the complex lorenz equations we have undertaken a study of the complex lorenz equations x x y y r z x ay z bz 12 x y xy where x and y are complex and z is real the complex parameters r and a are defined by r r1 ir2 a 1 ie and and b are real behaviour remarkably different from the real lorenz model occurs only the origin is a fixed point except for the special case e r2 0 we have been able to determine analytically two critical values of r1 namely r1c and r1c the origin is a stable fixed point for 0 r1c a hopf bifurcation to a limit cycle occurs we have an exact analytic solution for this limit cycle which is always stable if 1 then this limit is only stable in the region r1c rlc a transition to a finite amplitude oscillation about the limit cycle occurs the nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the stuart landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency this latter bifurcation is either a sub or super critical hopf like bifurcation to a doubly periodic motion the direction of bifurcation depending on the parameter values the nature of the bifurcation is complicated by the existence of a zero eigenvalue

painleve property and constants of the motion of the complex lorenz model the results of the painleve analysis and of a search for constants of the motion of the complex lorenz model are presented

parametric forcing of scroll wave patterns in three dimensional excitable media abstract the dynamics of axisymmetric and twisted scroll rings under homogeneous periodic forcing is studied numerically the collapse of axisymmetric rings can be enhanced or retarded significantly with resonant forcing twisted scroll rings exhibit the phenomenon of resonant drift in a direction normal to the axis of the central filament the scaling of mean and fluctuating axial drift due to forcing is determined evidence is provided showing that the appropriate symmetry group for the dynamics of the unforced twisted scroll ring is se 2 r ordinary differential equations based on symmetry considerations explain the dynamics without resorting to laws of filament motion and the local geometry hypothesis

on the nature of the torus in the complex lorenz equations the complex lorenz equations are a nonlinear fifth order set of physically derived differential equations which exhibit an exact analytic limit cycle which subsequently bifurcates to a torus in this paper we build upon previously derived results to examine a connection between this torus at high and low r1 bifurcation parameter and between zero and nonzero r2 complexity parameter in so doing we are able to gain insight on the effect of the rotational invariance of the system and on how extra weak dispersion r2 0 affects the chaotic behavior of the real lorenz system which describes a weakly dissipative dispersive instability

tracking pattern evolution through extended center manifold reduction and singular perturbations in this paper we develop an extended center manifold reduction method a methodology to analyze the formation and bifurcations of small amplitude patterns in certain classes of multi component singularly perturbed systems of partial differential equations we specifically consider systems with a spatially homogeneous state whose stability spectrum partitions into eigenvalue groups with distinct asymptotic properties one group of successive eigenvalues in the bifurcating group are widely interspaced while the eigenvalues in the other are stable and cluster asymptotically close to the origin along the stable semi axis the classical center manifold reduction provides a rigorous framework to analyze destabilizations of the trivial state as long as there is a spectral gap of sufficient width when the bifurcating eigenvalue becomes commensurate to the stable eigenvalues clustering close to the origin the center manifold reduction breaks down moreover it cannot capture subsequent bifurcations of the bifurcating pattern through our methodology we formally derive expressions for low dimensional manifolds exponentially attracting the full flow for parameter combinations that go beyond those allowed for the classical center manifold reduction i e to cases in which the spectral gap condition no longer can be satisfied our method also provides an explicit description of the flow on these manifolds and thus provides an analytical tool to study subsequent bifurcations our analysis centers around primary bifurcations of transcritical type that can be either of co dimension 1 or 2 in two and three component pde systems we employ our method to study bifurcation scenarios of small amplitude patterns and the possible appearance of low dimensional spatio temporal chaos we also exemplify our analysis by a number of characteristic reaction diffusion systems with disparate diffusivities

on a complex beam beam interaction model with random forcing abstract in recent years many studies have been devoted to complex differential equations cde which appear in very important applications in physics and engineering this paper aims to investigate one such cde containing a random forcing term z o 2 z e 2 g z e 2 f z z p o t n t where z t x t iy t a complex function i 1 n t is a broad band process with zero mean and e2 and small real parameters in particular we use eq to model the interaction between two colliding beams in particle accelerators setting g z z f z z z z 2 and p o t cos o t and extend the work we had started in an earlier publication mahmoud physica a 216 1995 445 we apply the stochastic averaging method to derive a fokker planck kolmogorov equation for this equation and obtain analytically the exact stationary probability density function and the first and second moments in the amplitude of the solutions numerical simulations are carried out to compare with the theoretical ones and excellent agreement is found

central configurations of the n body problem via equivariant morse theory in this paper we use the equivariant morse theory to give an estimate of the minimal number of central configurations in the n body problem in 3 in the case of equal masses we prove that the planar central configurations are saddle points for the potential energy from this we deduce the presence of non planar central configurations for every n 4

adapted numerical methods for advection reaction diffusion problems generating periodic wavefronts abstract the paper presents an adapted numerical integration for advection reaction diffusion problems the numerical scheme exploiting the a priori knowledge of the qualitative behaviour of the solution gains advantages in terms of efficiency and accuracy with respect to classic schemes already known in literature the adaptation is here carried out through the so called trigonometrical fitting technique for the discretization in space giving rise to a system of odes whose vector field contains both stiff and non stiff terms due to this mixed nature of the vector field an implicit explicit imex method is here employed for the integration in time based on the first order forward backward euler method the coefficients of the method here introduced rely on unknown parameters which have to be properly estimated in this work such an estimate is performed by minimizing the leading term of the local truncation error the effectiveness of this problem oriented approach is shown through a rigorous theoretical analysis and some numerical experiments

normal forms resonances and meandering tip motions near relative equilibria of euclidean group actions the equivariant dynamics near relative equilibria to actions of noncompact finite dimensional lie groups g can be described by a skew product flow on a center manifold dot g g textbf a v dot v varphi v with g in g with v in a slice transverse to the group action and a v in the lie algebra of g we present a normal form theory near relative equilibria varphi v 0 0 in this general case for the specific case of the euclidean groups se n the skew product takes the form dot r r textbf r v qquad dot s r textbf s v qquad dot v varphi v with textbf r v in so n textbf s v in mathr n we give a precise meaning to the intuitive idea of tip motion of a meandering spiral it corresponds to the dynamics of s t this clarifies the notion of meander radii and drift resonance in the plane n 2 for illustration we discuss the unbounded tip motions associated with a weak focus in v on the verge of hopf bifurcation in the case of resonant hopf and rotation frequencies of the spiral and study resonant relative hopf bifurcation we also encounter random brownian tip motions for trajectories v t rightarrow gamma which become homoclinic for t rightarrow infty we conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the bogdanov takens bifurcation which turn out to be small beyond any finite order

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

on the evolution of periodic plane waves in reaction diffusion systems of l w type lambda omega systems are a class of simple reaction diffusion equations with a limit cycle in the reaction kinetics the author considers the solution of the system given by lambda r lambda 0 r p omega r omega 0 r p on a semi infinite spatial domain with initial data decaying exponentially across the domain numerical evidence is presented showing that this initial condition induces a wave front moving across the domain with periodic plane waves behind the front these periodic waves can move in either direction depending on the parameter values the author uses intuitive criteria to derive an expression for the speed of the advancing front and by reducing the system to ordinary differential equations for similarity solutions the amplitude and speed of the periodic plane waves are determined the speed of these periodic waves varies continuously with the initial data perturbation theory is then used to obtain an analytical approximation to the solutions these anal

exponentially fitted singly diagonally implicit runge kutta methods it is the purpose of this paper to derive diagonally implicit exponentially fitted ef runge kutta methods for the numerical solution of initial value problems based on first order ordinary differential equations whose solutions are supposed to exhibit an exponential behaviour in addition to the standard approach for the derivation of ef methods we provide a revised constructive technique that takes into account the contribution to the error inherited from the computation of the internal stages the derived methods are then compared to those obtained by neglecting the contribution of the error associated to the internal stages as classically done in the standard derivation of multistage ef based methods standard and revised ef methods are then compared in terms of linear stability and numerical performances

the real and complex lorenz equations in rotating fluids and lasers abstract the lorenz equations are derived systematically from amplitude equations of weakly nonlinear dispersively unstable physical systems near criticality when weak dissipation is added this derivation is only valid if the undamped neutral curve is not destabilised by the addition of weak dissipation the addition of extra weak dispersive effects make some of the coefficients complex and yields a complex set of lorenz equations both sets of equations are derived in examples in laser optics and baroclinic instability

chaotic field theory a sketch abstract spatio temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio temporally periodic solutions the periodic orbit theory yields the global averages characterizing the chaotic dynamics as well as the starting semiclassical approximation to the quantum theory new methods for computing corrections to the semiclassical approximation are developed in particular a nonlinear field transformation yields the perturbative corrections in a form more compact than the feynman diagram expansions

counting objects which behave like irreducible brauer characters of finite groups let n 0 g where g is a finite group so that g acts on irr n the set of irreducible ordinary complex characters of n suppose 8 e irr n is g invariant and write irr g 10 to denote the set of those x e irr g such that x involves and hence is a multiple of 8 how large is irr g io the answer to this question is well known and not very hard to prove analogously to the fact that lirr g i is equal to the number of conjugacy classes of g it turns out that iirr gjb l is equal to the number of certain good classes in the group g n there are two equivalent descriptions of which classes of g n these are one characterization depends on a certain complex twisted group algebra of g n and in this formulation the result which counts irr g 1 t9 is due to i schur s the other formulation of this result due to p x gallagher z defines goodness internally to g it depends on the character theory of g and its subgroups specifically gallagher defines an element of 2 e g n to be o good if some extension of 8 to n g is invariant in c where c n c jg the property of being q good is preserved by conjugacy in gin and gallagher shows that lirr g i 13 l is equal to the number of classes of g in which consist of good elements somewhat less well known is the corresponding result for irreducible brauer characters with respect to some prime p if 8 e ibr n is g invariant we intend to count the elements of ibr g t9 i e the irreducible brauer characters of g whose restrictions to n are multiples of 8 in view of the fact that iibr g equals the number of p regular classes of g it is reasonable to guess that iibr gl f l is the number of e good p regular

on aref s vortex motions with a symmetry center abstract we study the dynamics of point vortices with discrete symmetry of rotation in the case of two rings of vortices the reduction method of marsden and weinstein enables qualitative and quantitative description of the phase portrait bifurcations of phase portrait topology are detected numerically using the ratio of vorticities as the bifurcation parameter steady rotating configurations exist for any number of rings the case of two rings of four vortices is peculiar we prove two conjectures of aref on the ratio of equilibria radii of two rings of opposite circulations as the number of vortices increases and on the nonintegrability of the three ring problem on a restricted version

note on a paper by g rowlands chaotic trajectories of ordinary differential equations this note discusses the possible use of perturbation methods in studying chaotic trajectories of ordinary differential equations with particular focus on a recent paper on this topic by rowlands ibid vol 16 p 585 1983

new exact solutions of the complex lorenz equations the author presents two classes of particular exact solutions to the complex lorenz equations these solutions possess the painleve property the parameter range for which the above solutions hold includes as special cases two of the three parameter ranges for which the general solution of the complex lorenz equations has the painleve property

on fully ramified brauer characters let z be a normal subgroup of a finite group let p 5 be a prime and let ibr z be an irreducible g invariant p brauer character of z suppose that g e for some ibr g then g z is solvable in other words a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group

variational method for finding periodic orbits in a general flow a variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems an initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop the newton descent partial differential equation that governs this evolution is an infinitesimal step version of the damped newton raphson iteration the feasibility of the method is demonstrated by its application to the henon heiles system the circular restricted three body problem and the kuramoto sivashinsky system in a weakly turbulent regime

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

stationary configurations of point vortices the motion of point vortices in a plane of fluid is an old problem of fluid mechanics which was given a hamiltonian formulation by kirchhoff stationary configurations are those which remain self similar throughout the motion results of two types are presented configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry which establish necessary and sufficient conditions for existence relative equilibria rigidly rotating configurations which lie on a line are studied using a topological construction applicable to other power law systems upper and lower bounds for such configurations are found for vortices with mixed circulations arrangements of three vortices which collide in finite time are well known one dimensional families of such configurations are shown to exist for more than three vortices stationary configurations of four vortices are examined in detail the motion of point vortices in a plane of fluid is an old problem of fluid mechanics which was given a hamiltonian formulation by kirchhoff stationary configurations are those which remain self similar throughout the motion and are of considerable physical interest it is known that a configuration of vortices in equilibrium must have total vortex angular momentum 0 a converse is proved namely that for almost every choice of circulations with zero vortex angular momentum there are exactly n 2 equilibrium configurations a similar statement is proved for rigidly translating configurations with total circulation zero the proofs involve ideas from algebraic geometry relative equilibria rigidly rotating configurations were studied by palmore in the case of positive circulations upper and lower bounds for the number of collinear relative equilibria for arbitrary circulations are obtained by means of a topological construction which is applicable to other power law systems arrangements of three vortices which collide in finite time are well known one dimensional families of such configurations are shown to exist for n vortices n 3 stationary configurations of four vortices are examined in detail chapter 0 introduction the motion of point vortices in the plane is an old problem in fluid mechanics it was first given a hamiltonian formulation by kirchhoff 1876 who proved that n vortices with distinct positions xi yi i 1 n in the x y plane and received by the editors july 31 1985 and in revised form june 13 1986 1980 mathematics subject classification 1985 revision primary 76c05 secondary 70h05 t1987 american mathematical society 0002 9947 87 1 00 25 per page

reconstruction equations and the karhunen loeve expansion for systems with symmetry we present a method for applying the karhunen loeve decomposition to systems with continuous symmetry the techniques in this paper contribute to the general procedure of removing variables associated with the symmetry of a problem and related ideas have been used in previous works both to identify coherent structures in solutions of pdes and to derive low order models via galerkin projection the main result of this paper is to derive a simple and easily implementable set of reconstruction equationswhich close the system of odes produced by galerkin projection the geometric interpretation of the method closely parallels techniques used in geometric phases and reconstruction techniques in geometric mechanics we apply the method to the kuramoto sivashinsky equation and are able to derive accurate models of considerably lower dimension than are possible with the traditional karhunen loeve expansion

new embedded explicit pairs of exponentially fitted runge kutta methods two new embedded pairs of exponentially fitted explicit runge kutta methods with four and five stages for the numerical integration of initial value problems with oscillatory or periodic solutions are developed in these methods for a given fixed w the coefficients of the formulae of the pair are selected so that they integrate exactly systems with solutions in the linear space generated by sinh wt cosh wt the estimate of the local error behaves as o h 4 and the high order formula has fourth order accuracy when the stepsize h 0 these new pairs are compared with another one proposed by franco j m franco an embedded pair of exponentially fitted explicit runge kutta methods j comput appl math 149 2002 407 414 on several problems to test the efficiency of the new methods

motion of three point vortices on a sphere abstract we solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius r the system is more general than the corresponding one in the plane 1 3 5 19 28 33 34 37 38 and reduces to it in the limit r as long as the three vortices remain sufficiently close to each other during the course of their motion instead of using spherical coordinates we use cartesian coordinates in which the vector i points from the center of the sphere to the vortex with strength i an important conserved quantity is the center of vorticity vector c i i i which with no loss of generality we align with the z axis based on the size of this vector relative to the radius of the sphere we classify the motion into one of five types super radial radial sub radial degenerate or a limiting super radial case this categorization allows us to draw several conclusions about the qualitative motion of the vortices we then fully characterize all fixed and relative equilibria on the sphere for fixed equilibria the vortices must lie on great circles geodesics if the strengths are equal they form an equilateral triangle otherwise the triangle shape is specified once the strength of the vortices is given the relative equilibria are classified as either degenerate c 0 or non degenerate c 0 for each type the shape of the vortex triangle is described and the frequency of rotation is computed as in the planar problem it is possible to introduce trilinear coordinates and study the motion in a phase plane which allows us to locate all the equilibria as well as to characterize more complex relative dynamics we then describe self similar vortex collapse on the sphere stating necessary and sufficient conditions for collapse to occur computing the collapse times and vortex trajectories on the route towards collapse comparisons are made with the collapse formulas in the plane derived in 1 30

continuous symmetry reduction and return maps for high dimensional flows we present two continuous symmetry reduction methods for reducing highdimensional dissipative flows to local return maps in the hilbert polynomial basis approach the equivariant dynamics is rewritten in terms of invariant coordinates in the method of moving frames or method of slices the state space is sliced locally in such a way that each group orbit of symmetry equivalent points is represented by a single point in either approach numerical computations can be performed in the original state space representation and the solutions are then projected onto the symmetry reduced state space the two methods are illustrated by reduction of the complex lorenz system a 5 dimensional dissipative flow with rotational symmetry while the hilbert polynomial basis approach appears unfeasible for high dimensional flows symmetry reduction by the method of moving frames offers hope

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

the real and complex lorenz equations and their relevance to physical systems abstract we summarize some recently obtained results on real and complex lorenz equations and discuss their possible significance in relation to real fluid dynamical processes

exponentially fitted imex methods for advection diffusion problems the paper is devoted to the numerical solution of advection diffusion problems of boussinesq type by means of adapted numerical methods the adaptation occurs at two levels along space by suitably semidiscretizing the spatial derivatives through finite differences based on exponential fitting along time integration through an adapted imex method based on exponential fitting itself stability analysis is provided and numerical examples showing the effectiveness of the approach also in comparison with the classical one are given

numerical solution of reaction diffusion systems of λ ω type by trigonometrically fitted methods the numerical solution of reaction diffusion equations of type which are known to possess a one parameter family of periodic plane wave solutions is object of this paper due to the periodic character of such solutions a special purpose numerical integration is here proposed based on adapted finite differences the adaptation occurs at the level of the problem by a suitable spatial semi discretization based on trigonometrically fitted finite differences numerical experiments confirming the effectiveness of the approach are given

a model for the transition to baroclinic chaos abstract the transition to chaos in a 2 layer baroclinic fluid with a slightly different viscosity in each layer is investigated a low order model is constructed by truncating the quasi geostrophic partial differential equations to include one azimuthal wave in each layer while retaining a general zonal flow correction when the viscosities are equal the model reduces to a modified form of the lorenz equations and displays a cusped return map for slightly different viscosities the transition to chaos follows the feigenbaum scenario the return map in the chaotic regime is nearly parabolic and highly contracted multiple attractors are found for the same control parameters the predicted route to chaos is similar to that observed in laboratory experiments although significant quantitative differences remain

relative normal modes for nonlinear hamiltonian systems an estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a hamiltonian system with continuous symmetry is given this result constitutes a generalization to the hamiltonian symmetric framework of a classical result by weinstein and moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium the estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighboring the stable relative equilibrium in question and with any prefixed spatiotemporal isotropy subgroup moreover it is easily computable in particular examples it is interesting to see how in our result the existence of non trivial relative periodic orbits requires generic conditions on the higher order terms of the taylor expansion of the hamiltonian function in contrast with the purely quadratic requirements of the weinstein moser theorem which emphasizes the highly non linear character of the relatively periodic dynamical objects

exponential fitted runge kutta methods of collocation type fixed or variable knot points two different classes of exponential fitted runge kutta collocation methods are considered methods with fixed points and methods with frequency dependent points for both cases we have obtained extensions of the classical two stage gauss radauiia and lobattoiiia methods numerical examples reveal important differences between both approaches

above the glauberman correspondence let s be a finite solvable group and suppose s acts on the finite group n and they have coprime orders then the celebrated glauberman correspondence provides a natural bijection from the set irrs n of irreducible characters of n which are invariant under the action of s to the set irr cn s of all irreducible characters of the centralizer of s in n suppose further that the semidirect product sn is a normal subgroup of a finite group g let irrs n and let irr cn s be its glauberman correspondent we prove that there is a bijection with good compatibility properties from the set irr g of the irreducible characters of g above to irr ng s such that in the case when s is a p group for some prime p it preserves fields of values and schur indices over qp the field of p adic numbers using this result we also prove a strengthening of the mckay conjecture for all p solvable groups

a new matrix method for solving two dimensional time dependent diffusion equations with dirichlet boundary conditions abstract this paper is devoted to develop a new matrix scheme for solving two dimensional time dependent diffusion equations with dirichlet boundary conditions we first transform these equations into equivalent integro partial differential equations pdes such these integro pdes contain both of the initial and boundary conditions and can be solved numerically in a more appropriate manner subsequently all the existing known and unknown functions in the latter equations are approximated by bernoulli polynomials and operational matrices of differentiation and integration together with the completeness of these polynomials can be used to reduce the integro pdes into the associated algebraic generalized sylvester equations for solving these algebraic equations an efficient krylov subspace iterative method i e bicgstab is implemented two numerical examples are given to demonstrate the efficiency accuracy and versatility of the proposed method

exponentially fitted methods applied to fourth order boundary value problems fourth order boundary value problems are solved by means of exponentially fitted methods of different orders these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe special attention is paid to the expression for the error term and to the choice of the parameter in order to make the error as small as possible some numerical examples are given to illustrate the practical implementation issues of these methods

trigonometrically fitted nonlinear two step methods for solving second order oscillatory ivps abstract the construction of a class of nonlinear two step methods for solving second order oscillatory ivps is analyzed these methods are exact for the linear space generated by the set of functions sin t cos t trigonometrically fitted methods but they do not require a previous knowledge of the frequency or a good approximation of it some trigonometrically fitted nonlinear two step schemes with algebraic order up to four are derived and their stability and phase properties are analyzed it is shown that the new schemes can be efficiently implemented by using a special vector operation the vector product and quotient the numerical experiments carried out show that the new nonlinear two step schemes are more efficient than other standard and nonlinear methods proposed in the scientific literature for solving second order oscillatory differential systems

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

exponentially fitted methods and their stability functions we investigate the properties of stability functions of exponentially fitted runge kutta methods and we show that it is possible to some extent to determine the stability function of a method without actually constructing the method itself to focus attention examples are given for the case of one stage methods we also make the connection with so called integrating factor methods and exponential collocation methods various approaches are given to construct these methods

relative equilibria of point vortices on the sphere we prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non rotating sphere the proofs use the rotational symmetry group so 3 and the resulting conservation laws the time reversing reflectional symmetries in o 3 and the finite symmetry group of permutations of identical vortices results include both global existence theorems and local results on bifurcations from equilibria a more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis the paper ends with discussions of the bifurcation diagrams for systems of 3 6 identical vortices

unstable recurrent patterns in kuramoto sivashinsky dynamics we undertake an exploration of recurrent patterns in the antisymmetric subspace of the one dimensional kuramoto sivashinsky system for a small but already rather turbulent system the long time dynamics takes place on a low dimensional invariant manifold a set of equilibria offers a coarse geometrical partition of this manifold the newton descent method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions the attracting set appears surprisingly thin its backbone consists of several smale horseshoe repellers well approximated by intrinsic local one dimensional return maps each with an approximate symbolic dynamics the dynamics appears decomposable into chaotic dynamics within such local repellers interspersed by rapid jumps between them

dynamics of spiral waves on unbounded domains using center manifold reductions abstract an equivariant center manifold reduction near relative equilibria ofg equivariant semiflows on banach spaces is presented in contrast to previous results the lie groupginduces a strongly continuous group action on the underlying function space in fact gmay act discontinuously the results are applied to bifurcations of stable patterns arising in reaction diffusion systems on the plane or in three space modeling chemical systems such as catalysis on platinum surfaces and belousov zhabotinsky reactions these systems are equivariant under the euclidean symmetry group hopf bifurcations from rigidly rotating spiral waves to meandering or drifting waves and from twisted scroll rings are investigated