global attractivity in the recursive sequence xn 1 α βxn γ xn 1 our aim in this paper is to investigate the global attractivity of the recursive sequencex n email protected bx email protected n 1 n 0 1 where a 0 b c 0 we show that one positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

global attractivity in a rational recursive sequence our aim in this paper is to investigate the periodic character invariant intervals and the global attractivity of all positive solutions of the nonlinear difference equationx n email protected email protected email protected n 1 n 0 1 where a 0 b c 0 we show that the positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

consensus tracking in sensor networks with periodic sensing and switching connected topologies abstract this paper studies the consensus tracking of general linear dynamical target in periodically sensing networks with switching and connected sensing topologies allowable upper bounds of sensing period are investigated for both time invariant sensing and time varying sensing networks for tracking system with constant sensing period discretization approach is applied and the tracking problem under switching connected topologies is converted to the robust stability problem of a discrete time uncertain system then an allowable upper bound of sensing period is given by solving optimal h control problem and an explicit bound which is composed of the eigenvalues of target s dynamic matrix and topology matrix is further provided for tracking system with time varying sensing period input delay approach is applied and a bound is given by using lyapunov krasovskii functional analysis and solving the feasibility of three lmis numerical examples are given to illustrate the results

on the difference equation yn 1 α βe yn γ yn 1 in this paper we investigate the convergence the boundedness and the periodic character of the positive solutions of the difference equationy n email protected email protected y email protected y n 1 n 0 1 2 where the parameters a b c are positive numbers and the initial conditions y 1 y 0 are arbitrary non negative numbers

on some solvable systems of difference equations abstract we show that the following systems of difference equations x n 1 u n 1 v n y n 1 w n 1 s n n n 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases two cases are left unsolved probably the most interesting is the result in the case u n x n v n x n w n x n s n y n where a fascinating formula is obtained in an elegant way by using some ad hoc ideas

on global periodicity of difference equations abstract in this research we consider a difference equation of order k 2 of the following form y n 1 f y n y n k 1 n 0 1 2 we develop necessary and or sufficient conditions on f so that every solution of the above mentioned difference equation is periodic of the same period p

maximum allowable loss probability for consensus of multi agent systems over random weighted lossy networks this note studies the consensus seeking problem for a team of general linear dynamical agents that communicate via a weighted random lossy network linear state feedback consensus protocols are applied and both the weights and feedback gain are treated as control parameters in the protocol it is shown that the weights and the link loss probabilities of a network have non negligible effects on the consensus seeking ability of multi agent systems firstly a weight condition characterized by the eigenvalues of the weighted laplacian matrix is given for systems over ideal communication networks without packet losses secondly based on stochastic stability analysis a maximum allowable loss probability bound is proposed for systems over random lossy networks as long as the link loss probabilities of the network are less than this bound and the mean topology has spanning trees there exist linear protocols solving the mean square consensus problem of the system

sensor networks with random links topology design for distributed consensus in a sensor network in practice the communication among sensors is subject to 1 errors that can cause failures of links among sensors at random times 2 costs and 3 constraints such as power data rate or communication since sensors and networks operate under scarce resources the paper studies the problem of designing the topology i e assigning the probabilities of reliable communication among sensors or of link failures to maximize the rate of convergence of average consensus when the link communication costs are taken into account and there is an overall communication budget constraint we model the network as a bernoulli random topology and establish necessary and sufficient conditions for mean square sense mss and almost sure a s convergence of average consensus when network links fail in particular a necessary and sufficient condition is for the algebraic connectivity of the mean graph topology to be strictly positive with these results we show that the topology design with random link failures link communication costs and a communication cost constraint is a constrained convex optimization problem that can be efficiently solved for large networks by semidefinite programming techniques simulations demonstrate that the optimal design improves significantly the convergence speed of the consensus algorithm and can achieve the performance of a non random network at a fraction of the communication cost

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

reach almost sure consensus with only group information this brief presents a new distributed scheme to solve the consensus problem for a group of agents if neither their absolute states nor inter agent relative states are available the new scheme considers a random partition of agents into two subgroups at each step and then uses the relative group representative state as feedback information for the consensus purpose it is then shown that almost sure consensus can be achieved under the proposed scheme in both discrete time and continuous time for the discrete time case almost sure consensus is achieved if and only if the weighting parameter for state update is greater than one for the continuous time case almost sure consensus is realized when the weighting parameter is positive moreover it is shown that if a uniform probability is considered for group selection then the group of agents can reach average consensus in mean

on the recursive sequence xn 1 α βxn k 1a bxn k 1 cxn 2k 1 abstract in this paper we study the global asymptotic stability invariant intervals the character of semicycles and the boundedness of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 a bx n k 1 cx n 2 k 1 n 0 1 where a b c and are positive k 1 2 3 and the initial conditions x 2k 1 x 1 x0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable finally we present some informative examples

covariance of moore penrose inverses with respect to an invertible matrix abstract given an invertible complex matrix t a description is given of those matrices a with moore penrose inverse a such that tat 1 ta t 1

existence uniqueness and attractivity of prime period two solution for a difference equation of exponential form abstract in this paper we study the existence uniqueness and attractivity of prime period two solution of the difference equation x n 1 a bx n 1 e x n where a b are positive constants and the initial values x 1 x 0 are positive numbers

oscillation and asymptotic behavior of a class of higher order nonlinear recursive sequences abstract we investigate the global attractivity the periodic behavior and the invariant interval with the semicycles character of solutions of the equation x n 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 for all admissible non negative values of the parameters b c and the initial conditions x 2 k 1 x 1 x 0 we show that the solutions exhibit a periodic attitude depending on the parameters of the family of the above mentioned difference equation in comparison with each other

global stability of a higher order rational recursive sequence abstract in this paper we study the qualitative behavior of solutions of the class of delay difference equation x n 1 x n k 1 x n 2 k 1 a bx n k 1 n 0 1 2 where the initial conditions x 2 k 1 x 1 x 0 are positive k 1 2 and the parameters a b are positive our concentration is on invariant intervals and the global stability of the above mentioned equation we obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients furthermore the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed it is also illustrated that for some special case of parameters the solution will be either a period 2 solution or will converge to the equilibrium point or will have unbounded solutions

asymptotic behavior of the positive solutions of an exponential type system of difference equations in this paper we study the existence of a unique positive equilibrium the boundedness persistence and global attractivity of the positive solutions of a system of the following two difference equations x n 1 ax n by n 1 e x n y n 1 cy n dx n 1 e y n n 0 1 where a b c d are positive constants and the initial values x 1 x 0 y 1 y 0 are positive numbers

on the covariance of the moore penrose inverse abstract it is readily seen that the moore penrose inverse a of a singular square matrix a does not for all regular matrices t satisfy the covariance condition tat 1 ta t 1 thus the problem arises to describe the class c a of all those matrices t for which this condition is valid the problem is solved for all matrices a of rank one and two

stability of equilibrium states for a stochastically perturbed exponential type system of difference equations in the paper it is shown how the known results of stability theory can be simply applied to stability investigation of equilibrium points of some systems of nonlinear difference equations with stochastic perturbations a system of two difference equations with exponential nonlinearity is considered and it is shown that instead of the zero equilibrium this system can have also a positive equilibrium sufficient conditions for stability in probability of the both equilibriums of the initial nonlinear system with stochastic perturbations are obtained numerical simulations and figures illustrate a convergence of the positive solutions of the considered system to one of two zero or positive equilibriums in deterministic and stochastic cases the proposed investigation procedure can be applied for arbitrary nonlinear equations with an order of nonlinearity higher than one

study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form in this paper we study the boundedness the persistence and the asymptotic behavior of the positive solutions of the following systems of two difference equations of exponential form xn 1 a be yn c yn 1 y n 1 d e xn f xn 1

dynamics of a rational difference equation using both theoretical and computational approaches this paper is concerned with the qualitative behavior of solutions to the difference equationx n 1 x n px n qx n k n 0 1 2 where the initial conditions x k x 1 x 0 are non negative email protected 1 2 3 and the parameters p q are non negative our concentration is on invariant intervals the character of semicycles the global stability and the boundedness of the above mentioned equation it is worth to mention that this difference equation was an open problem introduced by kulenovic and ladas in m r s kulenovic and g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002 our final comments are about informative examples

a necessary and sufficient condition for consensus over random networks we consider the consensus problem for stochastic discrete time linear dynamical systems the underlying graph of such systems at a given time instance is derived from a random graph process independent of other time instances for such a framework we present a necessary and sufficient condition for almost sure asymptotic consensus using simple ergodicity and probabilistic arguments this easily verifiable condition uses the spectrum of the average weight matrix finally we investigate a special case for which the linear dynamical system converges to a fixed vector with probability 1

distributed consensus algorithms in sensor networks with imperfect communication link failures and channel noise the paper studies average consensus with random topologies intermittent links and noisy channels consensus with noise in the network links leads to the bias variance dilemma running consensus for long reduces the bias of the final average estimate but increases its variance we present two different compromises to this tradeoff the a nd algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition slowly decaying to zero and the a nc algorithm where the weights are constant but consensus is run for a fixed number of iterations iota then it is restarted and rerun for a total of p runs and at the end averages the final states of the p runs monte carlo averaging we use controlled markov processes and stochastic approximation arguments to prove almost sure convergence of a nd to a finite consensus limit and compute explicitly the mean square error mse variance of the consensus limit we show that a nd represents the best of both worlds zero bias and low variance at the cost of a slow convergence rate rescaling the weights balances the variance versus the rate of bias reduction convergence rate in contrast a nc because of its constant weights converges fast but presents a different bias variance tradeoff for the same number of iterations iota p shorter runs smaller iota lead to high bias but smaller variance larger number p of runs to average over for a static nonrandom network with gaussian noise we compute the optimal gain for a nc to reach in the shortest number of iterations iota p with high probability 1 delta epsiv delta consensus epsiv residual bias our results hold under fairly general assumptions on the random link failures and communication noise

global attractivity for a class of higher order nonlinear difference equations our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equationx n 1 a bx email protected k i 0b ix n i n 0 1 where a b b email protected 0 a b 0 b k email protected 0 email protected 1 2 and the initial conditions x k x 0 are arbitrary real numbers we show that one positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

exponential consensus of general linear multi agent systems under directed dynamic topology this paper aims to investigate the consensus control of generic linear multi agent systems mass under directed dynamic topology nonnegative matrix theory in particular the product properties of infinite row stochastic matrices which are widely used for multiple integrator agents is explored to deal with the convergence analysis of generic linear mass it is finally shown that the exponential consensus can be reached under very relaxed conditions i e the directed interaction topology is only required to be repeatedly jointly rooted and the exponentially unstable mode of each individual system is weak enough moreover a least convergence rate and a bound for the unstable mode of the individual agent system both of which are independent of the switching mode can be explicitly specified

a coupled system of rational difference equations abstract we investigate the global stability properties and asymptotic behavior of solutions of the recursive sequence where the parameters a b c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers

on the dynamics of two exponential type systems of difference equations in this paper we study the asymptotic behavior of the positive solutions of the systems of the two difference equations i x n 1 a by n 1e x n y n 1 c dx n 1e y n ii x n 1 a by n 1e y n y n 1 c dx n 1e x n where the constants a b c d are positive real numbers and the initial values x 1 x 0 y 1 y 0 are also positive real numbers

on the covariance of the moore penrose inverse abstract given a square complex matrix a with moore penrose inverse a we describe the class of invertible matrices t such that tat 1 ta t 1

global asymptotic behavior of a two dimensional difference equation modelling competition we investigate the global asymptotic behavior of solutions of the system of difference equations xn 1 xn a cyn yn 1 yn b dxn n 0 1 where the parameters a and b are in 0 1 c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers we show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior in the case where a b we find an explicit equation for the stable manifold

eventually constant solutions of a rational difference equation we describe all the solutions of a rational difference equation from putnam s mathematical competition which are eventually equal to its positive equilibrium email protected 1 as a consequence we give a new elegant and short proof of the fact that the equation has a positive solution which is not eventually equal to one moreover we show that almost all solutions of the equation are not eventually equal to one

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

consensus seeking over random weighted directed graphs we examine the consensus problem for a group of agents that communicate via a stochastic information network communication among agents is modeled as a weighted directed random graph that switches periodically the existence of any edge is probabilistic and independent from the existence of any other edge we further allow each edge to be weighted differently sufficient conditions for asymptotic almost sure consensus are presented for the case of positive weights and for the case of arbitrary weights

a necessary and sufficient condition for permanence of a lotka volterra discrete system with delays abstract in this paper we consider the permanence of the following lotka volterra discrete competition system with delays k1 k2 l1 and l2 x n 1 x n exp r1 1 x n k1 1y n k2 y n 1 y n exp r2 1 2x n l1 y n l2 we show the system is permanent for all nonnegative integers k1 k2 l1 and l2 if and only if 1

leader follower containment control over directed random graphs the leader follower consensus problem for multi agent systems over directed random graphs is investigated motivated by the fact that inter agent communication can be subject to random failure when agents perform tasks in a complex environment a directed random graph is used to model the random loss of communication between agents where the connection of the directed edge in the graph is assumed to be probabilistic and evolves according to a two state markov model in the leader follower network the leaders maintain a constant desired state and the followers update their states by communicating with local neighbors over the random communication network based on convex properties and a stochastic version of lasalle s invariance principle almost sure convergence of the followers states to the convex hull spanned by the leaders states is established for the leader follower random network a numerical simulation is provided to demonstrate the developed result

on the recursive sequence xn 1 α βxn k 1 γxn 2k 1bxn k 1 cxn 2k 1 abstract our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients our concentration is on invariant intervals the character of semicycles and the boundedness of the above mentioned equation our final comments are about informative examples

the characteristics of a higher order rational difference equation in this paper we study the global behavior of the higher order nonlinear difference equationx n email protected email protected n k email protected n 2 k 1a bx n k 1 n 0 1 where a b and a b c are positive email protected 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we investigate the character of semicycles periodicity global stability and boundedness of above mentioned equation

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

on lavoie s determinantal inequality abstract let the complex matrix a be partitioned into r blocks a a 1 a r and let b a 1 a r where the superscript plus denotes moore penrose inverse then det ab 1 with equality if and only if ab i the matrices a b ab and ba all have the same rank which is equal to the number of nonzero eigenvalues of ab these nonzero eigenvalues are all real and positive when rank a r1 rank ai then the product of the nonzero eigenvalues of ab cannot exceed 1 equality occurs if and only if b a

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature

consensus condition for linear multi agent systems over randomly switching topologies abstract this paper studies both continuous and discrete time consensus problems for multi agent systems with linear time invariant agent dynamics over randomly switching topologies the switching is governed by a time homogeneous markov process whose state corresponds to a possible interaction topology among agents necessary and sufficient conditions are derived for achieving consensus under a common control protocol respectively it is shown that the effect of switching topologies on consensus is determined by the union of topologies associated with the positive recurrent states of the markov process moreover the effect of random link failures on discrete time consensus is investigated the implications and relationships with the existing results are discussed finally the theoretical results are validated via simulations

oscillation and global asymptotic stability in a discrete epidemic model abstract in this paper we study the oscillation global asymptotic stability and other properties of the positive solutions of the difference equation x n 1 1 j 0 k 1 x n j 1 e ax n where a 0 k 2 3 and the initial values x k 1 x k 2 x0 are arbitrary real positive numbers such that j 0k 1x j

synchronization and convergence of linear dynamics in random directed networks recently methods in stochastic control are used to study the synchronization properties of a nonautonomous discrete time linear system x k 1 g k x k where the matrices g k are derived from a random graph process the purpose of this note is to extend this analysis to directed graphs and more general random graph processes rather than using lyapunov type methods we use results from the theory of inhomogeneous markov chains in our analysis these results have been used successfully in deterministic consensus problems and we show that they are useful for these problems as well sufficient conditions are derived that depend on the types of graphs that have nonvanishing probabilities for instance if a scrambling graph occurs with nonzero probability then the system synchronizes

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper