jacobi matrices and transversality the paper deals with smooth nonlinear ode systems in n f x such that the derivative f x has a matrix representation of jacobi type not necessarily symmetric with positive off diagonal entries a discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system f x t y two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system it is also proved that the nonwandering points are critical points a new class of morse smale systems in n is then explicitly constructed

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

analysis and optimal control of time varying linear systems via block pulse functions succinct and clear sighted operational properties of block pulse functions are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index piecewise constant solutions equally distributed which are simple in form and convenient for use or implementation are consequently obtained another advantage of this method is that any positive integer can be chosen as the number of sub intervals whereas in the case of walsh function approximation the choice can only be made from 2 4 8 16 32 and so on the algorithms developed in the paper are illustrated by appropriate examples

block pulse series analysis of linear systems incorporating observers the paper considers the analysis of linear systems incorporating observers via block pulse functions and presents recursive algorithms convenient for computers compared with walsh function approaches the results are much simpler in form an example is given to illustrate the use of the method

on the design of discrete time optimal dynamic controllers using the partial state observer abstract this paper considers the problem of designing optimal dynamic controllers based on a partial state observer for discrete time linear time invariant systems with inaccessible state it is assumed that the initial system state is an unknown random vector with known mean and covariance the performance index is taken to be the expectation with respect to the initial state of the standard quadratic one for the discrete time regulator necessary and sufficient conditions for optimality are derived provided that the subspace which ensures the existence of a partial state observer is given a design algorithm is also obtained which can be directly solved for the parameters of the optimal observer and requires the solution of a matrix riccati equation of the observer order as in the continuous time case the separation property in the restricted sense is seen to hold finally an example is presented to illustrate the procedure

generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle in this paper we show that for scalar reaction diffusion equations on the circle s 1 the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non linearity in other words we prove that in an appropriate functional space of non linear terms in the equation the set of functions for which all equilibria and periodic orbits are hyperbolic is a countable intersection of open dense sets the main tools in the proof are the property of the lap number and the sard smale theorem

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

a permutation characterization of sturm global attractors of hamiltonian type abstract we consider neumann boundary value problems of the form u t u x x f on the interval 0 x for dissipative nonlinearities f f u a permutation characterization for the global attractors of the semiflows generated by these equations is well known even in the much more general case f f x u u x we present a permutation characterization for the global attractors in the restrictive class of nonlinearities f f u in this class the stationary solutions of the parabolic equation satisfy the second order ode v f v 0 and we obtain the permutation characterization from a characterization of the set of 2 periodic orbits of this planar hamiltonian system our results are based on a diligent discussion of this mere pendulum equation

long time behavior of almost periodically forced parabolic equations on the circle abstract we study the long time behavior of the skew product semiflow generated by scalar reaction diffusion equation on the circle with almost periodic forcing u t u x x f t u u x t 0 x s 1 r 2 z where f t u u x is uniformly almost periodic in t almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic contrary to the time periodic cases for which any limit set can be embedded into a periodically forced circle flow we show that for almost periodic forcing the problem that whether can be embedded into an almost periodically forced circle flow is strongly related to the dimension of the center space v c associated with on the one hand if dim v c 1 then is either spatially inhomogeneous or spatially homogeneous and moreover any spatially inhomogeneous can be embedded into an almost periodically forced circle flow on the other hand when dim v c 1 it is shown that the above embedding property cannot hold anymore these reveal that for such system there are essential differences between time periodic forcing and non periodic forcing

a novel dynamic model of pseudo random number generator an interesting hierarchy of random number generators is introduced in this paper based on the review of random numbers characteristics and chaotic functions theory the main objective of this paper is to produce an ergodic dynamical system which can be implemented in random number generators in order to check the efficacy of pseudo random number generators based on this map we have carried out certain statistical tests on a series of numbers obtained from the introduced hierarchy the results of the tests were promising as the hierarchy passed the tests satisfactorily and offers a great capability to be employed in a pseudo random number generator

the morse smale property for a semi linear parabolic equation on etudie la dynamique d equations aux derivees partielles paraboliques semi lineaires du type suivant u t u xx f x u 0 0 u 0 t u 1 t 0 t 0 u x 0 x

laguerre polynomial analysis in optimal control systems incorporating observers this paper considers a method of analysis using the laguerre polynomial approach for finding the optimal control of linear systems incorporating an observer a recursive algorithm convenient for computers is also derived in which small matrices are always used to find a continuous solution the unique property of laguerre polynomials leads to excellent results which are demonstrated with an example

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

transversality in scalar reaction diffusion equations on a circle abstract we prove that stable and unstable manifolds of hyperbolic periodic orbits for general scalar reaction diffusion equations on a circle always intersect transversally the argument also shows that for a periodic orbit there are no homoclinic connections the main tool used in the proofs is matano s zero number theory dealing with the sturm nodal properties of the solutions

algorithm for normal random numbers we propose a simple algorithm for generating normally distributed pseudorandom numbers the algorithm simulates n molecules that exchange energy among themselves following a simple stochastic rule we prove that the system is ergodic and that a maxwell like distribution that may be used as a source of normally distributed random deviates follows in the n limit the algorithm passes various performance tests including monte carlo simulation of a finite two dimensional ising model using wolff s algorithm it only requires four simple lines of computer code and is approximately ten times faster than the box muller algorithm s1063 651x 99 04409 8 pacs number s 02 70 lq 02 90 1p

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

simplicity of zeros in scalar parabolic equations soit le systeme u t u xx f x u x 0 1 u t 0 u t 1 0 f x 0 0 u 0 x u 0 x soit t 0 u t a seulement des zeros simples soit z u t le nombre de zeros de u 0 on suppose z u 0 fini alors est ouvert et dense dans l intervalle de temps maximal d existence de u t

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle abstract we investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction advection diffusion equations with periodic boundary conditions using zero number properties of the solutions and the phase shift equivariance of the equation we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves

entropy of continuous mixtures and the measure problem in its continuous version the entropy functional measuring the information content of a given probability density may be plagued by a measure problem that results from improper weighting of phase space this issue is addressed considering a generic collision process whereby a large number of particles or agents randomly and repeatedly interact in pairs with prescribed conservation law s we find a sufficient condition under which the stationary single particle distribution function maximizes an entropylike functional that is free of the measure problem this condition amounts to a factorization property of the jacobian associated with the binary collision law from which the proper weighting of phase space directly follows

shifted chebyshev series analysis of linear optimal control systems incorporating observers the analysis of optimal control systems incorporating observers has been approached by the shifted chebyshev series in this study the method simplifies the system of equations into the successive solution of a set of linear algebraic equations one example is illustrated and only a small number of the chebyshev polynomials is needed to obtain a very attractive and excellent result