A continuous map f from a compact interval I into itself is densely (resp. generically) chaotic if the set of points (x, y) such that lim supn ->+infinity f(n)(x) - f(n)(y) - > 0 and lim inf(n ->+infinity) vertical bar f(n)(x) - fn(y)vertical bar = 0 is dense (resp. residual) in I x I. We prove that if the interval map f is densely but not generically chaotic then there is a descending sequence of invariant intervals, each of which contains a horseshoe for f(2). It implies that every densely chaotic interval map is of type at most 6 for Sharkovskii's order (i.e. there exists a periodic point of period 6), and its topological entropy is at least (log 2)/2. We show that equalities can be obtained.
In this paper we determine classes of conservation laws admitted by 2 x 2 first-order hyperbolic nonhomogeneous systems by following the so-called direct method proposed by Bluman et al. Such mathematical models can describe several situations of interest in applications like isentropic fluid dynamics, viscoelastic media, traffic flows, and transmission lines. A full classification of all the possible local conservation laws is given in the case where the governing equations are homogeneous. The general results obtained have been used in two examples concerning the p-system and a 2 x 2 model describing transmission lines.
In this work we present spectral algorithms for the numerical scattering for the defocusing Davey–Stewartson (DS) II equation with initial data having compact support on a disk, i.e. for the solution of d-bar problems. Our algorithms use polar coordinates and implement a Chebychev spectral scheme for the radial dependence and a Fourier spectral method for the azimuthal dependence. The focus is placed on the construction of complex geometric optics (CGO) solutions which are needed in the scattering approach for DS. We discuss two different approaches: The first constructs a fundamental solution to the d-bar system and applies the CGO conditions on the latter. This is especially efficient for small values of the modulus of the spectral parameter k. The second approach uses a fixed point iteration on a reformulated d-bar system containing the spectral parameter explicitly, a price paid to have simpler asymptotics. The approaches are illustrated for the example of the characteristic function of the disk and are shown to exhibit spectral convergence, i.e. an exponential decay of the numerical error with the number of collocation points. An asymptotic formula for large is given for the reflection coefficient.
The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity of weak solutions for these equalities to hold. The method of proof is suitable for the case of periodic as well as homogeneous Dirichlet boundary conditions. In particular, by a careful analysis using the homogeneous Dirichlet boundary condition, no boundary layer assumptions are required when dealing with bounded domains with a boundary.
We study the top Lyapunov exponents of random products of positive 2 x 2 matrices and obtain an efficient algorithm for its computation. As in the earlier work of Pollicott (2010 Inventiones Math. 181 209-26), the algorithm is based on the Fredholm theory of determinants of trace-class linear operators. In this article we obtain a simpler expression for the approximations which only require calculation of the eigenvalues of finite matrix products and not the eigenvectors. Moreover, we obtain effective bounds on the error term in terms of two explicit constants: a constant which describes how far the set of matrices are from all being column stochastic, and a constant which measures the minimal amount of projective contraction of the positive quadrant under the action of the matrices.