long range spatial correlations for anisotropic zero range processes the spatial correlations are investigated for a homogeneous system of indistinguishable particles undergoing stochastic anisotropic hopping dynamics on the d dimensional lattice d or 2 the interaction is zero range i e the rate at which particles leave a given site only depends on the occupation number at that site a series expansion around the independent particle system is given for the equal time correlations and is shown to converge for small times t the formal t to infinity limiting expansion is analysed termwise from which a quadrupole type decay approximately r d is derived for the stationary two points function eta 0 eta r eta 0 2 this phenomenon of self organized criticality is a direct consequence of the anisotropy causing the system to violate the condition of detailed balance combined with the conservation law forcing a diffusive decay approximately t d2 of the temporal correlations

a necessary condition for feedback stabilization abstract we give a new necessary condition for the existence of a static or dynamic continuous asymptotically stabilizing feedback control for the system x dot f x u we apply our condition to give an example of a system in r 3 which is small time locally controllable satisfies brockett s second and third condition and cannot be locally asymptotically stabilized by employing a static or dynamic continuous feedback law

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

lattice gas model in random medium and open boundaries hydrodynamic and relaxation to the steady state we consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i i d bounded random variables in a bounded domain in contact with particles reservoir at different densities we show in dimensions d 3 that the rescaled empirical density field almost surely with respect to the random field converges to the unique weak solution of a quasilinear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir further we show that the rescaled empirical density field in the stationary regime almost surely with respect to the random field converges to the solution of the associated stationary transport equation

random walk of second class particles in product shock measures we consider shock measures in a class of conserving stochastic particle systems on these shock measures have a product structure with a step like density profile and include a second class particle at the shock position we show for the asymmetric simple exclusion process for the exponential bricklayers process and for a generalized zero range process that under certain conditions these shocks and therefore the second class particles perform a simple random walk some previous results including random walks of product shock measures and stationary shock measures seen from a second class particle are direct consequences of our more general theorem multiple shocks can also be handled easily in this framework similar shock structure is also found in a nonconserving model the branching coalescing random walk where the role of the second class particle is played by the rightmost or leftmost particle

similarity reductions from extended painleve expansions for nonintegrable evolution equations abstract the use of singular manifold expansions to find exact solutions to nonintegrable evolution equations is extended to include arbitrary resonance coefficients in such a way as to make the resulting infinite series exactly resummable the technique involves the use of a rescaling ansatz analogous to that used to analyze the psi series of nonintegrable ordinary differential equations the result is a similarity reduction of the equation in which the constrained singular manifold plays the role of a similarity variable the method is capable of yielding new solutions corresponding to either classical or nonclassical conditional lie symmetries

small deviations from local equilibrium for a process which exhibits hydrodynamical behavior i the symmetric simple exclusion process on with sources at l l e is considered the stationary measure l is studied in the limit asl diverges the first order correction to its limit is proven to be of order 1 l and it is explicitly computed the result is in agreement with the analysis of the model from the hydrodynamical point of view initiated in ref 1

quenched convergence and strong local equilibrium for asymmetric zero range process with site disorder we study asymmetric zero range processes on z with nearest neighbour jumps and site disorder the jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles we prove quenched strong local equilibrium at subcritical and critical hydrodynamic densities and dynamic local loss of mass at supercritical hydrodynamic densities our results do not assume starting from local gibbs states as byproducts of these results we prove convergence of the process from given initial configurations with an asymptotic density of particles to the left of the origin in particular we relax the weak convexity assumption of 7 8 for the escape of mass property 1 msc 2010 subject classification 60k35 82c22

hydrodynamics of stationary non equilibrium states for some stochastic lattice gas models we consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior which conserve the particle number and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials the unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non reversible we study the structure of the stationary measure in the hydrodynamic limit as the microscopic lattice size goes to infinity in particular we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by fick s law

escape from the unstable equilibrium in a random process with infinitely many interacting particles we consider a one dimensional version of the model introduced in ref 1 at each site of z there is a particle with spin 1 particles move according to the stirring process and spins change according to the glauber dynamics in the hydrodynamical limit with the stirring process suitably speeded up the local magnetic density m r is proven in ref 1 to satisfy the reaction diffusion equation 8tmt r 1 2 rmt r v mt v m 89 4 and fl 0 c and fl being determined by the parameters of the glauber dynamics in the present paper we consider an initial state with zero magnetization mo r o we then prove that at long times before taking the hydrodynamical limit the evolution departs from that predicted by and that the microscopic state becomes a nontrivial mixture of states with different magnetizations

stabilizability does not imply homogeneous stabilizability for controllable homogeneous systems this paper presents an example of a homogeneous planar controllable system which is stabilizable while not stabilizable by homogeneous feedback addition of an integrator to the system provides a three dimensional it affine system with the same properties

entropy of open lattice systems we investigate the behavior of the gibbs shannon entropy of the stationary nonequilibrium measure describing a one dimensional lattice gas of l sites with symmetric exclusion dynamics and in contact with particle reservoirs at different densities in the hydrodynamic scaling limit l the leading order o l behavior of this entropy has been shown by bahadoran to be that of a product measure corresponding to strict local equilibrium we compute the first correction which is o 1 the computation uses a formal expansion of the entropy in terms of truncated correlation functions for this system the k th such correlation is shown to be o l k 1 this entropy correction depends only on the scaled truncated pair correlation which describes the covariance of the density field it coincides in the large l limit with the corresponding correction obtained from a gaussian measure with the same covariance

application of the g g expansion to travelling wave solutions of the broer kaup and the approximate long water wave equations by using the g g expansion proposed recently the travelling wave solutions involving parameters of the broer kaup equations and the approximate long water wave equations are found out the travelling wave solutions are expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions when the parameters are taken as special values the solitary wave solutions are obtained c 2008 elsevier inc all rights reserved

nonequilibrium measures which exhibit a temperature gradient study of a model we give some rules to define measures which could describe heat flow in homogeneous crystals we then study a particular model which is explicitly solvable the one dimensional nearest neighborhood ising model we analyze two cases in the first one the spins at the two boundaries interact with reservoirs at different temperatures in the thermodynamical limit the measure we introduce converges locally to gibbs measures and a temperature profile is so derived we obtain an explicit expression for the thermal conductivity coefficient which depends on the temperature in the second case we study the asymptotic behavior starting from an initial state in which each half of the space is at a different temperature we find again a temperature profile which asymptotically obeys the heat equation with the thermal conductivity coefficient previously derived from a mathematical point of view the analysis of the invariant measure is made possible by studying a time reversed process related to a graphical representation of an associated process this provides us with an explicit formula for then fold correlation function and we study the limiting behavior using both this representation for proving an exchangeability result and a donsker type spacetime renormalization procedure

on the dirichlet problem for asymmetric zero range process on increasing domains in order to obtain hitting time estimates for the asymmetric zero range process azrp on d in dimensions d 3 we characterize the principal eigenvalue of the generator of the azrp with dirichlet boundary on special domains we obtain a donsker varadhan variational representation and show that the corresponding eigenfunction is unique in a natural class of functions

cluster growth and dynamic scaling in a two lane driven diffusive system using high precision monte carlo simulations and a mean field theory we explore coarsening phenomena in a simple driven diffusive system the model is reminiscent of vehicular traffic on a two lane ring road at sufficiently high density the system develops jams clusters which coarsen with time a key parameter is the passing probability for small values of the growing clusters display dynamic scaling with a growth exponent of 2 3 for larger values of the growth exponent must be adjusted suggesting the ordered jammed state is not a genuine phase but rather a finite size effect

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

a further improved extended fan sub equation method for 2 1 dimensional breaking soliton equations in this paper a further improved extended fan sub equation method is used to construct exact solutions of the 2 1 dimensional breaking soliton equations as a result many new and more general non travelling wave and coefficient function solutions are obtained including soliton like solutions triangular like solutions single and combined non degenerate jacobi elliptic wave function like solutions weierstrass elliptic doubly like periodic solutions each of which contains an arbitrary function of two variables the results show the proposed method give new and more general exact solutions more importantly the method with the aid of symbolic computation provides a very effective and powerful mathematical tool for solving a great many non linear partial differential equations in mathematical physics

large deviations for a stochastic model of heat flow we investigate a one dimensional chain of 2n harmonic oscillators in which neighboring sites have their energies redistributed randomly the sites n and n are in contact with thermal reservoirs at different temperature and kipnis et al j statist phys 27 65 74 1982 proved that this model satisfies fourier s law and that in the hydrodynamical scaling limit when n the stationary state has a linear energy density profile bar theta u u 1 1 we derive the large deviation function s u for the probability of finding in the stationary state a profile u different from bar theta u the function s has striking similarities to but also large differences from the corresponding one of the symmetric exclusion process like the latter it is nonlocal and satisfies a variational equation unlike the latter it is not convex and the gaussian normal fluctuations are enhanced rather than suppressed compared to the local equilibrium state we also briefly discuss more general models and find the features common in these two and other models whose s is known

discontinuous condensation transition and nonequivalence of ensembles in a zero range process we study a zero range process where the jump rates do not only depend on the local particle configuration but also on the size of the system rigorous results on the equivalence of ensembles are presented characterizing the occurrence of a condensation transition in contrast to previous results the phase transition is discontinuous and the system exhibits ergodicity breaking and metastable phases this leads to a richer phase diagram including nonequivalence of ensembles in certain phase regions the paper is motivated by results from granular clustering where these features have been observed experimentally

a remark on the hydrodynamics of the zero range processes the nonequilibrium stationary hydrodynamical properties of the symmetric nearest neighbor zero range processes are studied local equilibrium and fourier s law are proven to hold and the bulk diffusion coefficient and the equal time covariance of the limiting nonequilibrium stationary density fluctuations field are computed the result fits with those already known and confirms some conjectures derived from a time dependent macroscopic analysis the very simple proof is based on a result already published but may be not so well known in this context

large deviations of interacting particle systems in infinite volume we prove a large deviation theorem from the hydrodynamical limit for the empirical measure of ginzburg landau and zero range processes in infinite volume starting from deterministic initial configurations in the ginzburg landau case the main tool is the study of the evolution of the h 1 norm and in the zero range case the attractiveness which allows couplings

extended transformed rational function method and applications to complexiton solutions abstract the transformed rational function method provides a systematical and convenient handling of the solution process of nonlinear equations unifying the tanh function type methods the homogeneous balance method the exp function method the mapping method and the f expansion type methods in this paper the transformed rational function method is improved and the extended method is used to obtain complexiton solutions to some nonlinear differential equations

new solitary wave solutions for two nonlinear evolution equations abstract in this paper multiple exp function method is employed to investigate exact multiple wave solutions for 2 1 dimensional potential kadomtsev petviashvili equation and 3 1 dimensional jimbo miwa equation not only already known multiple wave solutions are recovered but also several new or more general multiple wave solutions are obtained

the extended g g expansion method and its applications to the whitham broer kaup like equations and coupled hirota satsuma kdv equations in this paper based on a new general ansatze the extended g g expansion method is proposed to seek exact solutions of nonlinear evolution equations being concise and straightforward this method is applied to construct travelling wave solutions of whitham broer kaup like equations and coupled hirota satsuma kdv equations by using this method new exact solutions involving parameters expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions are obtained when the parameters are taken as special values some solitary wave solutions are derived from the hyperbolic function solutions

thermodynamic transformations of nonequilibrium states we consider a macroscopic system in contact with boundary reservoirs and or under the action of an external field we discuss the case in which the external forcing depends explicitly on time and drives the system from a nonequilibrium state to another one in this case the amount of energy dissipated along the transformation becomes infinite when an unbounded time window is considered following the general proposal by oono and paniconi and using results of the macroscopic fluctuation theory we give a natural definition of a renormalized work we then discuss its thermodynamic relevance by showing that it satisfies a clausius inequality and that quasi static transformations minimize the renormalized work in addition we connect the renormalized work to the quasi potential describing the fluctuations in the stationary nonequilibrium ensemble the latter result provides a characterization of the quasi potential that does not involve rare fluctuations

large deviation of the density profile in the steady state of the open symmetric simple exclusion process we consider an open one dimensional lattice gas on sites i 1 n with particles jumping independently with rate 1 to neighboring interior empty sites the simple symmetric exclusion process the particle fluxes at the left and right boundaries corresponding to exchanges with reservoirs at different chemical potentials create a stationary nonequilibrium state sns with a steady flux of particles through the system the mean density profile in this state which is linear describes the typical behavior of a macroscopic system i e this profile occurs with probability 1 when n the probability of microscopic configurations corresponding to some other profile x x i n has the asymptotic form exp n f f is the large deviation functional in contrast to equilibrium systems for which f eq is just the integral of the appropriately normalized local free energy density the f we find here for the nonequilibrium system is a nonlocal function of this gives rise to the long range correlations in the sns predicted by fluctuating hydrodynamics and suggests similar non local behavior of f in general sns where the long range correlations have been observed experimentally

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

internal dla and the stefan problem generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as poisson sources of particles which then perform symmetric random walks with an attractive zero range interaction until they reach the first site which has been visited by fewer than alpha particles at which point they stop sites on which particles are frozen constitute the occupied set we prove that in appropriate regimes the particle density has a hydrodynamic limit which is the one phase stefan problem this is then used to study the asymptotic behavior of the occupied set in two dimensions when the walks are independent with one source at the origin and alpha 1 we obtain in particular that the occupied set is asymptotically a disc of radius k sqrt t where k is the solution of exp k 2 4 pi k 2 settling a conjecture of lawler bramson and griffeath

the improved g g expansion method and its applications to the broer kaup equations and approximate long water wave equations by introducing a new general ansatze the improved g g expansion method is proposed to construct exact solutions of both broer kaup equations and approximate long water wave equations as a result some new travelling wave solutions involving parameters expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions are obtained when the parameters are taken as special values the solitary wave solutions are derived from the hyperbolic function solutions the proposed method is straightforward concise and effective and can be applied to other nonlinear evolution equations in mathematical physics

homogeneous state feedback stabilization of homogenous systems we show that for any asymptotically controllable homogeneous system in euclidean space not necessarily lipschitz at the origin there exists a homogeneous control lyapunov function and a homogeneous possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system if the system satisfies the usual local lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate if the system satisfies a global lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate the control lyapunov function and the feedback are based on the lyapunov exponents of a suitable auxiliary system and admit a numerical approximation

large deviations in weakly interacting boundary driven lattice gases one dimensional boundary driven lattice gases with local interactions are studied in the weakly interacting limit the density profiles and the correlation functions are calculated to first order in the interaction strength for zero range and short range processes differing only in the specifics of the detailed balance dynamics furthermore the effective free energy large deviation function and the integrated current distribution are also found to this order from the former we find that the boundary drive generates long range correlations only for the short range dynamics while the latter provides support to an additivity principle recently proposed by bodineau and derrida

algebraic and topological classification of the homogeneous cubic vector fields in the plane tsutomu date and masao iri in di gave an algebraic classification of systems i p x y i q x y where p and q are homogeneous polynomials of degree 2 for this they used the classification of the binary cubic forms and also the simultaneous classification of a linear binary form and a cubic binary form given by the algebraic invariant theory we begin by doing a similar study for systems p x y q x y where p and q are homogeneous polynomials of degree three i e cubic systems the classification s theorem of such systems is based on the classification of fourth order binary forms gurevich in gu did the classification of fourth order binary forms on the field of complex numbers since we did not find the classification on the real domain we adapt gurevich s proof to obtain it in section 1 we give some definitions and preliminary results while in section 2 we give the theorem of classification of fourth order binary forms on the real domain the method used in the proof caley s method let us obtain canonical forms of the fourth order binary forms and the algebraic characteristics section 3 is devoted to obtaining the algebraic classification of systems p x y j q x y where p and q are homogeneous polynomials of degree 3 given an arbitrary system x p q with p and q homogeneous polynomials of degree 3 we can know the equivalence class at which it belongs through the algebraic characteristics in section 4 we study the phase portraits of systems 1 p x y j q x y where p and q are homogeneous polynomials of degree n and p and q have no common factor such systems have been studied by j argemi in a here we give a shorter new proof of his results by using

a new elliptic equation rational expansion method and its application to the shallow long wave approximate equations a new elliptic equation rational expansion method is presented by a new general ansatz which is a direct and unified algebraic method for constructing multiple and more general travelling wave solution for nonlinear partial differential equation and implemented in a computer algebraic system the proposed method is applied to consider the shallow long wave approximate equation and obtains rich new families of the exact solutions including rational form solitary wave rational form triangular periodic rational form jacobi and weierstrass doubly periodic solutions

an exact solution of a one dimensional asymmetric exclusion model with open boundaries a simple asymmetric exclusion model with open boundaries is solved exactly in one dimension the exact solution is obtained by deriving a recursion relation for the steady state if the steady state is known for all system sizes less thann then our equation 8 gives the steady state for sizen using this recursion we obtain closed expressions 48 for the average occupations of all sites the results are compared to the predictions of a mean field theory in particular for infinitely large systems the effect of the boundary decays as the distance to the power 1 2 instead of the inverse of the distance as predicted by the mean field theory

the compound g g expansion method and double non traveling wave solutions of 2 1 dimensional nonlinear partial differential equations abstract to seek the exact double non traveling wave solutions of nonlinear partial differential equations the compound g g expansion method is firstly proposed in this paper with the aid of symbolic computation this new method is applied to construct double non traveling wave solutions of 2 1 dimensional painleve integrable burgers equation and 2 1 dimensional breaking soliton equation as a result abundant double non traveling wave solutions including double hyperbolic function solutions double trigonometric function solutions double rational solutions and a series of complexiton solutions of these two equations are obtained via the proposed method these exact solutions contain arbitrary functions which may be helpful to explain some complex phenomena when the parameters are taken as special values the double solitary like wave solutions can be derived from double hyperbolic function solutions furthermore the time evolutions of double solitary like wave solutions are discussed in detail

the stationary state and the heat equation for a variant of davies model of heat conduction we study a variant of davies model of heat conduction consisting of a chain of classical or quantum harmonic oscillators whose ends are coupled to thermal reservoirs at different temperatures and where neighboring oscillators interact via intermediate reservoirs in the weak coupling limit we show that a unique stationary state exists and that a discretized heat equation holds we give an explicit expression of the stationary state in the case of two classical oscillators the heat equation is obtained in the hydrodynamic limit and it is proved that it completely describes the macroscopic behavior of the model

processus des misanthropes we consider a system of identical interacting particles moving on the lattice d the rate at which a particle at the site x jumps to the site y is p y x b x y where p is an irreducible probability on d and b x y is an increasing resp decreasing function of the number x resp y of particles at site x resp y we study the convergence of the system to equilibrium and describe the invariant measures

stabilizability conditions in terms of critical hamiltonians and symbols symmetric functions of critical hamiltonians called symbols were used in such problems of nonlinear control as the characterization of symmetries and feedback invariants we derive here a stabilizability condition in the class of almost continuous feedback controls based on symbols the methodology proposed consists of defining a selector of the multivalued covector field satisfying an extra degree condition on a sphere close to the origin then the above selector is extended to some neighborhood in order to get an exact differential form integration of that form gives us a control lyapunov function candidate theorem 3 1 the condition proposed is applied to the analysis of a planar control affine system with polynomial homogeneous vector fields unlike known results in the literature we consider the case when each vector field vanishes at the origin and the control is bounded we give stabilizability conditions explicitly in terms of the control system parameters theorem 4 1

heat flow in an exactly solvable model a chain of one dimensional oscillators is considered they are mechanically uncoupled and interact via a stochastic process which redistributes the energy between nearest neighbors the total energy is kept constant except for the interactions of the extremal oscillators with reservoirs at different temperatures the stationary measures are obtained when the chain is finite the thermodynamic limit is then considered approach to the gibbs distribution is proven and a linear temperature profile is obtained

stabilization of a class of nonlinear systems by a smooth feedback control we consider the stabilizability problem for systems of the form x f x ub in the neighborhood of an equilibrium point of f x first by means of center manifold theory a lower order system is introduced if this system is stabilizable then so is the original system results on the stabilization of the low order system are presented no resort is taken to liapunov theory the relation between stabilizability and controllability is discussed

a generalized new auxiliary equation method and its application to the 2 1 dimensional breaking soliton equations abstract in this paper the new auxiliary equation method sirendaoreji a new auxiliary equation and exact travelling wave solutions of non linear equations phys lett a 356 2006 124 130 is improved and a generalized new auxiliary equation method is proposed to construct more general exact solutions of non linear partial differential equations with the aid of symbolic computation we choose the 2 1 dimensional breaking soliton equations to illustrate the validity and advantages of the method as a result many new and more general non travelling wave and coefficient function solutions are obtained including soliton like solutions trigonometric function solutions exponential solutions

q zero range has random walking shocks but no other surprises in the zero range world we check all nearest neighbour 1 dimensional asymmetric zero range processes for random walking product shock measures as demonstrated already for a few cases in the literature we find the totally asymmetric version of the celebrated q zero range process as the only new example besides an already known model of doubly infinite occupation numbers and exponentially increasing jump rates we also examine the interaction of shocks which appears somewhat more involved for q zero range than in the already known cases

absorbing state phase transition for driven dissipative stochastic dynamics on ℤ we study the long time behavior of conservative interacting particle systems in the activated random walk model for reaction diffusion systems and the stochastic sandpile we prove that both systems undergo an absorbing state phase transition

asymptotic stabilizability of homogeneous polynomial systems of odd degree we give some tools for the construction of the homogeneous feedback witch stabilizes a generic class of single input two dimensional homogeneous systems

condensation in stochastic particle systems with stationary product measures we study stochastic particle systems with stationary product measures that exhibit a condensation transition due to particle interactions or spatial inhomogeneities we review previous work on the stationary behaviour and put it in the context of the equivalence of ensembles providing a general characterization of the condensation transition for homogeneous and inhomogeneous systems in the thermodynamic limit this leads to strengthened results on weak convergence for subcritical systems and establishes the equivalence of ensembles for spatially inhomogeneous systems under very general conditions extending previous results which focused on attractive and finite systems we use relative entropy techniques which provide simple proofs making use of general versions of local limit theorems for independent random variables

microscopic stationary states for stochastic systems with particle flux let t be the symmetric nearest neighbor simple exclusion process on 3 modified to allow the spontaneous creation and destruction of particles at the origin we study the invariant measures for the system in this non equilibrium situation for which a net current of particles flows from the origin we analyze the n point correlations of the invariant measures which for n 2 behave like max x1x2 4

feedback stabilization of homogeneous polynomial systems of odd degree in the plane in this paper we give some tools for the construction of homogeneous feedback which stabilizes homogeneous polynomial systems of odd degree in the plane

homogeneous feedback controls for homogeneous systems let x x j 2n aj x xj be a real analytic n dimensional vector field with x 0 0 which is homogeneous of degree m 0 with respect to a dilation erx er1x1 ernxn and y xj we show that control system x x x uy admits a continuous asymptotically stabilizing feedback control u x then a one dimensional dynamic extension of the system admits a homogeneous asymptotically stabilizing dynamic feedback control u x such that the vector field w x 1 m u x 1 j 2aj x xj describing the extended controlled system is homogeneous of degree m with respect to the extended dilation er x e er1x1 ernxn

auxiliary equation method for the mkdv equation with variable coefficients abstract by using solutions of an ordinary differential equation an auxiliary equation method is described to seek exact solutions of nonlinear evolution equations with variable coefficients being concise and straightforward this method is applied to the mkdv equation with variable coefficients as a result new explicit solutions including solitary wave solutions and trigonometric function solutions are obtained with the aid of symbolic computation