lie group symmetries and invariants of the henon heiles equations lie group symmetries and invariants of the generalized henon heiles equations are found the coupled second order equations are invariant under translations in time in general and the stretching group dilation if the linear terms in the force are absent the equivalent set of four coupled first order equations is found to be invariant under a one parameter group for three cases and the group generators are given three different approaches are reported the classical method for determining lie group symmetries a modified method for finding lie group symmetries with vector fields and the direct method for calculating the invariants for the henon heiles equations the direct method is the most efficient

on the lie symmetry algebra of a general ordinary differential equation we give a description of the lie symmetry algebra of a general ordinary differential equation which clarifies certain issues in its practical use in particular we present examples of symmetry algebras for well known equations which illustrate features of this algebra

global stability and chaos in a population model with piecewise constant arguments sufficient conditions are obtained for the global stability of the positive equilibrium of dx dt rx t 1 ax t b j 0 c jx t j it is shown that for certain special cases solutions of the equation can have chaotic behaviour through period doubling bifurcations

projective lifts and generalised ermakov and bernoulli systems we discuss a system which generalizes the bernoulli equation analogously to the way rogers and schief generalize the ermakov system and set both in a geometrical context

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

comparison theorems and oscillation criteria for difference equations abstract first we establish the equivalence of the oscillation of the delay difference equation x n p n x n k 0 and the second order difference equation 2 y n 1 2 k 1 k k k 1 p n k k k 1 k 1 y n 0 where pn is a sequence of nonnegative real numbers and k is a positive integer next we obtain some sharp conditions for oscillations and nonoscillations of the first equation

on weakly unitarily invariant norm and the λ aluthge transformation for invertible operator abstract let t b h be an invertible operator with polar decomposition t up and b b h commute with t in this paper we prove that p bup 1 bt where is a weakly unitarily invariant norm on b h and 0 1 as the consequence of this result we have f p up 1 f t for any polynomial f

hidden and contact symmetries of ordinary differential equations the lie algebra for the maximal contact symmetries of third order ordinary differential equations odes is examined for type i and ii hidden symmetries where the analysis of hidden symmetries for point symmetries is extended to contact symmetries ones invariant under the group associated with the ten dimensional maximal lie algebra may produce type i hidden symmetries for two parameter subgroups and type ii hidden symmetries for certain solvable non abelian three parameter subgroups in the third order odes when they are reduced in order a new class of type ii hidden symmetries is recognized in which contact symmetries transform to point symmetries in some reduction paths two examples of odes invariant under subgroups of the ten parameter group under which y 0 is invariant demonstrate the new class of type ii hidden symmetries

the persistence and global attractivity in general nonautonomous discrete single species kolmogorov model with delays in this paper general nonautonomous discrete time single species kolmogorov model with delays is studied some sufficient conditions on the boundedness persistence permanence and global attractivity of solutions are established as application the discrete time nonautonomous delayed allee effect model is discussed the sufficient conditions on the permanence and global attractivity are obtained

an equation admitting infinite true contact transformations abstract the linear wave equation is shown to possess the unique property that if w n is a true contact transformation admitted by the wave equation i e w n is not linear in the first derivatives of the dependent variable then so is n w n we comment of the physical implications

equivalence classes of second order ordinary differential equations with only a three dimensional lie algebra of point symmetries and linearisation there are seven equivalence classes of second order ordinary differential equations possessing only three lie point symmetries and hence not linearisible by means of a point transformation we examine the representatives of these classes for linearisibility by means of other types of transformation in particular we compare the potential for linearisibility and the possession of the painleve property the complete symmetry group is realised in the standard algebra for each of the equivalence classes

existence of positive periodic solutions of nonlinear discrete model exhibiting the allee effect in this paper we shall consider the discrete nonlinear delay population model with allee effectx n 1 x n exp a n b n x p n w c n x q n w where a n b n and c n are positive sequences of period w and p and q are positive integers by using the coincidence degree theorem as well as some priori estimates we will establish a sufficient condition for the existence of positive periodic solution x n

ermakov systems of arbitrary order and dimension structure and linearization ermakov systems of arbitrary order and dimension are constructed these inherit an underlying linear structure based on that recently established for the classical ermakov system as an application alignment of a 2 1 dimensional ermakov and integrable ernst system is shown to produce a novel integrable hybrid of a 2 1 dimensional sinh gordon system and of a conventional ermakov system

on solving a class of unbalanced ermakov pinney systems we use an intertwining property of linear differential operators to construct the general solution of a type of coupled ermakov pinney system with two distinct frequency terms

complete symmetry groups of ordinary differential equations and their integrals some basic considerations abstract the concept of the complete symmetry group of a differential equation introduced by j krause 1994 j math phys 35 5734 5748 is extended to integrals of such equations this paper is devoted to some aspects characterising complete symmetry groups the algebras of the symmetries of both differential equations and integrals are studied in the context of equations for which the elements are represented by point or contact symmetries so that there is no ambiguity about the group both algebras and groups are found to be nonunique

master partial differential equations for a type ii hidden symmetry an approach for determining a class of master partial differential equations from which type ii hidden point symmetries are inherited is presented as an example a model nonlinear partial differential equation pde reduced to a target pde by a lie symmetry gains a lie point symmetry that is not inherited hidden from the original pde on the other hand this type ii hidden symmetry is inherited from one or more of the class of master pdes the class of master pdes is determined by the hidden symmetry reverse method the reverse method is extended to determine symmetries of the master pdes that are not inherited we indicate why such methods are necessary to determine the genesis of type ii symmetries of pdes as opposed to those that arise in ordinary differential equations odes

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

oscillations of delay difference equations consider the delay difference equation n 1 n pn n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer new oscillation criteria for this equation are established an extension of the results to difference equations with several delays is also presented

the existence of contact transformations for evolution type equations we investigate the existence of a one parameter group of contact transformations for evolution type equations ut f t x u ux uxx u n subscripts denote differentiation unless otherwise indicated where u n is the nth derivative of u with respect to x we prove that contact transformations of evolution equations where f is expandable as a power series in terms of all derivatives of order higher than one are just extended lie point transformations this result is extended to the case with m independent space variables as a consequence we obtain an ansatz for determining lie point transformations for nth order evolution equations with m independent space variables examples are given to verify the results obtained as well as to show how lie point transformations of these evolution type partial differential equations can be calculated from this ansatz

symmetry and singularity properties of the generalised kummer schwarz and related equations abstract we examine the generalised kummer schwarz equation and some of its generalisations from the viewpoints of symmetry and singularity analyses we determine the complete symmetry group of the general equation and show that different forms of the fourth order representative illustrate the three possible classes of laurent series to be expected in the course of the singularity analysis

stability and periodicity in coupled pinney equations abstract we show that the coupled pinney equations with periodic coefficient constitute a class of nonlinear coupled hill s equations for which the problems of stability and periodicity of the solutions reduce to those for a pair of uncoupled linear hill s equations one of which is of ince type

oscillation and global attractivity in hematopoiesis model with periodic coefficients in this paper we shall consider the nonlinear delay differential equation p t email protected t p m email protected 1 p n email protected email protected t p t where k is a positive integer b t and c t are positive periodic functions of period w in the nondelay case we shall show that has a unique positive periodic solution email protected t and we will study the global attractivity of email protected t in the delay case we shall establish some sufficient conditions for oscillation of all positive solutions of about email protected t and establish some sufficient conditions for the global attractivity of email protected t our results in this paper extend as well as improve the results in the autonomous case

on the determination of non local symmetries the importance of non local symmetries of differential equations lies in their manifestation as lie point symmetries of the equations resulting from reduction of order the reason for the determination of these symmetries in second order equations with only one lie point symmetry is self evident however the disadvantage of non local symmetries is that no systematic approach to their determination exists we present such an approach applicable to differential equations of any order and apply it to some second order ordinary differential equations and show that they have a rich occurrence we also look at possible generalizations of the concept of non local symmetries

type ii hidden symmetries of the linear 2d and 3d wave equations type ii hidden symmetries of the linear two dimensional and three dimensional wave equations are analysed these hidden symmetries are lie point symmetries that appear in addition to the inherited point symmetries when the number of independent and dependent variables of a partial differential equation is reduced by a lie point symmetry the provenance of these hidden symmetries of partial differential equations is identified to be the same as found recently for some nonlinear partial differential equations the appearance of type ii hidden symmetries depends not only on the lie symmetries used but on the order in which the symmetries are applied the presence of type ii hidden symmetries of partial differential equations complicates the prediction of symmetry reductions based on the lie algebra associated with the original lie point symmetries

on weakly unitarily invariant norm and the aluthge transformation abstract in this paper we show that f p up 1 max f t u f t u f 0 i u u where t b h is a semi norm on b h which satisfies some conditions t up polar decomposition 0 1 and f is a polynomial as a consequence of this fact we will show that some semi norms including the radii 0 f p up 1 f t we also give some related results

oscillatory and asymptotic behavior of positive periodic solutions of nonlinear discrete model exhibiting the allee effect in this paper we shall consider the discrete nonlinear delay population model with allee effectx n 1 x n exp a n b n x p email protected c n x q email protected where a n b n and c n are positive sequences of period w and p and q are positive integers we will establish some sufficient conditions for the oscillation of all positive solutions about its positive periodic solution x n and prove that every nonoscillatory solution converges to x n monotonically as n

lie transformation group solutions of the nonlinear one dimensional vlasov equation the solution of the exact nonlinear one dimensional vlasov equation with a space and time dependent electric field is reduced to the solution of a nonlinear first order ordinary differential equation with two subsidiary equations the reduction holds for any electric field nonlinear in the spatial coordinate or for a subclass of electric fields linear in the spatial coordinate and is equivalent to the solution of a generalized bernstein greene kruskal bgk vlasov equation with a velocity dependent lorentz force the lie method for the solution of differential equations invariant under a transformation group has been used to calculate the group generator the canonical variables and the generalized bgk vlasov equation analytical forms for the functional dependence of the vlasov equation one particle distribution function are given

a λ symmetry based method for the linearization and determination of first integrals of a family of second order ordinary differential equations we present a characterization of the second order ordinary differential equations odes that can be linearized by means of certain nonlocal transformations this characterization is given in terms of the coefficients of the equation and also determines the second order odes that admit symmetries and first integrals of some specific forms systematic methods to find these first integrals symmetries and the linearizing transformations are also derived

oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics abstract in this paper we shall consider the nonlinear delay differential equation where m and n are positive integers and v t and t are positive periodic functions of period in the nondelay case we shall show that has a unique positive periodic solution x t and provides sufficient conditions for the global attractivity of x t in the delay case we shall present sufficient conditions for the oscillation of all positive solutions of about x t and establish sufficient conditions for the global attractivity of x t

infinitesimal symmetry transformations of the langevin equation the lie algebra associated with infinitesimal symmetry transformations of the equation of motion for a radiating charged particle embedded in the radiation field i e the so called langevin equation is determined it is shown that the invariance group is a five parameter group with a commutative three parameter subgroup

symmetries of linear ordinary differential equations we discuss the lie symmetry approach to homogeneous linear ordinary differential equations in an attempt to connect it with the algebraic theory of such equations in particular we pay attention to the fields of functions over which the symmetry vector fields are defined and by defining a noncharacteristic lie subalgebra of the symmetry algebra are able to establish a general description of all continuous symmetries we use this description to rederive a classical result on differential extensions for second order equations

on 2 1 dimensional ermakov systems ermakov type systems in 2 1 dimensions are introduced multiwave solutions of a 2 1 dimensional pinney equation and a modulated 2 1 dimensional sine gordon equation are thereby constructed

rational ermakov systems of fuchsian type studies a subclass of a class of systems of coupled nonlinear oscillators that have well behaved singularities which can be studied using an exact linearization this allows the author to classify such systems and to isolate those with special properties

symmetry lie algebras of nth order ordinary differential equations abstract we show that an nth n 3 order linear ordinary differential equation has exactly one of n 1 n 2 or n 4 the maximum point symmetries the lie algebras corresponding to the respective numbers of point symmetries are obtained then it is shown that a necessary and sufficient conditon for an nth n 3 order equation to be linearizable via a point transformation is that it must admit the n dimensional abelian algebra na1 a1 a1 a1 we discuss in detail the symmetry realizations of n 1 a1 s a1 finally we prove that an nth n 3 order equation q n h t q qn 1 cannot admit exactly an n 3 dimensional algebra of point symmetries which is a subalgebra of na1 gl 2 r

a generalization of λ symmetry reduction for systems of odes σ symmetries we consider a deformation of the prolongation operation defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones this maintains the invariants by differentiation property and can hence be used to reduce odes satisfying suitable invariance conditions in a fully algorithmic way similarly to what happens for standard prolongations and symmetries

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

hidden symmetries and linearization of the modified painleve ince equation the linearization and hidden symmetries of the modified painleve ince equation y yy y3 0 where and are constants are presented the linearization of this equation by a nonlocal transformation yields a damped stable or growing unstable harmonic oscillator equation for 0 hidden symmetries are analyzed by transforming the modified painleve ince equation to a third order ordinary differential equation ode which in general is invariant under a three parameter group by a riccati transformation a type i hidden symmetry is found of a second order ode found from the third order ode where a symmetry is lost in the reduction of order by the non normal subgroup invariants a type ii hidden symmetry occurs in the third order ode because the symmetries of a second order ode reduced from the third order ode by another set of normal subgroup invariants are increased

hidden symmetries associated with the projective group of nonlinear first order ordinary differential equations hidden symmetries those not found by the classical lie group method for point symmetries are reported for nonlinear first order ordinary differential equations odes which arise frequently in physical problems these are for the special class of the eight nonabelian two parameter subgroups of the eight parameter projective group the first order odes can be transformed by non local transformations to new separable first order odes which then can be reduced to quadratures the first order odes include riccati equations and equations which in particular cases are of the form of abel s equation the procedure demonstrates the feasibility of integrating nonlinear odes that do not show any apparent lie group point symmetry applications to the vlasov characteristic equation and the reaction diffusion equation are given

conservation laws and hierarchies of potential symmetries for certain diffusion equations abstract we show that the so called hidden potential symmetries considered in a recent paper m l gandarias new potential symmetries for some evolution equations physica a 387 2008 2234 2242 are ordinary potential symmetries that can be obtained using the method introduced by bluman and collaborators g w bluman s kumei symmetries and differential equations springer new york 1989 g w bluman g j reid s kumei new classes of symmetries for partial differential equations j math phys 29 1988 806 811 in fact these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics furthermore we classify the conservation laws for classes of porous medium equations and then using the corresponding conserved potential systems we search for potential symmetries this is the approach one needs to adopt in order to determine the complete list of potential symmetries the provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones in particular in this way the potential systems potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described infinite series of infinite dimensional algebras of potential symmetries are constructed for such equations

generating functions and existence of contact symmetries of third order scalar ordinary differential equations analyzing the structure of generating functions associated to the contact symmetry condition of third order scalar ordinary differential equations constraints guaranteeing that the ode does not admit contact symmetries are obtained further inspecting the subalgebras generated by contact symmetries having a specific form an alternative procedure to generate third order equations with irreducible contact symmetry algebra sp 4 is developed

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

multi dimensional quasi simple waves in weakly dissipative flows abstract a multi dimensional simple wave formalism is employed to formulate a multi dimensional quasi simple wave for a weakly dissipative fluid this is a natural but nontrivial generalization of the so called unidirectional quasi simple wave the method is more close to multi orders analysis which is different from the standard perturbation method detailed solving up to the second order is presented for a 2d sound simple wave a new 2d burgers equation is derived for the wave phase it essentially differs from the known 2d generalizations of the burgers equation e g from the zabolotskaya khokhlov equation the derived equation is investigated in the framework of group analysis of differential equations multi parameter families of its exact solutions are constructed simplest solutions are chosen for analysis of their physical relevance in the initial variables

on generalized numerical range of the aluthge transformation abstract in this paper the authors show that the aluthge transformation t of a matrix t and a polynomial f satisfy the inclusion relation w c f t w c f t for the generalized numerical range if c is a hermitian matrix or a rank one matrix

periodic solutions of a single species discrete population model with periodic harvest stock abstract we discuss a discrete population model describing single species growth with periodic harvest stock the theory of coincidence degree is applied to show that the model equation admits two periodic solutions under minor technical assumptions we show that one of these two periodic solutions is positive and attracts almost all positive solutions

constructing conditionally integrable evolution systems in 1 1 dimensions a generalization of invariant modules approach given a generalized lie backlund vector field satisfying certain nondegeneracy assumptions we explicitly describe all 1 1 dimensional evolution systems that admit this vector field as a generalized conditional symmetry the connection with the theory of symmetries of systems of odes and with the theory of invariant modules is discussed

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

on the linearization of the generalized ermakov systems a linearization procedure is proposed for ermakov systems with frequency depending on dynamic variables the procedure applies to a wide class of generalized ermakov systems which are shown to be linearizable in a manner similar to that applicable to usual ermakov systems the kepler ermakov systems belong to this category but other more generic systems are also included