lyapunov functions for riemann liouville like fractional difference equations abstract discrete memory effects are introduced by fractional difference operators asymptotic stabilities of nonlinear fractional difference equations are investigated in this paper a linear scalar fractional difference equality is utilized lyapunov second direct method is proposed for nonlinear discrete fractional systems asymptotic stability conditions are provided and some examples are given

kernel method and linear recurrence system based on the kernel method we present systematic methods to solve equation systems on generating functions of two variables using these methods we get the generating functions for the number of permutations which avoid 1234 and 12k k 1 3 and permutations which avoid 1243 and 12 k

stability analysis by krasnoselskii s fixed point theorem for nonlinear fractional differential equations this paper is concerned with the stability analysis of nonlinear fractional differential equations of order 1 2 our main results are obtained by using krasnoselskii s fixed point theorem in a weighted banach space an example and its corresponding simulation are presented to illustrate the main results

on the dimension and the index of the solution set of nonlinear equations we study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving fredholm maps of positive index the classes of maps under consideration are pseudo a proper and either approximation essential or equivariant approximation essential applications are given to semilinear elliptic bvp s

infinitely many solutions of a symmetric dirichlet problem here q is a bounded domain in r with smooth boundary and f i r is c and satisfies certain growth conditions if m 1 and f is an even function hence f r r is an odd function then ambrosetti and rabinowitz l 21 proved the existence of infinitely many weak solutions of d using a symmetric version of the mountain pass theorem a similar result is due to michalek 3 who considered the case where m 21 rm s c and f is invariant under a free action of the cyclic group z p on c here z p is considered as a subgroup of s c c acting essentially via scalar multiplication on c in this paper we want to investigate which kind of symmetries guarantee the existence of infinitely many solutions of d let g be a compact lie group and p g o m an orthogonal representation of g on lrm then g acts on irm via the map g x ir ir g u y gv p g v we write v for the space rm considered as a g space do there exist infinitely many solutions of d if f is invariant under this symmetry i e f gu f u for all g e g and u e v without further assumptions on p the answer to this question is no in general for example if p g id for all g e g then there is no symmetry at all and one cannot expect infinitely many solutions more generally one should exclude the existence of fixed points vg u e vi gu u for all g e g 0 our first result says that there exist infinitely many solutions of d if the following condition is satisfied

stability preservation analysis for frequency based methods in numerical simulation of fractional order systems in this paper the frequency domain based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation first the stability boundary curve is exactly determined for these methods then this boundary is analyzed and compared with an accurate ideal boundary in different frequency ranges also the critical regions in which the stability does not preserve are determined finally the analytical achievements are confirmed via some numerical illustrations

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

brief paper stability and resonance conditions of elementary fractional transfer functions elementary fractional transfer functions are studied in this paper some basic properties of elementary transfer functions of the first kind are recalled then two main results are presented regarding elementary fractional transfer functions of the second kind written in a canonical form and characterized by a commensurate order a pseudo damping factor and a natural frequency first stability conditions are established in terms of the pseudo damping factor and the commensurate order as a corollary to matignon s stability theorem they extend the previous result into conditions that are simpler to check then resonance conditions are established numerically in terms of the commensurate order and the pseudo damping factor and give interesting information on the frequency behavior of fractional systems it is shown that elementary transfer functions of the second kind might have up to two resonant frequencies moreover three abaci are given allowing to determine the pseudo damping factor and the commensurate order for respectively a desired normalized gain at each resonance a desired phase at each resonance and a desired normalized first or second resonant frequency

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

homoclinic solutions of hamiltonian systems with symmetry abstract we prove the existence of infinitely many homoclinic solutions for a first order hamiltonian system symmetric with respect to an action of a compact lie group by means of variational methods we make no convexity assumption on the hamiltonian

diffusion process modeling by using fractional order models this paper deals with a concept and description of a rc network as an electro analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions it is based on well known analogy between heat and electrical conduction in the paper are compared analytical solution together with numerical solution and experimentally measured data for the first time a fractional order model of diffusion process and its modeling via lumped rc network has been used simple examples of simulations measurements and their comparison are shown

the kernel method and systems of functional equations with several conditions we generalize the kernel method to equation systems in which the number of unknowns is allowed to exceed the number of equations with this generalization we derive the generating functions for several kinds of sequences and generating trees whose recursions are dependent on the parity of the indices

existence and exponential decay of homoclinics in a nonperiodic superquadratic hamiltonian system abstract this paper deals with existence and exponential decay of homoclinic orbits in the first order hamiltonian system z j h z t z where the hamiltonian function h t z is nonperiodic in t r and superquadratic in z r 2 n with certain mild conditions we obtain the solutions via variational methods for strongly indefinite problems

stability and bifurcation of a delayed generalized fractional order prey predator model with interspecific competition abstract the present paper considers a delayed generalized fractional order prey predator model with interspecific competition the existence of the nontrivial positive equilibrium is discussed and some sufficient conditions for global asymptotic stability of the equilibrium are developed meanwhile the existence of hopf bifurcation is discussed by choosing time delay as the bifurcation parameter finally some numerical simulations are carried out to support the analytical results

first order elliptic systems and the existence of homoclinic orbits in hamiltonian systems and h r x r 2 r is smooth by a homoclinic or a doubly asymptotic orbit to y where y is a periodic solution of 1 we mean a solution x y of 1 such that i x 0 y t l 0 as i tl oo finding homoclinic orbits in systems like 1 can be quite difficult in the case when y is a hyperbolic orbit this is equivalent to showing that the stable and unstable manifold of y intersect in some situation this can be done by melnikov s method recently some progress has been made by applying variational methods coti zelati and ekeland showed in 3 using dual variational methods that 1 has at least one homoclinic orbit if h satisfies some convexity and growth assumptions in their proof the convexity assumption enters in a crucial way already in the set up of the variational problem variational methods have been used by benci and giannoni 2 who studied homoclinic orbits on compact riemannian manifolds and by p rabinowitz in 9 who proved existence of heteroclinic orbits on the n dimensional torus as well as their multiplicity in 2 and 9 only special hamiltonian of lagrangian type are studied the associated variational problem is then at least semi definite the

a semilinear schrodinger equation in the presence of a magnetic field we consider the semilinear stationary schrodinger equation in a magnetic field i a 2u v x u g x u u in n where v is the scalar or electric potential and a is the vector or magnetic potential we study the existence of nontrivial solutions both in the critical and in the subcritical case respectively g x u u 2 2 and g x u c 1 u p 2 where 2 p 2 the results are obtained by variational methods for g critical we use constrained minimization and for subcritical g we employ a minimax type argument in the latter case we also study the existence of infinitely many geometrically distinct solutions

kernel method and system of functional equations introduced by knuth and subsequently developed by banderier et al prodinger and others the kernel method is a powerful tool for solving power series equations in the form of f z t a z t f z 0 t b z t and several variations recently hou and mansour q h hou t mansour kernel method and linear recurrence system j comput appl math 2007 in press presented a systematic method to solve equation systems of two variables f z t a z t f z 0 t b z t where a is a matrix and f and b are vectors of rational functions in z and t in this paper we generalize this method to another type of rational function matrices i e systems of functional equations since the types of equation systems we are interested in arise frequently in various enumeration questions via generating functions our tool is quite useful in solving enumeration problems to illustrate this we provide several applications namely the recurrence relations with two indices and counting descents in signed permutations

an algorithm for stabilization of fractional order time delay systems using fractional order pid controllers this technical note presents a solution to the problem of stabilizing a given fractional order system with time delay using fractional order pllambdadmu controllers it is based on determining a set of global stability regions in the kp ki kd space corresponding to the fractional orders lambda and mu in the range of 0 2 and then choosing the biggest global stability region in this set this method can be also used to find the set of stabilizing controllers that guarantees prespecified gain and phase margin requirements the algorithm is simple and has reliable result which is illustrated by an example and hence is practically useful in the analysis and design of fractional order control systems

fractional order systems and pi sup spl lambda d sup spl mu controllers dynamic systems of an arbitrary real order fractional order systems are considered the concept of a fractional order pi sup spl lambda d sup spl mu controller involving fractional order integrator and fractional order differentiator is proposed the laplace transform formula for a new function of the mittag leffler type made it possible to obtain explicit analytical expressions for the unit step and unit impulse response of a linear fractional order system with fractional order controller for both open and closed loops an example demonstrating the use of the obtained formulas and the advantages of the proposed pi sup spl lambda d sup spl mu controllers is given

homoclinic orbits in a first order superquadratic hamiltonian system convergence of subharmonic orbits abstract we consider the existence of homoclinic orbits for a first order hamiltonian system z jh z t z we assume h t z is of form h t z 1 2 az z w t z where a is a symmetric matrix with ja i r and w t z is 2 periodic in t and has superquadratic growth in z we prove the existence of a nontrivial homoclinic solution z t and subharmonic solutions zt t t n i e 2 t periodic solutions of hs such that zt t z t in cloc1 r r 2n as t

nonstationary homoclinic orbits for an infinite dimensional hamiltonian system we study the following system of diffusion equations on r rn tu xu v x u g t x v tv xv v x v f t x u where z u v r rn r2 v c rn r g t x v f t x u are periodic in t x and superlinear at infinity by a variant linking theorem and concentration compactness arguments we establish the existence of nonstationary homoclinic type solutions of least energy for this systems

nyquist based stability analysis of non commensurate fractional order delay systems abstract as a generalization of the first and second order models the elementary fractional order models have been widely used in various engineering fields however most of the previous studies only focus on commensurate fractional order models in this paper a general non commensurate elementary fractional order delay system is investigated first the stability of the studied fractional order delay system is analyzed based on nyquist theorem then a series of sufficient stability conditions are presented for different combinations of parameters including the fractional orders time delay pseudo damping factor and natural frequency 0 finally three examples are given to show the effectiveness of the presented results

critical point theory for indefinite functionals with symmetries letxbe a hilbert space and c1 x r be strongly indefinite assume in addition that a compact lie groupgacts orthogonally onxand that is invariant in order to find critical points of we develop the limit relative category of fournieret al in the equivariant context we use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem in the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension also other conditions of the classical symmetric mountain pass theorem forg z 2 due to ambrosetti and rabinowitz can be weakened considerably for example we are able to deal with infinite dimensional fixed point spaces the proofs consist of a direct reduction to a relative borsuk ulam type theorem this provides a new proof even for the classical mountain pass theorem the abstract critical point theorems are applied to an elliptic system with dirichlet boundary conditions we only need a weak version of the usual superquadraticity condition the linking theorem can be applied to asymptotically linear hamiltonian systems which are symmetric with respect to a generalized symplectic group action

closed loop time response analysis of irrational fractional order systems with numerical laplace transform technique abstract irrational transfer function has been widely used in modelling and identification but time response analysis of systems with irrational transfer functions is hard to be achieved in comparison with the rational ones one of the main reasons is that irrational transfer function generally has infinite poles or zero in this paper the closed loop time response of fractional system with irrational transfer function is analyzed based on numerical inverse laplace transform the numerical solutions and stability evaluation of irrational fractional order systems are presented several examples of fractional order systems with irrational transfer functions are shown to verify the effectiveness of the proposed algorithm in both time and frequency domain analysis the matlab codes developed to solve the fractional differential equations using numerical laplace transform are also provided the results of this paper can be used on analysis and design of control system described by irrational fractional order or integer order transfer function