a survey on oscillation of impulsive delay differential equations this paper presents a systematic development of the oscillatory results for the linear and nonlinear impulsive delay differential equations we also discuss linearized oscillation theory and give some applications to mathematical biology

finite size effects in roughness distribution scaling we study numerically finite size corrections in scaling relations for roughness distributions of various interface growth models the most common relation which considers the average roughness as a scaling factor is not obeyed in the steady states of a group of ballisticlike models in 2 1 dimensions even when very large system sizes are considered on the other hand good collapse of the same data is obtained with a scaling relation that involves the root mean square fluctuation of the roughness which can be explained by finite size effects on second moments of the scaling functions we also obtain data collapse with an alternative scaling relation that accounts for the effect of the intrinsic width which is a constant correction term previously proposed for the scaling of this illustrates how finite size corrections can be obtained from roughness distributions scaling however we discard the usual interpretation that the intrinsic width is a consequence of high surface steps by analyzing data of restricted solid on solid models with various maximal height differences between neighboring columns we also observe that large finite size corrections in the roughness distributions are usually accompanied by huge corrections in height distributions and average local slopes as well as in estimates of scaling exponents the molecular beam epitaxy model of das sarma and tamborenea in 1 1 dimensions is a case example in which none of the proposed scaling relations work properly while the other measured quantities do not converge to the expected asymptotic values thus although roughness distributions are clearly better than other quantities to determine the universality class of a growing system it is not the final solution for this task

backstepping controller synthesis and characterizations of incremental stability incremental stability is a property of dynamical and control systems requiring the uniform asymptotic stability of every trajectory rather than that of an equilibrium point or a particular time varying trajectory similarly to stability lyapunov functions and contraction metrics play important roles in the study of incremental stability in this paper we provide characterizations and descriptions of incremental stability in terms of existence of coordinate invariant notions of incremental lyapunov functions and contraction metrics respectively most design techniques providing controllers rendering control systems incrementally stable have two main drawbacks they can only be applied to control systems in either parametric strict feedback or strict feedback form and they require these control systems to be smooth in this paper we propose a design technique that is applicable to larger classes of control systems including a class of non smooth control systems moreover we propose a recursive way of constructing contraction metrics for smooth control systems and incremental lyapunov functions which have been identified as a key tool enabling the construction of finite abstractions of nonlinear control systems the approximation of stochastic hybrid systems source code model checking for nonlinear dynamical systems and so on the effectiveness of the proposed results in this paper is illustrated by synthesizing controllers rendering two non smooth control systems incrementally stable the first example aims to show how to recursively construct the incremental lyapunov functions the second example aims to show the key role of the computed incremental lyapunov function in constructing a finite abstraction that is equivalent to the system under study

roughness distribution of multiple hit and long surface diffusion length noise reduced discrete growth models abstract conventionally the universality class of a discrete growth model is identified via the scaling of interface width this method requires large scale simulations to minimize finite size effects on the results the multiple hit noise reduction techniques m 1 nrt and the long surface diffusion length noise reduction techniques l 1 nrt have been used to promote the asymptotic behaviors of the growth models lately an alternative method involving comparison of roughness distribution in the steady state has been proposed in this work the roughness distribution of the 2 1 dimensional das sarma tamborenea dt wolf villain wv and larger curvature lc models with and without nrts are calculated in order to investigate effects of the nrts on the roughness distribution additionally effective growth exponents of the noise reduced 2 1 dimensional dt wv and lc models are also calculated our results indicate that the nrts affect the interface width both in the growth and the saturation regimes in the steady state the nrts do not seem to have any impact on the roughness distribution of the dt model but it significantly changes the roughness distribution of the wv and lc models to the normal distribution curves

synchronization and transient stability in power networks and nonuniform kuramoto oscillators motivated by recent interest for multiagent systems and smart grid architectures we discuss the synchronization problem for the network reduced model of a power system with nontrivial transfer conductances our key insight is to exploit the relationship between the power network model and a first order model of coupled oscillators assuming overdamped generators possibly due to local excitation controllers a singular perturbation analysis shows the equivalence between the classic swing equations and a nonuniform kuramoto model here nonuniform kuramoto oscillators are characterized by multiple time constants nonhomogeneous coupling and nonuniform phase shifts extending methods from transient stability synchronization theory and consensus protocols we establish sufficient conditions for synchronization of nonuniform kuramoto oscillators these conditions reduce to necessary and sufficient tests for the standard kuramoto model combining our singular perturbation and kuramoto analyses we derive

oscillation properties of a second order impulsive delay differential equation abstract for the second order delay differential equation y t a t y t i 1 n p i t y g i t 0 t 0 t t k with impulsive conditions y t y t b y t y t y t b y t an explicit necessary and sufficient condition for all bounded solutions to be oscillatory is obtained by the comparison theorem on bounded oscillation of the impulsive differential equation with the corresponding nonimpulsive differential equation

convergent systems vs incremental stability two similar stability notions are considered one is the long established notion of convergent systems the other is the younger notion of incremental stability both notions require that any two solutions of a system converge to each other yet these stability concepts are different in the sense that none implies the other as is shown in this paper using two examples it is shown under what additional assumptions one property indeed implies the other furthermore this paper contains necessary and sufficient characterizations of both properties in terms of lyapunov functions

a contraction approach to the hierarchical analysis and design of networked systems this brief is concerned with the stability of continuous time networked systems using contraction theory a result is established on the network structure and the properties of the individual component subsystems and their couplings to ensure the overall contractivity of the entire network specifically it is shown that a contraction property on a reduced order matrix that quantifies the interconnection structure coupled with contractivity expansion estimates on the individual component subsystems suffices to ensure that trajectories of the overall system converge towards each other

a contraction theory approach to stochastic incremental stability we investigate the incremental stability properties of ito stochastic dynamical systems specifically we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system this bound can be expressed as a function of the noise intensity and the contraction rate of the noise free system we illustrate these results in the contexts of nonlinear observers design and stochastic synchronization

brief paper stability and robustness analysis of nonlinear systems via contraction metrics and sos programming a wide variety of stability and performance questions about linear dynamical systems can be reformulated as convex optimization problems involving linear matrix inequalities lmis these techniques have been recently extended to nonlinear systems with polynomial or rational dynamics through the use of sum of squares sos programming in this paper we further extend the class of systems that can be analyzed with convexity based methods we show how to analyze the robust stability properties of uncertain nonlinear systems with polynomial or rational dynamics via contraction analysis and sos programming since the existence of a global contraction metric is a sufficient condition for global stability of an autonomous system we develop an algorithm for finding such contraction metrics using sos programming the search process is made computationally tractable by relaxing matrix definiteness constraints the feasibility of which indicates the existence of a contraction metric to sos constraints on polynomial matrices we illustrate our results through examples from the literature and show how our contraction based approach offers advantages when compared with traditional lyapunov analysis

numerical solution of the higher order linear fredholm integro differential difference equation with variable coefficients the main aim of this paper is to apply the legendre polynomials for the solution of the linear fredholm integro differential difference equation of high order this equation is usually difficult to solve analytically our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted legendre polynomials with unknown coefficients the operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted legendre polynomials illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results

numerical study of the kardar parisi zhang equation we integrate numerically the kardar parisi zhang kpz equation in 1 1 and 2 1 dimensions using a euler discretization scheme and the replacement of nablah 2 by exponentially decreasing functions of that quantity to suppress instabilities when applied to the equation in 1 1 dimensions the method of instability control provides values of scaling amplitudes consistent with exactly known results in contrast to the deviations generated by the original scheme in 2 1 dimensions we spanned a range of the model parameters where transients with edwards wilkinson or random growth are not observed in box sizes 8 or l or 128 we obtain a roughness exponent of 0 37 or alpha or 0 40 and steady state height distributions with skewness s 0 25 0 01 and kurtosis q 0 15 0 1 these estimates are obtained after extrapolations to the large l limit which is necessary due to significant finite size effects in the estimates of effective exponents and height distributions on the other hand the steady state roughness distributions show weak scaling corrections and evidence of stretched exponential tails these results confirm previous estimates from lattice models showing their reliability as representatives of the kpz class

oscillation of the bounded solutions of impulsive differential difference equations of second order the present paper is devoted to a class of impulsive differential difference equations of second order the moments of impulse effect are fixed ones sufficient conditions are found for unboundedness of the solutions and oscillation of the bounded solutions

skewness and kurtosis of height distribution of thin films simulated by larger curvature model with noise reduction techniques abstract time varying skewness s and kurtosis q of height distribution of 2 1 dimensional larger curvature lc model with and without noise reduction techniques nrts are investigated in both transient and steady state regimes in this work effects of the multiple hit nrt m 1 nrt and the long surface diffusion length nrt l 1 nrt on the surface morphologies and characteristics of s and q are studied in the early growth time plots of s and q versus time of the m 1 morphologies show pronounced oscillation indicating the layer by layer growth our results show that s 0 and q 0 at every half layer while s 0 and q 0 at every complete layer the results are confirmed by the same plots of the results from the das sarma tamborenea dt model the l 1 lc model on the other hand has no evidence of the layer by layer growth mode due to the rapidly damped oscillation of s and q in the steady state the m 1 and l 1 nrts affect weakly on the values of s and q and the mounded morphologies of the film this lead to the evidence of universality of s and q in the steady state of the lc models with various m and l the finite size effect on the values of s and q is found to be very weak in the lc model by extrapolating to l we obtain s l 0 05 and q l 0 62 which are in agreement with the nrts results

certifying spatially uniform behavior in reaction diffusion pde and compartmental ode systems we present a condition that guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction diffusion pde with neumann boundary conditions this condition makes use of the jacobian matrix of the reaction terms and the second neumann eigenvalue of the laplacian operator on the given spatial domain and eliminates the global lipschitz assumptions commonly used in mathematical biology literature we then derive numerical procedures that employ linear matrix inequalities to certify this condition and illustrate these procedures on models of several biochemical reaction networks finally we present an analog of this pde result for the synchronization of a network of identical ode models coupled by diffusion terms from a systems biology perspective the main contribution of the paper is to blend analytical and numerical tools from nonlinear systems and control theory to derive a relaxed and verifiable condition for spatial uniformity of biological processes

contraction theory on riemannian manifolds abstract contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another in this work we present the fundamental results of contraction theory in an intrinsic coordinate free setting with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties we provide coordinate free proofs of the main results for autonomous vector fields and clarify the assumptions under which the results hold we state and prove several interesting corollaries to the main result study cascade and feedback interconnections of contracting systems study some simple examples and highlight how contraction theory has arisen independently in other scientific disciplines we conclude by illustrating the developed theory for the case of gradient dynamics

on approximating contractive systems we consider approximating the trajectories of a contractive system using a simpler system e g an lti system we derive bounds on the approximation error that depend on the trajectories of the simpler system and demonstrate their usefulness using several models from systems biology

bessel polynomial solutions of high order linear volterra integro differential equations abstract in this study a practical matrix method which is based on collocation points is presented to find approximate solutions of high order linear volterra integro differential equations vides under the mixed conditions in terms of bessel polynomials numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results the results show the efficiency and accuracy of the present work all of the numerical computations have been performed on the computer using a program written in matlab v7 6 0 r2008a

a new collocation method for solution of mixed linear integro differential difference equations numerical solution of mixed linear integro differential difference equation is presented using chebyshev collocation method the aim of this article is to present an efficient numerical procedure for solving mixed linear integro differential difference equations our method depends mainly on a chebyshev expansion approach this method transforms mixed linear integro differential difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system maple10

width distributions and the upper critical dimension of kardar parisi zhang interfaces simulations of restricted solid on solid growth models are used to build the width distributions of d 2 5 dimensional kardar parisi zhang kpz interfaces we find that the universal scaling function associated with the steady state width distribution changes smoothly as d is increased thus strongly suggesting that d 4 is not an upper critical dimension for the kpz equation the dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity

scaling cumulant ratios and height distribution of ballistic deposition in 3 1 and 4 1 dimensions we investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by alves et al phys rev e 90 052405 2014 in d 2 1 dimensions where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account in the present work we show that this concept holds for d 3 1 and 4 1 dimensions we have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models presenting small finite time corrections to the scaling that in principle belong to the kardar parisi zhang kpz universality class in both d 3 1 and 4 1 our results constitute further evidence that the upper critical dimension of the kpz class if it exists is larger than 4

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

logarithmic lipschitz norms and diffusion induced instability abstract this paper proves that ordinary differential equation systems that are contractive with respect to l p norms remain so when diffusion is added thus diffusive instabilities in the sense of the turing phenomenon cannot arise for such systems and in fact any two solutions converge exponentially to each other the key tools are semi inner products and logarithmic lipschitz constants in banach spaces an example from biochemistry is discussed which shows the necessity of considering non hilbert spaces an analogous result for graph defined interconnections of systems defined by ordinary differential equations is given as well

contraction after small transients contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation we consider three generalizations of contraction with respect to a norm that allow contraction to take place after small transients in time and or amplitude these generalized contractive systems gcss are useful for several reasons first we show that there exist simple and checkable conditions guaranteeing that a system is a gcs and demonstrate their usefulness using several models from systems biology second allowing small transients does not destroy the important asymptotic properties of contractive systems like convergence to a unique equilibrium point if it exists and entrainment to a periodic excitation third in some cases as we change the parameters in a contractive system it becomes a gcs just before it looses contractivity with respect to a norm in this respect generalized contractivity is the analogue of marginal stability in lyapunov stability theory

a taylor polynomial approach for solving differential difference equations the purpose of this study is to give a taylor polynomial approximation for the solution of mth order linear differential difference equations with variable coefficients under the mixed conditions about any point for this purpose taylor matrix method is introduced this method is based on first taking the truncated taylor expansions of the functions in the differential difference equations and then substituting their matrix forms into the equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed also we have discussed the accuracy of the method we use the symbolic algebra program maple to prove our results

a differential lyapunov framework for contraction analysis lyapunov s second theorem is an essential tool for stability analysis of differential equations the paper provides an analog theorem for incremental stability analysis by lifting the lyapunov function to the tangent bundle the lyapunov function endows the state space with a finsler structure incremental stability is inferred from infinitesimal contraction of the finsler metrics through integration along solutions curves