spatial disorder and pattern formation in lattices of coupled bistable elements abstract the spatio temporal dynamics of discrete lattices of coupled bistable elements is considered it is shown that both regular and chaotic spatial field distributions can be realized depending on parameter values and initial conditions for illustration we provide results for two lattice systems the fitzhugh nagumo model and a network of coupled bistable oscillators for the latter we also prove the existence of phase clusters with phase locking of elements in each cluster

differential geometric methods in variable structure control this article presents a differential geometric approach for the design of sliding modes in non linear variable structure feedback systems coordinate free characterizations of local existence conditions for sliding regimes and a geometric reformulation of some of its most salient features are presented the approach uses basic notions from differential geometry involving vector fields distributions and 1 forms both single and multiple input cases are treated with some illustrative examples

propagation of singularities for semilinear hyperbolic initial boundary value problems in one space dimension on etudie la propagation et la reflexion de la regularite pour des systemes semi lineaires strictement hyperboliques a 1 variable d espace

adjoint derived location and travel time probabilities for a multidimensional groundwater system backward location and travel time probabilities can be used to determine the former location of contamination in an aquifer for a contaminant parcel that was detected in an aquifer the backward location probability describes its position at some time prior to sampling and the backward travel time probability describes the amount of time required for it to travel to the sampling location from some upgradient position these probabilities which can provide information about the source of contamination are related to adjoint states of resident concentration the governing equations of the backward probabilities are adjoints of the forward governing equation e g the advection dispersion equation we derive these backward governing equations and their boundary and final conditions for both location and travel time probabilities in a multidimensional system each governing equation contains the adjoint of the advection dispersion operator and a load term that defines the particular adjoint state probability the load term depends on both the type of probability location or travel time and the sampling device pumping well or monitoring well with which the contamination was detected the adjoint equation can also be used to efficiently determine forward location and travel time probabilities describing the future location of groundwater contamination a feature most useful for delineating pumping well captures zones we illustrate the use of the backward model for obtaining location and travel time probabilities in a hypothetical two dimensional domain

surmounting oscillating barriers path integral approach for weak noise we consider the thermally activated escape of an overdamped brownian particle over a potential barrier in the presence of periodic driving a time dependent path integral formalism is developed which allows us to derive asymptotically exact weak noise expressions for both the instantaneous and the time averaged escape rate our results comprise a conceptually different systematic treatment of the rate prefactor multiplying the exponentially leading arrhenius factor moreover an estimate for the deviations at finite noise strengths is provided and a supersymmetry type property of the time averaged escape rate is verified for piecewise parabolic potentials the rate expression can be evaluated in closed analytical form while in more general cases as exemplified by a cubic potential an action integral remains to be minimized numerically our comparison with very accurate numerical results demonstrates an excellent agreement with the theoretical predictions over a wide range of driving strengths and driving frequencies

neumann problems with time singularities in this paper we study the existence and uniqueness of solutions to a nonlinear neumann problem for a scalar second order ordinary differential equation u atu f t u u where a 0 and f t x y satisfies the local caratheodory conditions on 0 t xrxr

robust tracking in nonlinear systems a nonlinear feedback multivariable controller is used to implement multivariable tracking in a nonlinear system the tracking error is measured by a general function of system state and the input command the controller is robust in the sense that the tracking error is ultimately bounded in the presence of modeling errors free parameters which affect the form of the controller allow flexibility in determining such factors as the size of the ultimate bound the rate of error decay excursion of the control conditions on the class of modeling errors and the level of system gain restrictive assumptions on the structure of the model and the modeling errors are required but they are treated in a transformation framework which allows the generalization of similar conditions which appear in the prior literature these assumptions hold for robotic manipulators this application is investigated at some length and it appears that the resulting control scheme may have advantages over others which have been proposed in the robotics literature

holder continuity of solutions to an anisotropic elliptic equation abstract for a class of anisotropic quasi linear elliptic equations with measurable coefficients following dibenedetto s intrinsic rescaling method we prove the holder continuity of solutions under the condition for which only local boundedness was known

a singular nonlinear differential equation arising in the homann flow abstract the existence and uniqueness of a positive solution of the singular nonlinear boundary value problem y 2 y 1 2 1 x 2 y 0 0 x 1 y 0 1 2 and y 1 0 which arises in the homann flow are studied

globally valid adaptive controllers of mechanical systems the adaptive controller design of a large class of mechanical systems is addressed in this paper the configuration space for the attitude of a vehicle is modelled as so 3 namely the rotation matrix group to avoid singularity problems some studies adopt the representation of unit quaternions for modeling system dynamics and for the design of controllers however the complicated formulation of quatrains makes the application of this method limited to simple mechanical systems here the strategy is to carry out the modeling and analysis in terms of the global representations of jet bundles of so 3 while the unit quaternion is chosen to implement the controller for its least number of parameters four to represent so 3 globally based on the transformation between the rotation matrix and the quaternion the method explored in this paper gives rise to an adaptive controller in a clean and efficient way

periodic solutions for damped differential equations with a weak repulsive singularity abstract this paper deals with the existence of positive t periodic solutions for the following damped differential equation x p t x q t x f t x c t where p q c l 1 r are t periodic functions and f car r r r is t periodic in the first variable we will prove that a weak repulsive singularity enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem

a singular nonlinear boundary value problem membrane response of a spherical cap the nonlinear boundary value problem y n x 2 32y 2 lambda 2 8 quad 0 0 nu 0 are constants models the large deflection membrane response of a spherical cap and has been solved numerically by various techniques in the engineering literature we prove existence and uniqueness of a solution of this problem provide qualitative information about the solution and use this information to suggest an approach to numerical computation

structural stability for variable exponent elliptic problems ii the p u laplacian and coupled problems abstract we study well posedness for elliptic problems under the form b u div a x u u f where a satisfies the classical leray lions assumptions with an exponent p that may depend both on the space variable x and on the unknown solution u a prototype case is the equation u div u p u 2 u f we have to assume that inf x z r p x z is greater than the space dimension n then under mild regularity assumptions on and on the nonlinearities we show that the associated solution operator is an order preserving contraction in l 1 in addition existence analysis for a sample coupled system for unknowns u v involving the p v laplacian of u is carried out coupled elliptic systems with similar structure appear in applications e g in modelling of stationary thermorheological fluids

elliptic problems with lack of compactness via a new fixed point theorem abstract this paper provides a new fixed point theorem for increasing self mappings g b b of a closed ball b x where x is a banach semilattice which is reflexive or has a weakly fully regular order cone x by means of this fixed point theorem we are able to establish existence results of elliptic problems with lack of compactness

periodic bvps in odes with time singularities abstract in this paper we show the existence of solutions to a nonlinear singular second order ordinary differential equation u t a t u t f t u t u t subject to periodic boundary conditions where a 0 is a given constant 0 is a parameter and the nonlinearity f t x y satisfies the local caratheodory conditions on 0 t r r here we study the case that a well ordered pair of lower and upper functions does not exist and therefore the underlying problem cannot be treated by well known standard techniques instead we assume the existence of constant lower and upper functions having opposite order analytical results are illustrated by means of numerical experiments

exact solutions to a class of feedback systems on so n this paper provides a novel approach to the problem of attitude tracking for a class of almost globally asymptotically stable feedback laws on so n the closed loop systems are solved exactly for the rotation matrices as explicit functions of time the initial conditions and the gain parameters of the control laws the exact solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix applications of these results are found in model predictive control problems where detailed insight into the transient attitude dynamics is utilized to approximately complete a task of secondary importance knowledge of the future trajectory of the states can also be used as an alternative to the zero order hold in systems where the attitude is sampled at discrete time instances

an efficient algorithm of logarithmic transformation to hirota bilinear form of kdv type bilinear equation abstract in this paper an efficient algorithm of logarithmic transformation to hirota bilinear form of the kdv type bilinear equation is established in the algorithm some properties of hirota operator and logarithmic transformation are successfully applied which helps to prove that the linear terms of the nonlinear partial differential equation play a crucial role in finding the hirota bilinear form experimented with various integro differential equations our algorithm is proven to be more efficient than the algorithm referred by zhou fu and li in getting the hirota bilinear form especially in achieving the coefficient of the logarithmic transformation

layered incompressible fluid flow equations in the limit of low mach number and strong stratification abstract we investigate the low mach number regimes of the complete navier stokes fourier system which describes an evolution of a viscous heat conducting gas with a heat source term interpreted as a radiation cooling in models of atmospheric flows this investigation is performed in the context of weak solutions on an arbitrary large time interval and for the ill prepared initial data the physically expected limiting equations as the mach and froude numbers tend to zero at the same rate are layered incompressible flow equations with strong stratification we give a rigorous mathematical proof of this result

darboux theory of integrability in cn taking into account the multiplicity abstract darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces called now the darboux theory of integrability in 1979 jouanolou successfully improved the darboux theory of integrability characterizing the existence of rational first integrals for this he used sophisticated tools of algebraic geometry the aim of this paper is to improve the classical result of darboux and the new one of jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces additionally our proof of the improved result of jouanolou is much simpler and elementary than the original one some examples show that the improved darboux theory of integrability with multiplicity is much useful than the classical one

numerical solutions of linear mixed boundary value problems using stochastic representations the application of ito s formula induces some probabilistic representations of solutions of deterministic linear problems with boundary conditions of dirichlet neumann fourier and mixed types these representations are used to establish some easily implementable algorithms which compute an approximate solution by means of simulation of reflected random walks they do not require selected configurations at the neighbourhood of the domain boundary nor a discretization mesh the associated simulation methods are obtained and applied to problems for each class of boundary conditions numerical experiments with distributed source in two or three dimensional geometries and computational results with estimation of error are reported 1997 by john wiley sons ltd

on two classes of autonomous third order nonlinear ordinary differential equations abstract two autonomous nonlinear third order ordinary differential equations whose dynamics can be represented by second order nonlinear ordinary differential equations for the first order derivative of the solution are studied analytically and numerically the analytical study includes both the obtention of closed form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations it is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first order derivative

solving nonlinear recursions a general method to map a polynomial recursion on a matrix linear one is suggested the solution of the recursion is represented as a product of a matrix multiplied by the vector of initial values this matrix is product of transfer matrices whose elements depend only on the polynomial and not on the initial conditions the method is valid for systems of polynomial recursions and for polynomial recursions of arbitrary order the only restriction on these recurrent relations is that the highest order term can be written in explicit form as a function of the lower order terms existence of a normal form a continuous analog of this method is described as well

nonlinear generalised dynamic inversion for aircraft manoeuvring control the nonlinear control problem of aircraft trajectory tracking is tackled in the framework of multiple linear time varying constrained control using the newly developed paradigm of generalised dynamic inversion the time differential forms of the multiple constraints encapsulate the control objectives and are inverted to obtain the reference trajectory realising control law the inversion process utilises the moore penrose generalised inverse and the associated nullspace projection and it predictably involves the problematic generalised inversion singularity thus a singularity avoidance scheme based on a new type of dynamically scaled generalised inverses is introduced that guarantees both asymptotically stable tracking and singularity avoidance the steady state closed loop system allows for two inherently noninterfering control actions working towards a unified goal to exploit the aircraft s control authority over the entire state space one control action is performed by the particular part of the c

group analysis of a conformal perfect fluid spacetime we find new exact solutions of the einstein field equations for a perfect fluid metric conformal to a spacetime of type d in the petrov classification scheme we analyse the complete system of equations using lie group analysis while previous work was confined to conformal factors of the form u u t x we investigate the complete situation u u t x y z as well as an auxiliary integrability condition new classes of solutions are generated via the symmetry generators the resulting solutions are examined for physical plausibility expressions for the energy density and pressure are obtained explicitly and empirical results suggest that these dynamical quantities are positive as expected

conley type index applied to hamiltonian inclusions in the paper we develop the theory of a cohomological index of the conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional hilbert space an application to the existence of periodic orbits to asymptotically linear hamiltonian inclusions is presented

stability and convergence of staggered runge kutta schemes for semilinear wave equations a staggered runge kutta staggered rk scheme is a runge kutta type scheme using a time staggered grid as proposed by ghrist et al in 2000 6 afterwards verwer in two papers investigated the efficiency of a scheme proposed by ghrist et al 6 for linear wave equations we study stability and convergence properties of this scheme for semilinear wave equations in particular we prove convergence of a fully discrete scheme obtained by applying the staggered rk scheme to the mol approximation of the equation

fast algorithms for spectral collocation with non periodic boundary conditions we present a method for the numerical solution of partial differential equations using spectral collocation by employing a structured representation of linear operators we are able to use fast algorithms without being restricted to periodic boundary conditions the underlying ideas are introduced and developed in the context of linearly implicit methods for stiff equations we show how different boundary conditions may be applied and illustrate the technique on the allen cahn equation and the diffusion equation

functional calculus for semigroup generators via transference abstract in this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators in particular it is proved that if a generates a c 0 semigroup on a hilbert space then for each 0 the operator a has a bounded calculus for the closed ideal of bounded holomorphic functions on a sufficiently large right half plane that satisfy f z o e re z as z the bound of this calculus grows at most logarithmically as 0 as a consequence f a is a bounded operator for each holomorphic function f on a right half plane with polynomial decay at then we show that each semigroup generator has a so called strong m bounded calculus for all m n and that this property characterizes semigroup generators similar results are obtained if the underlying banach space is a umd space upon restriction to so called bounded semigroups the hilbert space results actually hold in general banach spaces

differential and integral equations with henstock kurzweil integrable functions abstract in this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing henstock kurzweil integrable functions

on the solution of the abel equation of the second kind by the shifted chebyshev polynomials abstract this paper presents a new approximate method of abel differential equation by using the shifted chebyshev expansion of the unknown function abel differential equation is approximately transformed to a system of nonlinear equations for the unknown coefficients a desired solution can be determined by solving the resulting nonlinear system this method gives a simple and closed form of approximate solution of abel differential equation the solution is calculated in the form of a series with easily computable components the numerical results show the effectiveness of the method for this type of equation comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate

extremal solutions for nonlinear functional ϕ laplacian impulsive equations abstract in this paper we deal with some nonlinear laplacian second order boundary value problems with impulses subject to a general type of boundary value conditions which cover dirichlet and periodic boundary data as particular cases our approach is that of upper and lower solutions together with growth restrictions of nagumo s type the impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear in both cases functional dependence on the solution is allowed

explicit magnus expansions for nonlinear equations in this paper we develop and analyse new explicit magnus expansions for the nonlinear equation y a t y y defined in a matrix lie group in particular integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules the structure of the algorithm allows us to change the step size and even the order during the integration process thus improving its efficiency several examples are considered including isospectral flows and highly oscillatory nonlinear differential equations

the application of runge kutta schemes to singular initial value problems a theory for explicit runge kutta schemes applied to the initial value problem for a first order system of differential equations with a singularity of the first kind is developed it is shown that in general the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems 1 introduction in this paper we examine the application of explicit runge kutta schemes to the numerical solution of the problem

robustness of nonlinear state feedback a survey a survey of robustness of nonlinear state feedback is given for series perturbations there are fairly complete results showing that under mild restrictions an optimal controller can tolerate an infinite increase in gain for gain reductions there are some results for systems linear in the control in particular there is a 50 reduction tolerance if the control penalty is quadratic usually the optimal controller cannot be computed exactly there are some results showing the effects of truncation on the robustness essentially robustness is maintained but in a reduced computable part of the state space

one dimensional quaternion homogeneous polynomial differential equations in this paper we study the dynamics of real four dimensional polynomial differential equations of the forms q aqkq m or q qkaqm where q a are quaternions q is the conjugate of q and k and m are non negative integers in many parts of the work the quaternion structure is used along the proofs

riemannian geometry of quantum computation abstract a review is given of some recent developments in the differential geometry of quantum computation for which the quantum evolution is described by the special unitary unimodular group s u 2 n using the lie algebra s u 2 n detailed derivations are given of a useful riemannian geometry of s u 2 n including the connection and the geodesic equation for minimal complexity quantum computations

spatiotemporal accessible solitons in fractional dimensions we report solutions for solitons of the accessible type in globally nonlocal nonlinear media of fractional dimension fd viz for self trapped modes in the space of effective dimension 2 d 3 with harmonic oscillator potential whose strength is proportional to the total power of the wave field the solutions are categorized by a combination of radial orbital and azimuthal quantum numbers n l m they feature coaxial sets of vortical and necklace shaped rings of different orders and can be exactly written in terms of special functions that include gegenbauer polynomials associated laguerre polynomials and associated legendre functions the validity of these solutions is verified by direct simulations the model can be realized in various physical settings emulated by fd spaces in particular it applies to excitons trapped in quantum wells

nonlinear second order equations with functional implicit impulses and nonlinear functional boundary conditions in this paper we deal with nonlinear second order boundary value problems with impulses the impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear in both cases functional dependence on the solution is allowed the existence results follow from the existence of a pair of well ordered lower and upper solutions

symmetry properties and solutions of nonlinear dispersive thin film equations in three dimensions the lie point symmetries are obtained for each one of two nonlinear dispersive thin film equations in 2 1 variables by means of the invariants of the one and two dimensional subgroups all reductions to ordinary differential equations and to partial differential equations in two variables are obtained solutions are provided for a number of the reduced equations it is also ascertained that neither one of the two considered thin film equations is integrable in some cases solitary and periodic wave solutions having curved wavecrests are obtained

homogenization of surface and length energies for spin systems abstract we study the homogenization of lattice energies related to ising systems of the form e e u i j c i j e u i u j with u i a spin variable indexed on the portion of a cubic lattice e z d by computing their limit in the framework of surface energies in a bv setting we introduce a notion of homogenizability of the system c i j e that allows to treat periodic almost periodic and random statistically homogeneous models the latter in dimension two when the coefficients are positive ferromagnetic energies in which case the limit energy is finite on bv 1 and takes the form f u u 1 d h d 1 is the normal to u 1 where is characterized by an asymptotic formula in the random case can be expressed in terms of first passage percolation characteristics the result is extended to coefficients with varying sign under the assumption that the areas where the energies are antiferromagnetic are well separated finally we prove a dual result for discrete curves

long time dynamics of an extensible plate equation with thermal memory abstract this work is concerned with long time dynamics of solutions of extensible plate equations with thermal memory the problem corresponds to a model of thermoelasticity based on a theory of non fourier heat flux by considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system and guarantees the existence of a finite dimensional global attractor in addition the existence of exponential attractors is also considered

exact nonlinear control of large angle rotational maneuvers the rigid body attitude control problem with external torques is transformed into an equivalent linear form implementable by three double integrators the linearizing transformations themselves are formulated in vector algebra requiring no integrators for implementation it is thereby shown that optimal command generation for fast slewing maneuvers can be carried out exactly in the transformed system together with regulator design without gain scheduling for correction of unmodeled disturbances in particular a general rest to rest maneuver is computed in closed form and coupled with a previously obtained exact detumbling maneuver so that arbitrary initial conditions can be accommodated an illustrative simulation is appended

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

multiple solutions for nonlocal system of p x q x kirchhoff type abstract in this paper we study the existence of solutions for a neumann elliptic system with variable exponents under some suitable conditions and by applying an equivalent variational approach to a recent ricceri s three critical points theorem we established the existence of at least three weak solutions

global transformations of nonlinear systems recent results have established necessary and sufficient conditions for a nonlinear system of the form dot x t f x t u t g x t with f 0 0 to be locally equivalent in a neighborhood of the origin in rnto a controllable linear system we combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system in doing so we introduce a technique for constructing a transformation under the assumptions that g ldot f dotg ad n 1 f ldotg span an n dimensional space and that g ldot f ldot g ad n 2 f ldotg is an involutive set

blowup and solitary wave solutions with ring profiles of two component nonlinear schrodinger systems abstract blowup ring profiles have been investigated by finding non vortex blowup solutions of nonlinear schrodinger equations nlses cf fibich et al 2005 7 and fibich et al 2007 8 however those solutions have infinite l 2 norm so one may not maintain the ring profile all the way up to the singularity to find h 1 non vortex blowup solutions with ring profiles we study the blowup solutions of two component systems of nlses with nonlinear coefficients and j j 1 2 when 0 and 1 2 0 the two component system can be transformed into a multi scale system with fast and slow variables which may produce h 1 blowup solutions with non vortex ring profiles we use the localized energy method with symmetry reduction to construct these solutions rigorously on the other hand these solutions may describe steady non vortex bright ring solitons various types of ring profiles including m ring and ring ring profiles are presented by numerical solutions

collocation for singular boundary value problems of second order the collocation methods for solving linear boundary value problems with a singularity of the first kind are investigated the stability and convergence results are established particular attention is paid to the superconvergence properties of the methods which are different as in the classical case when the singularity is not present the results are extended to the nonlinear case and illustrated by numerical examples

structural stability for variable exponent elliptic problems i the p x laplacian kind problems abstract we study the structural stability i e the continuous dependence on coefficients of solutions of the elliptic problems under the form b u n div a n x u n f n the equation is set in a bounded domain of r n and supplied with the homogeneous dirichlet boundary condition on here b is a non decreasing function on r and a n x n is a family of applications which verifies the classical leray lions hypotheses but with a variable summability exponent p n x 1 p p n p the need for varying p x arises for instance in the numerical analysis of the p x laplacian problem uniqueness and existence for these problems are well understood by now we apply the stability properties to further generalize the existence results the continuous dependence result we prove is valid for weak and for renormalized solutions notice that besides being interesting on its own the renormalized solutions framework also permits us to deduce optimal convergence results for the weak solutions our technique avoids the use of a fixed duality framework like the w 0 1 p x w 1 p x duality and thus it is suitable for the study of problems where the summability exponent p also depends on the unknown solution itself in a local or in a non local way the sequel of this paper will be concerned with well posedness of some p u laplacian kind problems and with existence of solutions to elliptic systems with variable solution dependent growth exponent

existence and comparison results for operator and differential equations in abstract spaces in this paper we apply the chain methods developed in heikkila and lakshmikantham 1994 to obtain new fixed point theorems and new existence and comparison results for operator equations in partially ordered sets these results are then applied to discontinuous implicit functional differential equations in ordered banach spaces

existence results for operator and differential equations and inclusions abstract this paper contains a brief summary of existence results for operator and differential equations and inclusions derived recently via a recursion method and generalized iteration methods introduced in 11 12 23

complete sets of invariants for dynamical systems that admit a separation of variables consider a classical hamiltonian h in n dimensions consisting of a kinetic energy term plus a potential if the associated hamilton jacobi equation admits an orthogonal separation of variables then it is possible to generate algorithmically a canonical basis q p where p1 h p2 pn are the other second order constants of the motion associated with the separable coordinates and qi qj pi pj 0 qi pj ij the 2n 1 functions q2 qn p1 pn form a basis for the invariants we show how to determine for exactly which spaces and potentials the invariant qj is a polynomial in the original momenta we shed light on the general question of exactly when the hamiltonian admits a constant of the motion that is polynomial in the momenta for n 2 we go further and consider all cases where the hamilton jacobi equation admits a second order constant of the motion not necessarily associated with orthogonal separable coordinates or even separable coordinates at all in each of these cases we construct an addition

planar nonautonomous polynomial equations the riccati equation abstract we give a few sufficient conditions for the existence of two periodic solutions of the riccati ordinary differential equation in the plane we give also examples of the equation without periodic solutions

a boundary element numerical scheme for the two dimensional convection diffusion equation in this study the theoretical and numerical fundamentals of biem techniques for the two dimensional convection diffusion problem are presented after an extended presentation of the basic integral formulation the discretizing and iterative processes for its resolution are introduced interesting remarks on general expressions versus previously published particularized results are worth mentioning a numerical solution scheme is provided which has been completely developed and designed to the physical problem posed a novel scheme based in the simultaneous solving of the potential and the gradient of the potential boundary integral equations is included a diversity of problems is tested to prove the possibilities of the method copyright 2008 john wiley sons ltd

asymptotically correct error estimation for collocation methods applied to singular boundary value problems we discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind as an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts this requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class

a globally convergent angular velocity observer for rigid body motion the problem of obtaining the angular velocity of a rigid body from orientation and torque measurements only without noisy numerical differentiation is considered a novel angular velocity angular momentum observer for rigid body motion is presented using euler quaternions and a mechanical energy function approach it is shown that the observer estimates converge globally and that the convergence is eventually exponential it is hoped that a mechanical energy function approach to rigid body control can be combined with the observer presented to lead to a globally stable nonlinear observer based rigid body controller in which the observer and controller errors can be separated in much the same way as one can separate controller and observer poles in the output feedback controllers of linear system theory

asymptotic integration of nonlinear ϕ laplacian differential equations abstract the aim of the present paper is to study the existence of solutions to initial values problems for a laplace like operator we generalize the results of agarwal 2007 1 section 4 theorem 3 and the result of philos 2004 20 theorem 5 to the laplacian like problems

the monotone method for neumann functional differential equations with upper and lower solutions in the reverse order abstract in this paper we show that the monotone technique produces two monotone sequences that converge uniformly to extremal solutions of second order functional differential equations and laplacian equations with neumann boundary value conditions moreover we obtain existence results assuming upper and lower solutions in the reverse order

linearization of attitude control error dynamics some useful properties of direction cosine and quaternion attitude formulations that include specification of attitude error and the inversion of rigid body rotational dynamics are reviewed the inversion procedure is then applied to a measure of attitude error to realize a new model follower control system that exhibits linear attitude error dynamics error analyses and simulation results for spacecraft attitude control systems are presented to demonstrate the more robust performance obtainable from an exact linear error formulation over that obtained from either direction cosine or quaternion formulations with simple linear feedback control laws

the attitude control problem a general framework for the analysis of the attitude tracking control problem for a rigid body is presented a large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body the controllers share the common structure of a proportional derivative feedback plus some feedforward which can be zero the model independent case the coriolis torque compensation or an adaptive compensation these controller structures are compared in terms of the requirement on the a priori model information guaranteed transient performance and robustness the global stability of the luh walker paul robot end effector controller is also analyzed in this framework

orientation control on so 3 with piecewise sinusoids this paper addresses rigid body orientation control for the case that angular velocity is restricted to a class of piecewise sinusoidal signals the orientation kinematics are represented by the group of rotation matrices so 3 we present a new closed form solution of the rotation matrix kinematics and we use this solution to prove controllability we then present kinematic level orientation feedback controllers for setpoint tracking and command following

on a singular boundary value problem arising in the theory of shallow membrane caps abstract we investigate the following singular boundary value problem which originates from the theory of shallow membrane caps t 3 u t t 3 1 8 u 2 t a 0 u t b 0 t 2 4 0 lim t 0 t 3 u t 0 u 1 0 where a 0 b 0 and are given constants we show the existence of a positive solution to the above problem by means of a generalized lower and upper functions method involving limiting processes we illustrate the theory by numerical experiments in which we used the new version of the matlab code sbvp based on polynomial collocation to approximate the solution of the membrane problem

on an instability exhibited by the ballistic diffusive heat conduction model of xu and hu we show that the constitutive relation for the thermal flux proposed by xu hu 2011 admits an unconditional instability we also highlight the difference between mathematical models containing delay and those that include relaxation effects

the 16th hilbert problem on algebraic limit cycles abstract for real planar polynomial differential systems there appeared a simple version of the 16th hilbert problem on algebraic limit cycles is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m in j llibre r ramirez n sadovskaia on the 16th hilbert problem for algebraic limit cycles j differential equations 248 2010 1401 1409 llibre ramirez and sadovskaia solved the problem providing an exact upper bound in the case of invariant algebraic curves generic for the vector fields and they posed the following conjecture is 1 m 1 m 2 2 the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have in this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves this result includes the llibre et al s as a special one for the polynomial vector fields having only non dicritical invariant algebraic curves we answer the simple version of the 16th hilbert problem

a comparison between hmlp and hrbf for attitude control in this paper the problem of controlling the attitude of a rigid body such as a spacecraft in three dimensional space is approached by introducing two new control strategies developed in hypercomplex algebra the proposed approaches are based on two parallel controllers both derived in quaternion algebra the first is a feedback controller of the proportional derivative pd type while the second is a feedforward controller which is implemented either by means of a hypercomplex multilayer perceptron hmlp neural network or by means of a hypercomplex radial basis function hrbf neural network several simulations show the performance of the two approaches the results are also compared with a classical pd controller and with an adaptive controller showing the improvements obtained by using neural networks especially when an external disturbance acts on the rigid body in particular the hmlp network gave better results when considering trajectories not presented during the learning phase

computation of lie transfer maps for perturbed hamiltonian systems time evolution of a hamiltonian system can be viewed as a canonical transformation therefore perturbations giving rise to near identity deviations from an unperturbed solution can be represented by products of lie transformations or together with the unperturbed solution lie transfer maps in this paper i broaden the applicability to all perturbed hamiltonian systems the method of dragt and finn and subsequent co workers who developed a representation using a product of lie transformations factored by phase space variable order in the present paper perturbation parameters are no longer necessarily associated with the phase space variables this method treats both internal and external perturbations on an equal footing and a rank is assigned to each variable to reflect the degree of perturbation it represents with the companion program pglt analytic development of the lie transfer maps is relatively easy for many systems

existence results for operator equations in abstract spaces and an application in this paper we apply the recursion method presented in heikkila and lakshmikantham 1994 to derive fixed point theorems and existence results for operator equations in partially ordered sets the obtained results combined with related results of heikkila 2002 are then applied to operator equations in ordered hilbert spaces and to a functional elliptic boundary value problem

two dimensional dynamical system associated with abel s nonlinear differential equation a two dimensional dynamical system is proposed that is described by a pair of nonlinear ordinary differential equations odes with a complex parameter it reduces to abel s nonlinear ode of the first kind by an appropriate transformation using this fact the properties of solutions are investigated in detail with the aid of numerical computations it is found that various types of bifurcation phenomena occur depending on the values of the parameter in particular the solution is shown to blow up in finite time under certain conditions in order to visualize the behaviors of dynamical motions the trajectories of solutions are depicted in the plane finally a discussion is made on some generalizations of the proposed system