on a backward parabolic problem with local lipschitz source abstract we consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form u t a u t f u t t u 1 where a is a positive self adjoint unbounded operator and f is a local lipschitz function as known it is ill posed and occurs in applied mathematics e g in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology a new version of quasi reversibility method is described we show that the regularized problem with a regularization parameter 0 is well posed and that its solution u t converges on 0 1 to the exact solution u t as 0 these results extend some earlier works on the nonlinear backward problem

a piecewise quadratic maximum entropy method for the statistical study of chaos abstract let the frobenius perron operator p s l 1 0 1 l 1 0 1 related to a nonsingular transformation s 0 1 0 1 have an invariant density f we propose a piecewise quadratic maximum entropy method for the numerical approximation of f the role of the partition of unity property of the quadratic functions for the numerical recovery of f has been depicted the proposed algorithm overcomes the ill conditioning shortage of the traditional maximum entropy method which only employs polynomials so that any number of moments can be used to increase the accuracy of the computed invariant density the convergence rate of the method is shown to be of order 3 numerical results are presented to justify the theoretical analysis of the method

piecewise linear least squares approximations of frobenius perron operators abstract we propose a piecewise linear numerical method based on least squares approximations for computing stationary density functions of frobenius perron operators associated with piecewise c2 and stretching mappings of the unit interval we prove the weak convergence of the method for a class of frobenius perron operators and the numerical results show that it is also norm convergent and has a better convergence rate than the piecewise linear markov approximation method

symmetry breaking and semilinear elliptic equations it is a readily observable fact that many physical and mathematical systems possess a degree of symmetry and that a study of this symmetry may give us valuable insight into their behaviour it is particularly interesting that symmetric systems exist which possess non symmetric solutions and where this solution branch arises from a symmetry breaking bifurcation on a branch of symmetric solutions in this paper we shall study an example of such a system namely we shall study the symmetry breaking bifurcations henceforth denoted as sbb s which occur on the radially symmetric solution branches of the following semilinear elliptic equation

convergence analysis of projection methods for frobenius perron operators based on matrix norm techniques abstract using matrix norm techniques we give a unified convergence analysis of a general projection method for computing stationary density functions of frobenius perron operators associated with piecewise c 2 and stretching transformations of the unit interval

a finite element method for the frobenius perron operator equation we present a finite element framework to numerically solve the absolutely continuous invariant measure associated with a non singular transformation both convergence analysis and error estimates of the method are given for a class of mappings in r n

a maximum entropy method for solving frobenius perron operator equations let s 0 1 0 1 be a nonsingular transformation that preserves an absolutely continuous invariant measure with density f in this paper we develop a general algorithm for approximating f using the maximum entropy approach

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

approximations of frobenius perron operators via interpolation abstract let s 0 1 0 1 be a chaotic map and let p l1 0 1 l1 0 1 be the corresponding frobenius perron operator we propose a piecewise linear approximations method based on interpolation that can be efficiently used to compute a fixed density of p the convergence of the method for a class of mappings is proved and numerical results are also presented

semilinear elliptic equations with near critical growth rates on etudie les solutions symetriques de l equation elliptique semi lineaire u u u u p 1 0 u b 0 ou b est la boule unite dans r 3 on prend p proche de 5 on obtient une description asymptotique des solutions avec une grande norme

constructive approximations of markov operators we construct piecewise linear markov finite approximations of markov operators defined on l1 0 1 n and we study various properties such as consistency stability and convergence for the purpose of numerical analysis of markov operators

high order approximation of the frobenius perron operator we formulate a general convergence theory for the finite dimensional projection approximation of the fixed point of a class of linear operators in particular we explore the relation with the standard framework of stability and consistency of the approximation imply convergence applications include the frobenius perron operator p s corresponding to a nonsingular measurable transformation from 0 1 to itself numerical experiments are also given to support the theoretical analysis

invariant densities and escape rates rigorous and computable approximations in the l norm abstract in this article we study piecewise linear discretization schemes for transfer operators perron frobenius operators associated with interval maps we show how these can be used to provide rigorous pointwise approximations for invariant densities of markov interval maps we also derive the order of convergence of the approximate invariant density to the real one in the l norm the outcome of this paper complements recent results on the formulae of escape rates of open dynamical systems keller and liverani 2009 7 in particular the novelty of our work over previous results on b v and l approximations is that it provides a method for explicit computation of the approximation error

stability of solutions of a class of nonlinear fractional laplacian parabolic problems abstract in this paper we study the forward and the backward in time problems for a class of nonlinear partial differential equations of fractional laplace operator with nonlinear source function the equations are generalized from lots of well known equations such as the burger equation the cahn hilliard equation herein we investigate the stability of the solution of the forward problem depended on the fractional order of the operator and the instability of the backward problem and then we establish a regularization method some numerical experiments are performed to verify efficiency and accuracy of our method

pattern selection with o 3 symmetry abstract for each irreducible representation of so 3 and o 3 we determine up to conjugacy all isotropy subgroups and identify in particular the maximal isotropy subgroups each isotropy subgroup corresponds to a possible planform for the spherical benard problem using an equivariant branching lemma of cicogna 1 we prove for each of these representations the existence of solutions corresponding to a number of different planforms thus extending substantially the work of busse 2 3 and sattinger 4 we also give a useful criterion for showing when solutions obtained by the equivariant branching lemma must be unstable

discretization of the frobenius perron operator using a sparse haar tensor basis the sparse ulam method the global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated frobenius perron operator for systems with low dimensional long term dynamics efficient techniques exist for a numerical approximation of the most important eigenfunctions cf m dellnitz and o junge siam j numer anal 36 1999 pp 491 515 they are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor ulam s method in this paper we develop a numerical technique which makes ulam s approach applicable to systems with higher dimensional long term dynamics it is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids c zenger sparse grids in parallel algorithms for partial differential equations kiel 1990 vieweg braunschweig 1991 pp 241 251 h j bungartz and m griebel acta numer 13 2004 pp 147 269 here we use a sparse haar tensor basis as the underlying approximation space we develop the technique establish statements about its complexity and convergence and present two numerical examples

time and space fractional partial differential equations the fundamental solution for time and space fractional partial differential operator dt a2 2 0 is given in terms of the fox s h function here the time fractional derivative in the sense of generalized functions distributions dt is defined by the convolution dt f t t f t where t t 1 and f t 0 as t 0 and the fractional n dimensional laplace operator 2 is defined by its fourier transform with respect to spatial variable f 2g x f g x the solutions for initial value problems for time and space fractional partial differential equation in the sense of caputo and riemann liouville time fractional derivatives respectively are obtained by the fundamental solution

a modified piecewise linear markov approximation of markov operators in this paper we propose a modified piecewise linear markov method for approximating markov operators defined on l 1 0 1 we present numerical results with the new method as compared to those using the original piecewise linear markov method

computing the invariant measure and the lyapunov exponent for one dimensional maps using a measure preserving polynomial basis we consider a generalisation of ulam s method for approximating invariant densities of one dimensional maps rather than use piecewise constant polynomials to approximate the density we use polynomials of degree n which are defined by the requirement that they preserve the measure on n 1 neighbouring subintervals over the whole interval this results in a discontinuous piecewise polynomial approximation to the density we prove error results where this approach is used to approximate smooth densities we also consider the computation of the lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself together with using cubic polynomials in the density approximation this yields a very efficient method for computing highly accurate estimates of the lyapunov exponent we illustrate the theoretical findings with some examples

finite approximations of markov operators in this paper we survey some recent developments in the numerical analysis of markov operators and in particular frobenius perron operators associated with chaotic discrete dynamical systems

on the norm convergence of a piecewise linear least squares method for frobenius perron operators abstract in this paper we prove both the l 1 norm and the bv norm convergence for a piecewise linear least squares approximation method that was recently proposed for computing stationary density functions of frobenius perron operators associated with piecewise c 2 and stretching mappings of the unit interval we also give a convergence rate analysis under the bv norm

markov finite approximation of frobenius perron operator with doubly stochastic tridiagonal action matrix we extend the set of finite rank approximations of frobenius perron operators from projections to general finite rank operators by indicating an arbitrary action matrix we prove the convergence in the case for which the action matrix is double stochastic and tridiagonal and we find the rate of convergence

computing invariant measures of piecewise convex transformations lets 0 1 0 1 be a piecewise convex transformation satisfying some conditions which guarantee the existence of an absolutely continuous invariant probability measure we prove the convergence of a class of markov finite approximations for computing the invariant measure using a compactness argument forl1 spaces

symmetry breaking of solutions of non cooperative elliptic systems abstract in this article we study the symmetry breaking phenomenon of solutions of non cooperative elliptic systems we apply the degree for g invariant strongly indefinite functionals to obtain simultaneously a symmetry breaking and a global bifurcation phenomenon

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

the resonance spectrum of the cusp map in the space of analytic functions we prove that the frobenius perron operator u of the cusp map f 1 1 1 1 f x 1 2 x which is an approximation of the poincare section of the lorenz attractor has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1 we also prove that for any q 0 1 the spectrum of u in the hardy space in the disk z c z q 1 q is the union of the segment 0 1 and some finite or countably infinite set of isolated eigenvalues of finite multiplicity

the projection method for computing multidimensional absolutely continuous invariant measures we present an algorithm for numerically computing an absolutely continuous invariant measure associated with a piecewisec2 expanding mappings on a bounded region rn the method is based on the galerkin projection principle for solving an operator equation in a banach space with the help of the modern notion of functions of bounded variation in multidimension we prove the convergence of the algorithm

markov finite approximation of frobenius perron operator the problem of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory 3 lasota and yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations s from 0 l to itself 4 later in 6 li and yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping the classical birkhoff s individual ergodic theorem says that if u is an ergodic invariant probability measure under s then for any measurable set a c 0 11 the time average lim 1 n c ax s x which measures the average time spent in a under iteration of s exists to be i for p almost all x where xa is the characteristic function of a 1 on a and 0 off a this seems to suggest that we may use the cesaro sum of the above form to calculate the invariant measure a simple and important example in 5 however shows that the computer round off error can completely dominate the calculation and make the implementation difficult in order to overcome this difficulty li 5 proposed a rigorous numerical procedure which can be implemented on a computer with negligible round off error problem a mapping s 0 l 0 l is called piecewise c2 if there is a partition 0 a a a 1 of 0 l such that for k 1 r the restriction sk of s on uk a is a c2 function which can be extended to the closed interval ak i ak as a c2 function s need not be continuous at the point ak for a c 0 11 we write s a for x s x e a throughout the paper we assume s is a nonsingular measurable transformation i e s is measurable and for any measurable subset a c 0 l with m a 0 m s a 0 here m denotes the lebesgue measure on 0 11 the operator ps l 0 1 l 0 1 defined by