generalized drinfel d sokolov hierarchies a general approach is adopted to the construction of integrable hierarchies of partial differential equations a series of hierarchies associated to untwisted kac moody algebras and conjugacy classes of the weyl group of the underlying finite lie algebra is obtained the generalized kdv hierarchies of v g drinfel d and v v sokolov are obtained as the special case for the coxeter element various examples of the general formalism are treated in some detail including the fractional kdv hierarchies

integrable deformations of the mkdv and sg hierarchies and quasigraded lie algebras abstract we construct a new family of quasigraded lie algebras that admit the kostant adler scheme they coincide with special quasigraded deformations of twisted subalgebras of the loop algebras using them we obtain new hierarchies of integrable equations in partial derivatives they coincide with the deformations of integrable hierarchies associated with the loop algebras we consider the case g g l 2 in detail and obtain integrable hierarchies that could be viewed as deformations of mkdv sine gordon and derivative non linear shrodinger hierarchies and some other integrable hierarchies such as the w3 non linear shrodinger hierarchy and its doubled form

unique normal forms for vector fields and hamiltonians on construit des formes canoniques pour tous les champs vectoriels resp hamiltoniens en equilibre a 0 r n relativement au groupe des diffeomorphismes formels resp des transformations canoniques tangentiels a l identite

deformations of loop algebras and integrable systems hierarchies of integrable equations using special quasigraded lie algebras that could be viewed as deformations of loop algebras we obtain new hierarchies of integrable nonlinear equations admitting zero curvature representations in particular we obtain integrable hierarchies that generalize the heisenberg magnet landau lifshitz and anisotropic chiral field hierarchies we also obtain a new type of so 3 anisotropic chiral field equation along with its higher rank generalization

lqg optimal scalar static output feedback an eigenproblem based technique is given for computing the stabilizing scalar static output feedback gain that minimizes the standard linear quadratic performance index in the presence of a gaussian white noise disturbance the method adapts recent work on eigenproblem based computation of h norms

spectral sequences and parametric normal forms abstract this paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases in this paper the method is first applied to two simple examples parametric single zero and parametric resonant saddle singularities and then to parametric generalized hopf singularity we provide the simplest parametric normal form for these cases

further reductions of normal forms for dynamical systems abstract we propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields with concrete and interesting applications we use lower order nonlinear terms in the normal form for the simplifications of higher order terms our approach is applicable for both the non nilpotent and the nilpotent cases for dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal or unique with respect to equivalence by formal near identity transformations we can compute at the same time a formal diffeormorphism that realizes the normalization comparisons with other methods are given for several examples

analysis of carleman representation of analytical recursions we study a general method to map a nonlinear analytical recursion onto a linear one the solution of the recursion is represented as a product of matrices whose elements depend only on the form of the recursion and not on initial conditions first we consider the method for polynomial recursions of arbitrary degree and then the method is generalized to analytical recursions some properties of these matrices such as the existence of an inverse matrix and diagonalization are also studied

transfer function matrix of 2 d systems a formula is given which allows the determination of the transfer function matrix of linear multivariable two dimensional 2 d systems directly in terms of the state transition matrix and the characteristic equation

quasigraded lie algebras on hyperelliptic curves and classical integrable systems a new family of infinite dimensional lie algebras on hyperelliptic curves is constructed we show them to be quasigraded and explicitly find their central extensions we also show that constructed algebras in the case of zero central charge possess infinite number of invariant functions besides they admit a decomposition into the direct sum of two subalgebras these two facts together enables one to use them to construct new integrable systems in such a way we find new integrable hamiltonian systems which are direct higher rank generalizations of the integrable systems of steklov liapunov associated with the e 3 algebra and steklov veselov associated with the so 4 algebra besides we give hyperelliptic lax representation for the generalized euler tops

infinite dimensional lie algebras classical r matrices and lax operators two approaches for each finite dimensional simple lie algebra g starting from a general g g valued solutions r u v of the generalized classical yang baxter equation we construct infinite dimensional lie algebras gr of g valued meromorphic functions we outline two ways of embedding of the lie algebra gr into a larger lie algebra with kostant adler symmes decomposition the first of them is an embedding of gr into lie algebra g u 1 u of formal laurent power series the second is an embedding of gr as a quasigraded lie subalgebra into a quasigraded lie algebra gr gr gr gr such that the kostant adler symmes decomposition is consistent with a chosen quasigrading we construct dual spaces gr gr and explicit form of the lax operators l u l u as elements of these spaces we develop a theory of integrable finite dimensional hamiltonian systems and soliton hierarchies based on lie algebras gr gr we consider examples of such systems and soliton equations and obtain the most general form of

new non skew symmetric classical r matrices and twisted quasigraded lie algebras we construct a family of quasigraded lie algebras that admit the kostant adler scheme they coincide with quasigraded deformations of the loop algebras in different gradings using them we explicitly construct new non skew symmetric classical r matrices with spectral parameters we consider examples of the constructed algebras and constructed r matrices corresponding to the homogeneous quasigrading z2 quasigrading and to the principle quasigrading in detail

vector potential normal form classification for completely integrable solenoidal nilpotent singularities abstract we introduce an sl 2 invariant family of polynomial vector fields with an irreducible nilpotent singularity in this paper we are concerned with characterization and normal form classification of these vector fields we show that the family is a lie subalgebra and each vector field from this family is volume preserving completely integrable and rotational all such vector fields share a common quadratic invariant we provide a poisson structure for the lie subalgebra from which the second invariant for each vector field can be readily derived we show that each vector field from this family can be uniquely characterized by two alternative representations which can be found in applications one uses a vector potential while the other uses two functionally independent clebsch potentials our normal form results are designed to preserve these structures and representations

normal forms resonance and bifurcation analysis via the carleman linearization abstract equivalence between the normal form nf and the carleman linearization of a nonlinear dynamic system is established the near identity nonlinear coordinate transformation which brings a system to its nf is shown to be the similarity transformation bringing a carleman system to a jordan canonical form it is shown that the steady state multiplicity is given by the nullity of the first carleman matrix with noncomplete nullspace the stability of the limit cycles at hopf bifurcation is determined by the direction of the i generalized eigenvector of the first odd order carleman matrix which has a noncomplete i eigenspace the coefficients of the resonant terms that are retained in the nf are explicitly determined

an efficient algorithm for computing the coefficients of the characteristic polynomial of 2 d state space models for 2 d systems described by a state space structure two different methods for computing the coefficients of their characteristic polynomial are introduced the relative efficiencies of these methods are compared to that of faddeeva s method an efficient method is then chosen which can be used to compute the coefficients of the characteristic polynomial with a minimum number of multiplications

hierarchies of integrable equations associated with hyperelliptic lie algebras using a family of special quasigraded lie algebras on hyperelliptic curves we construct new hierarchies of integrable nonlinear equations admitting zero curvature representations we show that in the case of the rational degeneration of the curve they coincide with heisenberg magnet hierarchies

steady state bifurcations and exact multiplicity conditions via carleman linearization the problem of steady state bifurcations of vector fields under parameter perturbation is resolved by a linear algebraic method exact multiplicity conditions for any steady state are obtained in terms of the system parameters no reduction of the steady state system to one equation is required instead the one dimensional case is included as a subspace in this generalized framework the key point that this paper highlights is that the order of the steady multiplicity at bifurcation can be determined by examining the dimension of the kernel of the successive carleman linear operators for all cases of practical interest in particular the dimension of the kernel of any carleman linear operator of order l equals l if l is less than the multiplicity however the th order carleman operator retains a 1 dimensional kernel

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

quasigraded bases in loop algebras and classical rational r matrices in the present paper we construct quasigraded bases in the loop algebras associated with each classical rational r matrix we show that they are orthogonal and agreed with the decomposition of the loop algebra into sum of subalgebras that corresponds to this rational r matrix using the quasigraded character of the constructed algebras we define the infinite sequence of the embedded ideals in the each subalgebra of the decomposition and apply this result to the theory of finite dimensional integrable systems

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented