersed in a uniform magnetic field. The dynamics of such solitons is studied numerically within the discrete spin model as well as analytically within a continuum approximation based on a suitable extension of the nonlinear model. Vortex dynamics is affected rather profoundly by the applied field and acquires the characteristic features of the Hall effect of electrodynamics or the Magnus effect of fluid dynamics. In particular, a single vortex is always spontaneously pinned, two like vortices form a rotating bound state, and a vortex-antivortex pair undergoes Kelvin motion.
TL;DR: In this paper, the Bautin index was used to bound the number of complex periodic solutions on a neighborhood of prescribed size for the polynomial and periodic Abel equations of any degree.
Abstract: An approach to the centre-focus problem for homogeneous perturbations proposed by Cherkas yields a transformation to periodic Abel equations of degree 3. In this paper we consider both the polynomial and periodic Abel equations of any degree. We define the Bautin ideal for these two classes of Abel equations. Recently an approach based on the use of a 1-parameter integrating factor allowed to find the successive derivatives of the return map for a polynomial system which is a homogeneous perturbation of the rotation at the origin. We present the same type of results for the Abel equations. For the polynomial Abel equations, we show that there is an integrating factor defined by a convergent series expansion with polynomial coefficients which satisfy a simple linear recurrency relation. We solve this recurrency relation for low degrees of the perturbation and compute the Bautin index. We then use our previous findings based on the Bernstein inequality and Bautin index to bound the number of complex periodic solutions on a neighbourhood of prescribed size. For the periodic Abel equations, we show that the existence of the integrating factor is equivalent to the periodicity of all other orbits.
TL;DR: In this paper, a Markov approximation scheme based on a partition of into equal intervals produces a probability measure whose total variation norm distance from the exact absolutely continuous invariant measure is bounded by ; C is a constant depending only on the map f.
Abstract: Let f be a sufficiently expanding circle map. We prove that a certain Markov approximation scheme based on a partition of into equal intervals produces a probability measure whose total variation norm distance from the exact absolutely continuous invariant measure is bounded by ; C is a constant depending only on the map f.
TL;DR: In this article, the q-difference Painleve VI equation is investigated on the basis of Hirota forms and it is shown that it admits solutions expressed by Casorati determinants whose entries are given by basic hypergeometric functions.
Abstract: The q-difference Painleve VI equation is investigated on the basis of Hirota forms. It is shown that it admits solutions expressed by Casorati determinants whose entries are given by basic hypergeometric functions.
TL;DR: In this article, the authors study the long-time behavior of solutions of the non-stationary Ginzburg Landau equation of the Schrodinger type and show that the vortex pair produces a shock wave.
Abstract: In this paper we continue our study of the long-time behaviour of solutions of the non-stationary Ginzburg Landau equation of the Schrodinger type. Here we consider initial conditions corresponding to either two vortices of charges +1 each or two vortices of opposite charges +1 and -1. We show that in the first case the vortices radiate while rotating around each other. As a result of this radiation they move apart at the rate of asymptotically. In the second case we show that the radiation is absent for large vortex separations which conforms with previous results. For separations of orders O (1) and less we argue that the vortex pair produces a shock wave. As a result it loses energy and eventually collapses.
TL;DR: In this article, it was shown that the radial d-dimensional sine-Gordon equation has localized metastable breather-like solutions and analyzed some properties of these solutions concentrating on their stability.
Abstract: We show that the radial d-dimensional sine-Gordon equation has localized metastable breather-like solutions. We analyse some properties of these solutions concentrating on their stability.
TL;DR: In this paper, it was shown that the stability result does not hold if p(t) is just a continuous function by providing two examples of instability, and the second example also answers the question posed in Levi M and Zehnder E (1995 Boundedness of solutions for quasiperiodic potentials SIAM J. Math. Anal. 26 1233-56) about instability in the ''squash player's' problem.
Abstract: The motion of a classical particle bouncing elastically between two parallel walls, with one of the walls undergoing a periodic motion is considered. This problem, called Fermi-Ulam `ping-pong', is known to possess only bounded solutions if the motion of the wall is sufficiently smooth , where p(t) is the position of the wall. It is shown that the stability result does not hold if p(t) is just a continuous function by providing two examples of instability. The second example also answers the question posed in Levi M and Zehnder E (1995 Boundedness of solutions for quasiperiodic potentials SIAM J. Math. Anal. 26 1233-56) about instability in the `squash player's' problem. Both examples are constructed for an equivalent system with motionless walls. The reduced system is obtained using the transformation, developed in the heat equation theory to solve the moving boundary problem.
TL;DR: In this paper, the notion of connectedness for compact metric spaces is reformulated in a way that may be implemented computationally, which can distinguish sets that are connected, have a finite number of connected components, have infinitely many connected components or are totally disconnected.
Abstract: We reformulate the notion of connectedness for compact metric spaces in a manner that may be implemented computationally. In particular, our techniques can distinguish between sets that are connected, have a finite number of connected components, have infinitely many connected components, or are totally disconnected. We hope that this approach will prove useful for studying structures in the phase space of dynamical systems.