stability analysis for positive singular systems with time varying delays this paper is concerned with the stability analysis for continuous time positive singular systems with time varying delays first an auxiliary system is introduced to establish the positivity condition for singular time delay systems based on the positivity condition the stability criterion is obtained for the positive singular systems with constant delays by analyzing the monotonic property of the system trajectory we extend the stability condition to the cases with time varying delays a numerical simulation is employed to illustrate the effectiveness of our results

on the decay rates of homogeneous positive systems of any degree with time varying delays this technical note studies the stability problem of homogeneous positive systems of any degree with time varying delays delay independent conditions are derived for asymptotic and finite time stability estimates on the decay rates which reveal how the system delays affect the rates of convergence are obtained more precisely this technical note features three contributions first we derive a necessary and sufficient condition for global polynomial stability of continuous time homogeneous cooperative systems with time varying delays when the degree of homogeneity is greater than one second we characterize finite time stability of continuous time homogeneous cooperative delay free systems of degree smaller than one finally for discrete time positive systems with time varying delays a local exponential stability criterion is established when the vector fields are order preserving and homogeneous of degree greater than one an illustrative example is given to show the effectiveness of our results

a continuous family of equilibria in ferromagnetic media are ground states we show that a foliation of equilibria a continuous family of equilibria whose graph covers all the configuration space in ferromagnetic transitive models are ground states the result we prove is very general and it applies to models with long range and many body interactions as an application we consider several models of networks of interacting particles including models of frenkel kontorova type on mathbb z d and one dimensional quasi periodic media the result above is an analogue of several results in the calculus of variations fields of extremals and in pde s since the models we consider are discrete and long range new proofs need to be given we also note that the main hypothesis of our result the existence of foliations of equilibria is the conclusion using kam theory of several recent papers hence we obtain that the kam solutions recently established are minimizers when the interaction is ferromagnetic and transitive these concepts are defined later

analytical methods for solving the time fractional swift hohenberg s h equation in this paper the most effective methods the homotopy perturbation method hpm and the differential transform method dtm are applied to obtain the approximate solutions of the nonlinear time fractional swift hohenberg s h equation the basic philosophy of these methods does not involve linearization weak nonlinearity assumptions or perturbation theory numerical solutions for various combinations of the parameters a eigenvalue parameter l length and a fractional index are obtained the solutions of the s h equation are useful for studies of shear thinning effects in non newtonian fluid flows at the end the solutions obtained are also presented graphically

the structure of general interference functions and applications this paper provides a theoretical framework for the analysis of interference coupled multiuser systems the fundamental behavior of such a system is modeled by interference functions defined by axioms ldquononnegativity rdquoscale invariance rdquo and ldquomonotonicity rdquo it is shown that every interference function has an interpretation as the optimum of a min max problem where the optimization is over a closed comprehensive positive coefficient set this provides new insight into the structure of general interference functions and its elementary building blocks conversely it is shown that every closed comprehensive positive set can be expressed as a level set of an interference function this shows a close connection between the analysis of interference functions and multiuser performance regions which are typically closed comprehensive the generality of this framework allows for a wide range of potential applications as an example we analyze the problem of interference balancing

positive forms and stability of linear time delay systems we consider the problem of constructing lyapunov functions for linear differential equations with delays for such systems it is known that exponential stability implies the existence of a positive lyapunov function which is quadratic on the space of continuous functions we give an explicit parameterization of a sequence of finite dimensional subsets of the cone of positive lyapunov functions using positive semidefinite matrices this allows stability analysis of linear time delay systems to be formulated as a semidefinite program

model reduction via parametrized locally invariant manifolds some examples a method for model reduction in non linear ode systems is demonstrated through computational examples the method does not require an implicit separation of time scales in the fine dynamics to be effective from the computational standpoint the method has the potential of serving as a subgrid modeling tool from the physical standpoint it provides a model for interpreting and describing history dependence in coarse grained response of an autonomous system

existence of dynamical hull functions with two variables for the ac driven frenkel kontorova model abstract in this paper we are concerned with the dynamics of incommensurate structures in the overdamped limit case of the frenkel kontorova model with periodic driving force we show in the infinite dimensional phase space there exists an ordered circle on which both the poincare map and the space shift map induce orientation preserving circle homeomorphisms with rotation numbers and respectively the latter characterizing the mean spacing of particles furthermore we prove that the average velocity for the particle chain with irrational exists and t where t denotes the period of the driving force and it is continuous and nondecreasing with respect to f the average of the external driving force if the shift map on the invariant ordered circle is ergodic we demonstrate the existence of two variable dynamical hull functions and that there are no mode locking behaviors

d stability and delay independent stability of homogeneous cooperative systems we introduce a nonlinear definition of d stability extending the usual concept for positive linear time invariant systems we show that globally asymptotically stable cooperative systems homogeneous of any order with respect to arbitrary dilation maps are d stable we also prove a strong stability result for delayed cooperative homogeneous systems finally we show that both of these results also hold for planar cooperative systems without the restriction of homogeneity

on the decay rate of discrete time linear delay systems with cone invariance this technical note is concerned with the decay rate constrained stability analysis for linear delay systems that possess cone invariant property in order to capture the decay rate of such systems we introduce a nondecreasing positive function whose reciprocal represents the decay rate under mild assumptions on the growth rate of this function an explicit condition is given to ensure that a cone preserving linear system with unbounded time varying delays is asymptotically stable with a given decay rate as typical cases necessary and sufficient conditions are given to characterize the decay rate when the delay is restricted by a linear sublinear or logarithmic growth rate finally some numerical examples are given to illustrate the effectiveness of the theoretical results

extension of the perron frobenius theorem to homogeneous systems this paper deals with a particular class of positive systems the state components of a positive system are positive or zero for all positive times these systems are often encountered in applied areas such as chemical engineering or biology it is shown that for this particular class the first orthant contains an invariant ray in its interior an invariant ray generalizes the concept of an eigenvector of linear systems to nonlinear homogeneous systems then sufficient conditions for uniqueness of this ray are given the main result states that the vector field on an invariant ray determines the stability properties of the zero solution with respect to initial conditions in the first orthant the asymptotic behavior of the solutions is examined finally we compare our results to the perron frobenius theorem which gives a detailed picture of the dynamical behavior of positive linear systems

stability and performance of continuous time power control in wireless networks this paper develops a comprehensive stability analysis framework for general classes of continuous time power control algorithms under heterogeneous time varying delays our first set of results establish global asymptotic stability of power control laws involving two sided scalable interference functions and include earlier work on standard interference functions as a special case we then consider contractive interference functions and demonstrate that the associated continuous time power control laws always have unique fixed points which are exponentially stable even under bounded heterogeneous time varying delays for this class of interference functions we present explicit bounds on the decay rate that allow us to quantify the impact of delays on the convergence time of the algorithm when interference functions are linear we also prove that contractivity is necessary and sufficient for exponential stability of continuous time power control algorithms with time varying delays finally numerical simulations illustrate the validity of our theoretical results

stability analysis for continuous time positive systems with time varying delays this note addresses the stability problem of continuous time positive systems with time varying delays it is shown that such a system is asymptotically stable for any continuous and bounded delay if and only if the sum of all the system matrices is a hurwitz matrix the result is a time varying version of the widely known asymptotic stability criterion for constant delay positive systems a numerical example illustrates the correctness of our result

absolute exponential l1 gain analysis and synthesis of switched nonlinear positive systems with time varying delay this paper investigates absolute exponential stability and stabilization with l1 gain performance of switched nonlinear positive systems with time varying delay nonlinear functions considered in the paper are located in a sector field and the time varying delay is unknown but bounded the notion of absolute exponential l1 stability of switched nonlinear positive systems is first introduced a nonlinear lyapunov krasovskii functional is constructed for the underlying systems by using the lyapunov krasovskii functional a sufficient condition for absolute exponential l1 stability of the systems is established in terms of linear programming then absolute exponential l1 control synthesis of the systems is addressed a feedback control law containing nonlinear functions and a state feedback control law are designed respectively by comparing with existing results it is shown that the present design is less conservative finally an illustrative example is provided to illustrate the effectiveness of the results

on exponential stability of switched homogeneous positive systems of degree one in this paper we investigate the stability problem of switched homogeneous positive systems of degree one we first show that the positiveness of the considered system is guaranteed when each switched subsystem is cooperative then an integral excitation condition is formulated to ensure the exponential stability of the switched homogeneous positive system of degree one in particular the proposed condition allows that some or even all switched subsystems are only lyapunov stable rather than asymptotically stable in the second part of this paper we further consider the cases of sub homogeneous systems time varying delay systems and discrete time systems in the sub homogeneous case we first show that the previous integral excitation condition is also sufficient such that the exponential stability is achieved we also derive an exponential stability result by relaxing the cooperative and integral excitation conditions in the time varying delay case a delay independent exponential stability result is shown under the extended integral excitation condition finally in the discrete time case based on the fact that each switched subsystem is order preserving instead of cooperative the exponential stability of the considered system is shown under a weak excitation condition

brief paper unconditional stability of the foschini miljanic algorithm in this note we prove the unconditional stability of the foschini miljanic algorithm our results show that the foschini miljanic algorithm is unconditionally stable convergent even in the presence of bounded time varying communication delays and in the presence of topology changes the implication of our results may be important for the design of code division multiple access cdma based wireless networks

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

reachability of a class of discrete time positive switched systems in this paper monomial reachability and reachability properties for the special class of single input discrete time positive systems that switch among p subsystems sharing the same state transition matrix are investigated necessary and sufficient conditions for these properties to hold together with some examples are provided

reproducing kernel method for the numerical solution of the 1d swift hohenberg equation abstract the swift hohenberg equation is a nonlinear partial differential equation of fourth order that models the formation and evolution of patterns in a wide range of physical systems we study the 1d swift hohenberg equation in order to demonstrate the utility of the reproducing kernel method the solution is represented in the form of a series in the reproducing kernel space and truncating this series representation we obtain the n term approximate solution in the first approach we aim to explain how to construct a reproducing kernel method without using gram schmidt orthogonalization as orthogonalization is computationally expensive this approach will therefore be most practical for obtaining numerical solutions gram schmidt orthogonalization is later applied in the second approach despite the increased computational time as this approach will prove theoretically useful when we perform a formal convergence analysis of the reproducing kernel method for the swift hohenberg equation we demonstrate the applicability of the method through various test problems for a variety of initial data and parameter values

exponential stability of homogeneous positive systems of degree one with time varying delays while the asymptotic stability of positive linear systems in the presence of bounded time delays has been thoroughly investigated the theory for nonlinear positive systems is considerably less well developed this technical note presents a set of conditions for establishing delay independent stability and bounding the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case specifically when the time delays have a known upper bound we derive necessary and sufficient conditions for exponential stability of a continuous time positive systems whose vector fields are homogeneous and cooperative and b discrete time positive systems whose vector fields are homogeneous and order preserving we then present explicit expressions that allow us to quantify the impact of delays on the decay rate and show that the best decay rate of positive linear systems that our bounds provide can be found via convex optimization finally we extend the results to general linear systems with time varying delays

analytical and numerical results for the swift hohenberg equation the problem of the swift hohenberg equation is considered in this paper using homotopy analysis method ham the series solution is developed and its convergence is discussed and documented here for the first time in particular we focus on the roles of the eigenvalue parameter a and the length parameter l on the large time behaviour of the solution for a given time t we obtain analytical expressions for eigenvalue parameter a and length l which show how different values of these parameters may lead to qualitatively different large time profiles also the results are presented graphically the results obtained reveal many interesting behaviors that warrant further study of the equations related to non newtonian fluid phenomena especially the shear thinning phenomena shear thinning reduces the wall shear stress

on the aubry mather theory in statistical mechanics we generalize aubry mather theory for configurations on the line to general sets with a group action cocycles on the group play the role of rotation numbers the notion of birkhoff configuration can be generalized to this setting under mild conditions on the group we show how to find birkhoff ground states for many body interactions which are ferromagnetic invariant under the group action and having periodic phase space

stability of positive differential systems with delay we first prove an explicit criterion for positive linear time varying differential systems with distributed delay then some simple criteria for exponential stability of positive linear time invariant differential systems with delay are presented finally we extend obtained results to linear differential systems with time varying delay and to nonlinear differential systems with delay the results given in this technical note are extensions of some recent results in 7 11 and 14

hyperbolic cantori have dimension zero it is proved that cantori for area preserving twist maps have dimension zero only finitely many orbits of gaps and gap widths go to zero exponentially in each orbit if they are hyperbolic

stability of positive switched linear systems weak excitation and robustness to time varying delay this article investigates the stability of positive switched linear systems we start from motivating examples and focus on the case when each switched subsystem is marginally stable in the sense that all the eigenvalues of the subsystem matrix are in the closed left half plane with those on the imaginary axis simple instead of asymptotically stable a weak excitation condition is first proposed such that the considered positive switched linear system is exponentially stable an extension to the case without dwell time assumption is also presented then we study the influence of time varying delay on the stability of the considered positive switched linear system we show that the proposed weak excitation condition for the delay free case is also sufficient for the asymptotic stability of the positive switched linear system under unbounded time varying delay in addition it is shown that the convergence rate is exponential if there exists an upper bound for the delay irrespective of the magnitude of this bound the motivating examples are revisited to illustrate the theoretical results

homogenization of fully overdamped frenkel kontorova models in this paper we consider the fully overdamped frenkel kontorova model this is an infinite system of coupled first order odes each ode represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential after a proper rescaling a macroscopic model describing the evolution of densities of particles is obtained we get this homogenization result for a general class of frenkel kontorova models the proof is based on the construction of suitable hull functions in the framework of viscosity solutions

brief paper on linear co positive lyapunov functions for sets of linear positive systems in this paper we derive necessary and sufficient conditions for the existence of a common linear co positive lyapunov function for a finite set of linear positive systems both the state dependent and arbitrary switching cases are considered our results reveal an interesting characterisation of linear stability for the arbitrary switching case namely the existence of such a linear lyapunov function can be related to the requirement that a number of extreme systems are metzler and hurwitz stable examples are given to illustrate the implications of our results

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

new results on stability of discrete time systems with time varying state delay this note is concerned with the stability analysis of discrete time systems with time varying state delay by defining new lyapunov functions and by making use of novel techniques to achieve delay dependence several new conditions are obtained for the asymptotic stability of these systems the merit of the proposed conditions lies in their less conservativeness which is achieved by circumventing the utilization of some bounding inequalities for cross products between two vectors and by paying careful attention to the subtle difference between the terms sigmam k dk k 1 middot and sigma m k dm k 1 middot which is largely ignored in the existing literature these conditions are shown via several examples to be much less conservative than some existing result

planelike minimizers in periodic media we show that given an elliptic integrand in d that is periodic under integer translations and given any plane in d there is at least one minimizer of that remains at a bounded distance from this plane this distance can be bounded uniformly on the planes we also show that when folded back to d d the minimizers we construct give rise to a lamination one particular case of these results is minimal surfaces for metrics invariant under integer translations the same results hold for other functionals that involve volume terms small and average zero in such a case the minimizers satisfy the prescribed mean curvature equation a further generalization allows the formulation and proof of similar results in manifolds other than the torus provided that their fundamental group and universal cover satisfy some hypotheses 2001 john wiley sons inc

optical hamiltonians and symplectic twist maps abstract this paper concentrates on optical hamiltonian systems of t t n i e those for which hpp is a positive definite matrix and their relationship with symplectic twist maps we present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems the novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system

multiplicity results for interfaces of ginzburg landau allen cahn equations in periodic media the ginzburg landau allen cahn equation is a variational model for phase coexistence and for other physical problems it contains a term given by a kinetic part of elliptic type plus a double well potential we assume that the functional depends on the space variables in a periodic way we show that given a plane with rational normal there are minimal solutions satisfying the following properties these solutions are asymptotic to the pure phases and are separated by an interface the convergence to the pure phases is exponentially fast the interface lies at a finite distance m from the chosen plane where m is a universal constant furthermore these solutions satisfy some monotonicity properties with respect to integer translations namely integer translations are always comparable to the function we then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane like properties we also consider the case of possibly irrationally oriented planes we show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions one which is a minimizer and another one which is not these solutions also have interfaces bounded by a universal constant they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers

mu stability of nonlinear positive systems with unbounded time varying delays the stability of the zero solution plays an important role in the investigation of positive systems in this brief we discuss the mu stability of positive nonlinear systems with unbounded time varying delays the system is modeled by the continuous time ordinary differential equation under some assumptions on the nonlinear functions such as homogeneous cooperative and nondecreasing we propose a novel transform by which the nonlinear system reduces to a new system thus we analyze its dynamics which can simplify the nonlinear homogenous functions with respect to the arbitrary dilation map to those with respect to the standard dilation map we finally get some new criteria for the global mu stability taking the degree into consideration a numerical example is given to demonstrate the validity of obtained results

one dimensional infinite horizon variational problems arising in continuum mechanics in this paper we study a variational problem for real valued functions defined on an infinite semiaxis of the line to wit given x 2 we seek a minimal solution to the problem minimize the functional given by i left w left cdot right right int limits 0 infty f left w left s right ddot w left s right right ds w in ax left upsilon in w loc 2 1 left 0 infty right left upsilon left 0 right dot upsilon left 0 right x right right p

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

on linear copositive lyapunov functions and the stability of switched positive linear systems we consider the problem of common linear copositive lyapunov function existence for positive switched linear systems in particular we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time invariant systems several applications of this result are also given

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

the bounded real lemma for internally positive systems and h infinity structured static state feedback we consider the bounded real lemma for internally positive linear time invariant systems we show that the h norm of such systems can be evaluated by checking the existence of a certain diagonal quadratic storage function taking advantage of this fact the problem of designing a structured static state feedback controller achieving internal stability contractiveness and internal positivity in closed loop becomes convex and tractable

aubry mather sets and birkhoff s theorem for geodesic flows on the two dimensional torus in this paper we state the graph property for incompressible continuouse tori invariant under goedesic flows of riemannian metrics on the two dimensional torus also our method gives a new proof of birkhoff s theorem for twist maps of the cylinder we prove that if there exist an invariant incompressible torus of geodesic flow with irrational rotation number then it necessarily contains the aubry mather set with this rotation number

stability analysis of positive switched linear systems with delays this technical note addresses the stability problem of delayed positive switched linear systems whose subsystems are all positive both discrete time systems and continuous time systems are studied in our analysis the delays in systems can be unbounded under certain conditions several stability results are established by constructing a sequence of functions that are positive monotonically decreasing and convergent to zero as time tends to infinity additionally continuous for continuous time systems it turns out that these functions can serve as an upper bound of the systems trajectories starting from a particular region finally a numerical example is presented to illustrate the obtained results

strange non chaotic attractor in a quasiperiodically forced circle map abstract we show that in the quasiperiodically forced circle map strange non chaotic attractors can appear for non linearities far from the border of chaos the destruction of a two frequency quasiperiodic torus connected with the appearance of a strange non chaotic attractor is described in detail this strange non chaotic attractor is characterized by logarithmically slow diffusion of the phase it is shown that in this regime the high order phase locking states disappear and the rotation number varies rather smoothly with the parameters

exponential stability of two dimensional homogeneous monotone systems with bounded directional delays one dimensional 1 d monotone systems have received considerable attention recently due to their wide applicability and interesting mathematical properties one of these special properties is that for lti monotone systems exponential stability is insensitive to time delays some extensions to 1 d nonlinear monotone systems based on conditions of homogeneity have also been reported in this paper we study the problem of exponential stability of discrete time 2 d nonlinear monotone systems described by the roesser model with time varying delays specifically based on the monotone property and homogeneity of the associated vector fields necessary and sufficient delay independent exponential stability conditions are derived the magnitudes of delays are also taken into deriving an explicit estimation of the exponential decay rate which correlates the impact of delays on the system performance two examples are given to demonstrate the effectiveness of the obtained results

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

stability theory for nonnegative and compartmental dynamical systems with time delay abstract nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations and involve the exchange of nonnegative quantities between subsystems or compartments these models are widespread in biological and physical sciences and play a key role in understanding these processes a key physical limitation of such systems is that transfers between compartments is not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments in this paper we present necessary and sufficient conditions for stability of nonnegative and compartmental dynamical systems with time delay specifically asymptotic stability conditions for linear and nonlinear nonnegative dynamical systems with time delay are established using linear lyapunov krasovskii functionals

destruction of invariant tori in pendulum type equations abstract in this paper we prove that if there exists an invariant torus with the rotation number 1 in the pendulum type equation x q 0 x t x for a given potential q 0 q 0 t x c t 2 and is a liouville number then for any neighborhood n q 0 of q 0 in the c topology there exists a potential q q t x n q 0 such that the system x q x t x does not admit any invariant torus with the rotation number 1 this confirms j moser s suggestion in bol soc brasil mat 20 1981 29 45

on the influence of the kinetic energy on the stability of equilibria of natural lagrangian systems natural lagrangian systems t on r2 described by the equation frac d dt frac partial t partial dot q frac partial t partial q frac partial pi partial q are considered where t q q is a positive definite quadratic form in q and q has a critical point at 0 it is constructively proved that there exist a c potential energy and two c kinetic energies t and t such that the equilibrium q t 0 is stable for the system t and unstable for the system t equivalently it is established that for c natural systems the kinetic energy can influence the stability in the analytic category this is not true

asymptotic stability and decay rates of homogeneous positive systems with bounded and unbounded delays there are several results on the stability of nonlinear positive systems in the presence of time delays however most of them assume that the delays are constant this paper considers time varying possibly unbounded delays and establishes asymptotic stability and bounds the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case specifically we present a necessary and sufficient condition for delay independent stability of continuous time positive systems whose vector fields are cooperative and homogeneous we show that global asymptotic stability of such systems is independent of the magnitude and variation of the time delays for various classes of time delays we are able to derive explicit expressions that quantify the decay rates of positive systems we also provide the corresponding counterparts for discrete time positive systems whose vector fields are nondecreasing and homogeneous

percival variational principle for invariant measures and commensurate incommensurate phase transitions in one dimensional chains we prove for a one dimensional system of classical particles with potential energy u alpha gamma sum limits n left alpha v x n f x n 1 x n gamma right the existence of such a smooth function 0 0 that the system with potential energyu has the equilibrium state at the temperaturet 0 this is the incommensurate phase with the ratio of periods equal to the prescribed irrational number badly approximated by rational ones a simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing percival s energy functional

input output gain analysis for linear systems on cones abstract this paper is concerned with the input output gain analysis for dynamic systems that are defined on cones for such monotone systems we characterize their input output gain induced by the so called cone linear absolute norm in terms of generalized inequalities it is further shown that for a linear time invariant system represented by a convolution operator that maps one proper cone to another in the l 1 space its operator norm is governed by the static gain of the system as a byproduct the spectral radius of a cone preserving convolution operator turns out to be the same as that of the static gain matrix moreover dual results based on the cone max norm are presented finally the theoretical results are illustrated via linear systems over polyhedral cones or second order cones as well as a class of stochastic systems

spatial chaos in weakly dispersive and viscous media a nonperturbative theory of the driven kdv burgers equation abstract the asymptotic travelling wave solution of the kdv burgers equation driven by the long scale periodic driver is constructed the solution represents a shock train in which the quasi periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions it is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes two types of chaotic behavior are considered the first type is a weak chaos where only a small chaotic deviation from the periodic solution occurs the second type corresponds to the developed chaos where the solution ignores the driver period and represents a random sequence of uncorrelated shocks in the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor while in the case of developed chaos it moves on a repeller both solutions depend on a parameter indicating the reference shock position in the shock train the structure of a one dimensional set to which this parameter belongs is investigated this set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos the capacity dimension of this set is calculated

observations on the stability properties of cooperative systems we extend two fundamental properties of positive linear time invariant lti systems to homogeneous cooperative systems specifically we demonstrate that such systems are d stable meaning that global asymptotic stability is preserved under diagonal scaling we also show that a delayed homogeneous cooperative system is globally asymptotically stable gas for all non negative delays if and only if the system is gas for zero delay

connection between cooperative positive systems and integral input to state stability of large scale systems we consider a class of continuous time cooperative systems evolving on the positive orthant r n we show that if the origin is globally attractive then it is also globally stable and furthermore there exists an unbounded invariant manifold where trajectories strictly decay this leads to a small gain type condition which is sufficient for global asymptotic stability gas of the origin we establish the following connection to large scale interconnections of integral input to state stable iss subsystems if the cooperative system is integral iss and arises as a comparison system associated with a large scale interconnection of i iss systems then the composite nominal system is also i iss we provide a criterion in terms of a lyapunov function for integral input to state stability of the comparison system furthermore we show that if a small gain condition holds then the classes of systems participating in the large scale interconnection are restricted in the sense that certain iiss systems cannot occur moreover this small gain condition is essentially the same as the one obtained previously by dashkovskiy ruffer and wirth 2007 in press for large scale interconnections of iss systems

amount of rotation about a point and the morse index for the case of exact area preserving monotone twist diffeomorphisms we give formulas relating the amount of rotation about an orbit and certain morse indices

angular dynamical spectra a new method for determining frequencies weak chaos and cantori we define the invariant spectra of rotation angles and twist angles angular dynamical spectra and study the properties of their moments angular moments in a model two dimensional map the angular moments give the main frequencies of the orbits a main frequency is defined both for a regular and chaotic orbit for kam curves around a centre this frequency corresponds to the rotation number inside islands of stability we obtain both the main frequency rotation number and the epicyclic frequency a fast detection of thin chaotic layers is obtained on the basis of the behaviour of the frequency curve we explore the resonant structure near a last kam boundary the secondary islands of various multiplicities forming farey sequences and the noble tori between them are located we find a criterion to determine whether various resonant chaotic zones communicate or are separated by invariant tori

stabilization of positive linear systems we consider stabilization of equilibrium points of positive linear systems which are in the interior of the first orthant the existence of an interior equilibrium point implies that the system matrix does not possess eigenvalues in the open right half plane this allows to transform the problem to the stabilization problem of compartmental systems which is known and for which a solution has been proposed already we provide necessary and sufficient conditions to solve the stabilization problem by means of affine state feedback a class of stabilizing feedbacks is given explicitly

aubry mather theory and hamilton jacobi equations we review and extend some results of jauslin kreiss moser and weinan e on the smooth approximation of mather sets

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

technical communique delay range dependent stability for systems with time varying delay this paper is concerned with the stability analysis for systems with time varying delay in a range an appropriate type of lyapunov functionals is proposed to investigate the delay range dependent stability problem the present results may improve the existing ones due to a method to estimate the upper bound of the derivative of lyapunov functional without ignoring some useful terms and the introduction of additional terms into the proposed lyapunov functional which take into account the range of delay numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method

modular smoothing of action the central quantity in the theory of transport for hamiltonian systems and in particular the area preserving twist maps is the action of rotational periodic orbits usually this is a complicated discontinuous function of two arguments some perturbation parameter k and a rational rotation number denoted by we applied the idea of modular smoothing to this complicated fractal our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable

on the solutions of fractional swift hohenberg equation with dispersion in this article the approximate solutions of the non linear swift hohenberg equation with fractional time derivative in the presence of dispersive term have been obtained the fractional derivative is described in caputo sense time fractional nonlinear partial differential equations in the presence of dispersion and bifurcation parameters have been computed numerically to predict hydrodynamic fluctuations at convective instability for different particular cases and results are depicted through graphs

on algebraic proofs of stability for homogeneous vector fields we prove that if a homogeneous continuously differentiable vector field is asymptotically stable then it admits a lyapunov function which is the ratio of two polynomials i e a rational function we further show that when the vector field is polynomial the lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and hence such a lyapunov function can always be found by semidefinite programming this generalizes the classical fact that an asymptotically stable linear system admits a quadratic lyapunov function which satisfies a certain linear matrix inequality in addition to homogeneous vector fields the result can be useful for showing local asymptotic stability of nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component this paper also includes some negative results we show that in absence of homogeneity globally asymptotically stable polynomial vector fields may fail to admit a global rational lyapunov function and in presence of homogeneity the degree of the numerator of a rational lyapunov function may need to be arbitrarily high even for vector fields of fixed degree and dimension on the other hand we also give a family of homogeneous polynomial vector fields that admit a low degree rational lyapunov function but necessitate polynomial lyapunov functions of arbitrarily high degree this shows the potential benefits of working with rational lyapunov functions particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones

interface interactions in modulated phases and upsilon points certain features in frenkel kontorova and other models of phases with a one dimensional modulation can be analyzed by assuming parallel interfaces separating sets of lattice planes belonging to two different phases and treating the free energy to create interfaces as well as the interaction of two three or more interfaces as phenomenological parameters a strategy employed by fisher and szpilka for interacting defects can be extended to the case of interfaces allowing a systematic study of the phase diagram by ignoring all interface interactions and then successively taking into account pair triple and higher order terms the possible phase diagrams which can occur near the point where 0 include various sorts of endpoints analogous to critical endpoints an accumulation point of first order transitions and triple points and a self similar structure which we call an upsilon point which turns out to be an accumulation point of an infinite number of segments of first order transition lines each of which terminates in two upsilon points

on the stability of positive linear switched systems under arbitrary switching laws we consider n dimensional positive linear switched systems a necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is hurwitz several researchers conjectured that for positive linear switched systems this condition is also sufficient recently gurvits shorten and mason showed that this conjecture is true for the case n 2 but is not true in general their results imply that there exists some minimal integer np such that the conjecture is true for all n np but is not true for n np we show that np 3

pattern selection of solutions of the swift hohenberg equation abstract we study the large time behaviour of solutions u x t of the swift hohenberg equation on a one dimensional domain 0 l focusing in particular on the role of the eigenvalue parameter and the length l of the domain on the selection of the limiting profiles v we show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles and prove the existence of a collection n of disjoint intervals which depend on the value of such that if l n then solutions all converge to the trivial solution v 0 we show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches we establish global bounds for the solutions u x t and identify the global attractor finally we derive an estimate about the shape of the final state

qualitative theory of compartmental systems dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations many nonlinear that are called compartmental systems in this paper the authors define compartmental systems specify their relations to other nonnegative systems and discuss examples of applications the authors review the qualitative results on linear and nonlinear compartmental systems including their relation to cooperative systems they review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems providing a framework for unifying them under a few general theorems in the course of that they complete the solution of a problem posed by bellman and show that closed nonlinear autonomous n compartment systems can show the full gamut of possible behaviors of systems of odes finally to provide additional structure to this study the authors show how to partiti

on the destruction of minimal foliations monotone variational recurrence relations arise in solid state physics conservative lattice dynamics and the theory of hamiltonian twist maps an example is the frenkel kontorova lattice for such recurrence relations aubry mather theory guarantees the existence of solutions of every rotation number 2 r when is irrational they are the action minimizers that constitute the aubry mather set this aubry mather set is either connected or a cantor set in the rst case it is called a minimal foliation and otherwise a minimal lamination in this paper we prove that when the rotation number of a minimal foliation is easy to approximate by rational numbers then the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation this generalizes a theorem of mather for twist maps to a large class of monotone variational problems we moreover provide a proof that may allow for further generalizations

lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions the globally positive diffeomorphisms of the 2n dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite for these globally positive diffeomorphisms an aubry mather theory was developed by garibaldi and thieullen that provides the existence of some minimizing measures using the two green bundles g and g that can be defined along the support of these minimizing measures we will prove that there is a deep link between the angle between g and g along the support of the considered measure mu the size of the smallest positive lyapunov exponent of mu the tangent cone to the support of mu

recent developments in the theory of hamiltonian systems area preserving mappings of an annulus occur as poincare mappings of hamiltonian systems they were studied extensively by g d birkhoff recently aubry and mather investigated the subclass of so called monotone twist mappings for which they constructed independently closed invariant cantor sets their work led to important new results their theory is discussed and related to hamiltonian systems satisfying a legendre condition the connection of this theory with the stability problem kam theory and in particular the disintegration of invariant tori is discussed various constructions of mather sets are explained in which minimal solutions of variational problems play a central role this theory has a close relation to the differential geometric investigations by morse and hedlund on geodesics on two dimensional surfaces

positive linear observers for linear compartmental systems linear compartmental systems are mathematical systems that are frequently used in biology and mathematics the inputs states and outputs of such systems are positive because they denote amounts or concentrations of material for linear dynamic systems the observer problem has been solved the purpose of the observer problem is to determine a linear observer such that the state can be approximated the difference between the state and its estimate should converge to zero the interpretation in terms of a physical system requires that an estimate of the state be positive like the state itself in this paper conditions on the system matrices are presented that guarantee that there exists a positive linear observer such that both the error converges to zero and the estimate is positive

rational spline nonstandard finite difference scheme for the solution of time fractional swift hohenberg equation abstract in this paper we introduce a new scheme based on rational spline function and nonstandard finite difference technique to solve the time fractional swift hohenberg equation in the sense of riemann liouville derivative via fourier method the method is convergent and unconditionally stable also we investigated the existence and uniqueness of the proposed method numerical results are demonstrated to validate the applicability and the theoretical results

total destruction of invariant tori for the generalized frenkel kontorova model we consider generalized frenkel kontorova models on higher dimensional lattices we show that the invariant tori which are parameterized by continuous hull functions can be destroyed by small perturbations in the cr topology with r 1