control of chaotic dynamical systems using radial basis function network approximators abstract this paper presents a general control method based on radial basis function networks rbfns for chaotic dynamical systems for many chaotic systems that can be decomposed into a sum of a linear and a nonlinear part under some mild conditions the rbfn can be used to well approximate the nonlinear part of the system dynamics the resulting system is then dominated by the linear part with some small or weak residual nonlinearities due to the rbfn approximation errors thus a simple linear state feedback controller can be devised to drive the system response to a desirable set point in addition to some theoretical analysis computer simulations on two representative continuous time chaotic systems the duffing and the lorenz systems are presented to demonstrate the effectiveness of the proposed method

first integrals integrating factors and λ symmetries of second order differential equations for a given second order ordinary differential equation ode several relationships among first integrals integrating factors and symmetries are studied the knowledge of a symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first order linear systems of partial differential equations if two nonequivalent symmetries of the equation are known then an algorithm to find two functionally independent first integrals is provided these methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or in the prelle singer method these results are applied to several odes that appear in the study of relevant equations of mathematical physics

λ symmetries for discrete equations following the usual definition of symmetries of differential equations we introduce the analogous concept for difference equations and apply it to some examples

first integrals and parametric solutions of third order odes admitting mathfrak sl 2 mathbb r a complete set of first integrals for any third order ordinary differential equation admitting a lie symmetry algebra isomorphic to is explicitly computed these first integrals are derived from two linearly independent solutions of a linear second order ode without additional integration the general solution in parametric form can be obtained by using the computed first integrals the study includes a parallel analysis of the four inequivalent realizations of and it is applied to several particular examples these include the generalized chazy equation as well as an example of an equation which admits the most complicated of the four inequivalent realizations

improved superorthogonal codes through generalized rotations concatenation of orthogonal space time block codes ostbc with an outer trellis has led to simple and powerful codes known as superorthogonal codes or space time block trellis coded modulation in this letter we generalize these codes by finding new code supersets and corresponding set partitioning resulting in improved coding gain we provide design guidelines for the labeling of the generalized code trellises and demonstrate the gains by several example designs for two and four transmit antennas

high rate concatenated space time block code m tcm designs in this paper a new technique to design improved high rate space time st codes is proposed based on the concept of concatenated st block code stbc and outer trellis coded modulation m tcm encoder constructions unlike the conventional rate lossy stbc mtcm schemes the proposed designs produce higher rate st codes by expanding the codebook of the inner orthogonal stbc the classic set partitioning concept is adopted to realize the stbc mtcm designs with large coding gains the proposed expanded stbc mtcm designs for the two three and four transmitter cases are illustrated simulation results show the proposed stbc mtcm designs significantly outperform the traditional st tcm schemes furthermore decoding complexity of the proposed scheme is low because signal orthogonality is exploited to ease data decoding

geometric aspects of reduction of order using the differential geometry of vectorfields and forms we reinterpret and extend the traditional idea of an integrating factor for a first order differential equation with symmetry in particular we provide a simple and manifestly geometric approach to reduction of order via symmetry for ordinary differential equations which largely obviates the necessity for canonical coordinates and the associated quotient manifolds in so doing some new results which generalise the class of lie group actions which can be used to solve ordinary differential equations are developed

recursive hybrid algorithm for non linear system identification using radial basis function networks recursive identification of non linear systems is investigated using radial basis function networks a novel approach is adopted which employs a hybrid clustering and least squares algorithm the recursive clustering algorithm adjusts the centres of the radial basis function network while the recursive least squares algorithm estimates the connection weights of the network because these two recursive learning rules are both linear rapid convergence is guaranteed and this hybrid algorithm significantly enhances the real time or adaptive capability of radial basis function models the application to simulated real data are included to demonstrate the effectiveness of this hybrid approach

variational c symmetries and euler lagrange equations abstract a generalization of the concept of variational symmetry based on prolongations allows us to construct new methods of reduction for euler lagrange equations an adapted formulation of the noether s theorem for the new class of symmetries is presented some examples illustrate how the method works in practice

new symmetry solution techniques for first order non linear pdes using recent extensions of work of s lie vorlesungen uber differentialgleichungen mit bekannten infinitesimalen transformationen leipzig teubner 1891 and e cartan lecons sur les invariants integreaux hermann paris 1922 les systems differentials exterieures et leurs applications geometrique hermann paris 1945 for integrating frobenius integrable vector field distributions via symmetry we examine some symmetry techniques for finding local solutions of first order non linear partial differential equations pdes in the language of exterior differential systems we develop a technique for solving first order quasilinear pdes that is then applied to general first order non linear partial differential equations our results are significant inasmuch as we give algorithms for solving first order pdes in the presence of symmetry that are essentially mechanical in nature and done in the original coordinate system

on the relation between standard and μ symmetries for pdes we give a geometrical interpretation of the notion of prolongations of vector fields and of the related concept of symmetry for partial differential equations extending to pdes the notion of symmetry for odes we give in particular a result concerning the relationship between symmetries and standard exact symmetries the notion is also extended to the case of conditional and partial symmetries and we analyse the relation between local symmetries and nonlocal standard symmetries

deformation of lie derivative and μ symmetries we introduce in the spirit of witten s gauging of exterior differential a deformed lie derivative that allows a geometrical interpretation of and symmetries in complete analogy with standard symmetries the case of variational symmetries both for odes and for pdes is also considered in this approach leading to and conservation laws

super orthogonal space time trellis codes we introduce a new class of space time codes called super orthogonal space time trellis codes these codes combine set partitioning and a super set of orthogonal space time block codes in a systematic way to provide full diversity and improved coding gain over earlier space time trellis code constructions we also study the optimality of our set partitioning and provide coding gain analysis codes operating at different rates up to the highest theoretically possible rate for different number of states can be designed by using our optimal set partitioning super orthogonal space time trellis codes can provide a tradeoff between rate and coding gain simulation results show more than 2 db improvements over the codes presented in the literature while providing a systematic design methodology

symmetry and lie frobenius reduction of differential equations twisted symmetries widely studied in the last decade have proved to be as effective as standard ones in the analysis and reduction of nonlinear equations we explain this effectiveness in terms of a lie frobenius reduction this requires focus not just on the prolonged symmetry vector fields but on the distributions spanned by these and on systems of vector fields in involution in the frobenius sense not necessarily spanning a lie algebra

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

internal external and generalized symmetries backlund s theorem which characterizes contact transformations is generalized to give an analogous characterization of internal symmetries of systems of differential equations for a wide class of systems of differential equations we prove that every internal symmetry comes from a first order generalized symmetry and conversely every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry we analyze the contact conditions in detail deducing powerful necessary conditions for a system of differential equations that admit genuine internal symmetries i e ones which do not come from classical external symmetries applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to hilbert and cartan are the noncompact real form of the exceptional simple lie group g2

on the reduction methods for ordinary differential equations as tandard method based on the use of differential invariants of a lie group g e nables us to reduce any ordinary differential equation invariant under the action of g w e show that this method is applicable to vector fields more general than those associated with lie symmetries we characterize all such vector fields and study their relationship with nonlocal symmetries and symmetries govinder k s and leach p g l 1995 j phys a math gen 28 5349 59 muriel c and romero l 2001 ima j appl math 66 111 25

solvable structures and hidden symmetries we show that some examples in the literature of non standard symmetry reductions of ordinary differential equations can be understood using the concept of a solvable structure of one forms

on the geometry of lambda symmetries and pde reduction we give a geometrical characterization of prolongations of vector fields and hence of symmetries of odes this allows an extension to the case of pdes and systems of pdes in this context the central object is a horizontal 1 form and we speak of prolongations of vector fields and symmetries of pdes we show that these are as good as standard symmetries in providing symmetry reduction of pdes and systems and explicit invariant solutions

nonlocal transformations and linearization of second order ordinary differential equations the class of nonlinear second order equations that are linearizable by means of generalized sundman transformations s transformations is identified as the class of equations admitting first integrals that are polynomials of first degree in the first order derivative this class is also characterized in terms of the coefficients of the equations and constructive methods to derive the linearizing s transformations are presented only the equations of a well defined subclass can also be linearized by invertible point transformations these invertible point transformations can be constructed by using the algorithms for the calculation of linearizing s transformations several examples illustrate that both types of linearization are strictly different

parallel recursive prediction error algorithm for training layered neural networks a new recursive prediction error algorithm is derived for the training of feedforward layered neural networks the algorithm enables the weights in each neuron of the network to be updated in an efficient parallel manner and has better convergence properties than the classical back propagation algorithm the relationship between this new parallel algorithm and other existing learning algorithms is discussed examples taken from the fields of communication channel equalization and nonlinear systems modelling are used to demonstrate the superior performance of the new algorithm compared with the back propagation routine

a group theoretic approach to a class of second order ordinary differential equations not possessing lie point symmetries we consider the solution of a class of second order ordinary differential equations not possessing lie point symmetries by group theoretic means the method involves increasing the order of these equations using homogeneity symmetry the solution of this new third order equation is then sought in the instances that it and the new reduced second order equation possess additional symmetries as a result the number of second order equations solvable by the lie theory of extended groups is increased

local and nonlocal solvable structures in the reduction of odes solvable structures likewise solvable algebras of local symmetries can be used to integrate scalar ordinary differential equations odes by quadratures solvable structures however are particularly suitable for the integration of odes with a lack of local symmetries in fact under regularity assumptions any given ode always admits solvable structures even though finding them in general could be a very difficult task in practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra in this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order i e classical and higher in particular we introduce the notion of nonlocal solvable structure

nonlocal aspects of λ symmetries and odes reduction a reduction method of odes not possessing lie point symmetries makes use of the so called symmetries muriel and romero 2001 ima j appl math 66 111 the notion of covering for an ode is used here to recover symmetries of as nonlocal symmetries in this framework by embedding into a suitable system determined by the function any symmetry of can be recovered by a local symmetry of as a consequence the reduction method of muriel and romero follows from the standard method of reduction by differential invariants applied to

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples