a confidence limit for the empirical mode decomposition and hilbert spectral analysis the confidence limit is a standard measure of the accuracy of the result in any statistical analysis most of the confidence limits are derived as follows the data are first divided into subsections and then under the ergodic assumption the temporal mean is substituted for the ensemble mean next the confidence limit is defined as a range of standard deviations from this mean however such a confidence limit is valid only for linear and stationary processes furthermore in order for the ergodic assumption to be valid the subsections have to be statistically independent for non stationary and nonlinear processes such an analysis is no longer valid the confidence limit of the method here termed emd hsa for empirical mode decomposition hilbert spectral analysis is introduced by using various adjustable stopping criteria in the sifting processes of the emd step to generate a sample set of intrinsic mode functions imfs the emd technique acts as a pre processor for hsa on the original data producing a set of components imfs from the original data that equal the original data when added back together each imf represents a scale in the data from smallest to largest the ensemble mean and standard deviation of the imf sample sets obtained with different stopping criteria are calculated and these form a simple random sample set the confidence limit for emd hsa is then defined as a range of standard deviations from the ensemble mean without evoking the ergodic assumption subdivision of the data stream into short sections is unnecessary hence the results and the confidence limit retain the full frequency resolution of the full dataset this new confidence limit can be applied to the analysis of nonlinear and non stationary processes by these new techniques data from length of day measurements and a particularly violent recent earthquake are used to demonstrate how the confidence limit is obtained and applied by providing a confidence limit for this new approach a stable range of stopping criteria for the decomposition or sifting phase emd has been established making the results of the final processing with hsa and the entire emd hsa method more definitive

the quantum beating and its numerical simulation abstract we examine the suppression of quantum beating in a one dimensional non linear double well potential made up of two focusing nonlinear point interactions the investigation of the schrodinger dynamics is reduced to the study of a system of coupled nonlinear volterra integral equations for various values of the geometric and dynamical parameters of the model we give analytical and numerical results on the way states which are initially confined in one well evolve we show that already for a nonlinearity exponent well below the critical value there is complete suppression of the typical beating behavior characterizing the linear quantum case

diagonal stability on cactus graphs and application to network stability analysis a square matrix e is said to be diagonally stable if there exists a diagonal matrix d 0 satisfying de etd 0 this notion has been instrumental in recent studies on stability of interconnected system models in communication and biological networks in which the subsystems satisfy passivity properties and the matrix e combines this passivity information with the interconnection structure this paper presents a necessary and sufficient condition for diagonal stability when the digraph describing the network conforms to a cactus structure which means that a pair of distinct simple circuits in the graph have at most one common vertex in the special case of a single circuit this diagonal stability test recovers the secant criterion that was recently derived for cyclic networks the paper then incorporates the new diagonal stability test in network stability analysis where the diagonal entries of the matrix d serve as weights in a lyapunov function constructed from storage functions that verify passivity properties of the components finally the paper illustrates this stability test on examples motivated by gene networks and population dynamics

ionization for three dimensional time dependent point interactions we study the time evolution of a three dimensional quantum particle under the action of a time dependent point interaction fixed at the origin we assume that the strength of the interaction t is a periodic function with an arbitrary mean under very weak conditions on the fourier coefficients of t we prove that there is complete ionization as t starting from a bound state at time t 0 moreover we prove also that under the same conditions all the states of the system are scattering states

the point like limit for a nls equation with concentrated nonlinearity in dimension three abstract we consider a scaling limit of a nonlinear schrodinger equation nls with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a nls equation with nonlinearity concentrated at a point the regularized dynamics is described by the equation i t e t e t g e e e t 2 e e t e where e 0 weakly and the function g embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics the limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well posedness blow up and asymptotic properties of solutions our result is the first justification of the model as the point limit of a regularized dynamics

synchronization of interconnected systems with applications to biochemical networks an input output approach this paper provides synchronization conditions for networks of nonlinear systems the components of the network referred to as compartments in this paper are made up of an identical interconnection of subsystems each represented as an operator in an extended l2 space and referred to as a species the compartments are in turn coupled through a diffusion like term among the respective species the synchronization conditions are provided by combining the input output properties of the subsystems with information about the structure of the network the paper also explores results for state space models as well as biochemical applications the work is motivated by cellular networks where signaling occurs both internally through interactions of species and externally through intercellular signaling the theory is illustrated by providing synchronization conditions for networks of goodwin oscillators

brief paper diagonal stability of a class of cyclic systems and its connection with the secant criterion we consider a class of systems with a cyclic interconnection structure that arises among other examples in dynamic models for certain biochemical reactions we first show that a secant criterion for local stability derived earlier in the literature is in fact a necessary and sufficient condition for diagonal stability of the corresponding class of matrices we then revisit a recent generalization of this criterion to output strictly passive systems and recover the same stability condition using our diagonal stability result as a tool for constructing a lyapunov function using this procedure for lyapunov construction we exhibit classes of cyclic systems with sector nonlinearities and characterize their global stability properties

hurwitz stability of metzler matrices in this note a simple method is presented that is both necessary and sufficient for determining whether a given metzler matrix a is hurwitz the method is based on the well known fact that a hurwitz metzler matrix is also diagonally stable by using this fact very simple conditions are derived for the hurwitz stability of a metzler matrix the conditions are stated in terms of the signs of the diagonal entries of a sequence of lower dimensional matrices the efficacy of the conditions is demonstrated by applying them to determine stability in several examples

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

existence criteria of periodic oscillations in cyclic gene regulatory networks in this paper we consider the existence conditions of periodic oscillations in large scale cyclic gene regulatory networks using the poincare bendixson theorem for cyclic systems we first show that the local instability of an equilibrium point implies the existence of periodic oscillations we then derive the graphical and its equivalent analytic criteria for the existence of periodic oscillations based on local instability analysis these criteria have a remarkable feature that they can be applied systematically to a large scale cyclic gene regulatory network consisting of any number of genes the latter part of this paper is devoted to analyze the relation between an equilibrium point and the biochemical parameters this leads to an analytic existence criterion that explicitly takes the dependence of the equilibrium state on biochemical parameters into account which is often overlooked in nonlinear system analyses in particular the novel physical quantities that are essential for determining the existence of periodic oscillations are obtained based on the rigorous analytic criterion

a sufficient condition for additive d stability and application to reaction diffusion models abstract the matrix a is said to be additively d stable if a d remains hurwitz for all non negative diagonal matrices d in reaction diffusion models additive d stability of the matrix describing the reaction dynamics guarantees the stability of the homogeneous steady state thus ruling out the possibility of diffusion driven instabilities we present a new criterion for additive d stability using the concept of compound matrices we first give conditions under which the second additive compound matrix has non negative off diagonal entries we then use this metzler property of the compound matrix to prove additive d stability with the help of an additional determinant condition this result is then applied to investigate the stability of cyclic reaction networks in the presence of diffusion finally a reaction network structure that fails to achieve additive d stability is exhibited

passivity gains and the secant condition for stability a generalization of the classical secant condition for the stability of cascades of scalar linear systems is provided for passive systems the key is the introduction of a quantity that combines gain and phase information for each system in the cascade for linear one dimensional systems the known result is recovered exactly 2005 elsevier b v all rights reserved

a general trotter kato formula for a class of evolution operators in this article we prove new results concerning the existence and various properties of an evolution system ua b t s 0 s t t generated by the sum a t b t of two linear time dependent and generally unbounded operators de ned on time dependent domains in a complex and separable banach space b in particular writing l b for the algebra of all linear bounded operators on b we can express ua b t s 0 s t t as the strong limit in l b of a product of the holomorphic contraction semigroups generated by a t and b t respectively thereby proving a product formula of the trotter kato type under very general conditions which allow the domain d a t b t to evolve with time provided there exists a fi xed set d incl t in 0 t d a t b t everywhere dense in b we obtain a special case of our formula when b t 0 which in effect allows us to reconstruct ua t s 0 s t t very simply in terms of the semigroup generated by a t we then illustrate our results by considering various examples of non autonomous parabolic initial boundary value problems including one related to the theory of time dependent singular perturba tions of self adjoint operators we nally mention what we think remains an open problem for the corresponding equations of schrodinger type in quantum mechanics

robust stability analysis of gene protein regulatory networks with cyclic activation repression interconnections abstract this paper investigates analytic robust stability criteria for large scale cyclic gene protein regulatory network systems consisting of genes with heterogeneous dynamics combined with parametric or unstructured uncertainties from a control theoretic viewpoint we first consider a class of gene expressions which is described as an uncertain linear transcription translation model lttm with feedback loops from translation products to transcription next we show that such a model belong to a class of large scale dynamical linear network systems with a generalized frequency variable then we derive robust stability criteria systematically for the following two types of biological uncertain lttm i lttms consisting of heterogeneous gene dynamics with parametric uncertainties and ii lttms consisting of homogeneous nominal gene dynamics with unstructured uncertainties these criteria provide fairly simple analysis methodologies which can be readily applied to the analysis for large scale genetic regulatory networks and give some biological insight

delay robustness of interconnected passive systems an integral quadratic constraint approach we consider networks of passive systems with time delays in the interconnections and present a stability analysis technique with the help of the integral quadratic constraint iqc framework unlike the classical passivity approach that fails to characterize delay robustness and the small gain approach that conservatively accounts for arbitrarily large delays the new technique gives sharp stability estimates that depend on the duration of delay since the effect of delay depends on its duration relative to the time scales of the system we make use of a roll off iqc that captures magnitude roll off at high frequencies thus providing the critical time scale information we then combine this roll off iqc with an output strict passivity iqc that incorporates gain and phase information and demonstrate the benefit of this combined iqc approach on a cyclic interconnection structure with delay finally we develop a technique to verify these iqcs for classes of nonlinear state space models and present an example from internet congestion control

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

diagonal stability of matrices with cyclic structure and the secant condition an existence result on diagonal solutions of a linear matrix inequality is used to study diagonal hurwitz and schur stability and to derive the secant condition for systems with cyclic structure

a passivity based approach to stability of spatially distributed systems with a cyclic interconnection structure a class of distributed systems with a cyclic interconnection structure is considered these systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns in this paper a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified for these systems global stability results hold if the ldquosecantrdquo criterion is satisfied in the linear case it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic lyapunov function which extends a recent diagonal stability result to partial differential equations for reaction diffusion equations with nondecreasing coupling nonlinearities global asymptotic stability of the origin is established all of the derived results remain true for both linear and nonlinear positive diffusion terms similar results are shown for compartmental systems

diagonal riccati stability and the hadamard product we present an extension of a recent characterisation of diagonal riccati stability and using this extend a result of kraaijevanger on diagonal lyapunov stability to riccati stability of time delay systems we also describe a class of transformations that preserve the property of being diagonally riccati stable and apply these two results to provide novel stability results for classes of time delay systems

stability certification of large scale stochastic systems using dissipativity in this paper we analyse the stability of large scale nonlinear stochastic systems represented as an interconnection of lower order stochastic subsystems stochastic stability in probability and noise to state stability are addressed and sufficient conditions for the latter are provided the method proposed proves network stability by using appropriate stochastic passivity properties of its subsystems and the structure of its interactions stability properties are established by the diagonal stability of a dissipativity matrix which incorporates information about the passivity properties of the systems and their interconnection next we derive equilibrium independent conditions for the verification of the relevant passivity properties of the subsystems finally we illustrate the proposed approach on a class of biological reaction networks

biochemical oscillations in delayed negative cyclic feedback existence and profiles in this paper we propose a theoretical framework to systematically analyze the existence and the profiles of chemical oscillations in gene regulatory networks with negative cyclic feedback in particular we analytically derive the existence conditions and the profiles of oscillations in terms of reaction kinetic parameters and reveal dimensionless quantities that essentially characterize the oscillatory dynamics these discoveries then allow us to provide general biological insights that are useful for the design of synthetic gene circuits and wet lab experiments we point out that time delays due to splicing and transport play an important role for both of the existence and the profiles of oscillations to this end we first show that local instability leads to oscillations in cyclic gene regulatory networks and we derive the existence conditions based on local instability analysis then we analyze the period phase and amplitude of oscillations using multivariable harmonic balance analysis these results are demonstrated with two existing biochemical networks the pentilator and a self repression network of a hes protein

equilibrium independent passivity we extend the traditional notion of passivity to a forced system whose equilibrium is dependent on the control input by defining equilibrium independent passivity a system property characterized by a dissipation inequality centered at an arbitrary equilibrium point we provide a necessary input output condition which can be tested for systems of arbitrary dimension and sufficient conditions to certify this property for scalar systems an example from network stability analysis is presented which demonstrates the utility of this new definition we then proceed to numerical certification of equilibrium independent passivity using sum of squares programming finally through numerical examples we show that equilibrium independent passivity is less restrictive than incremental passivity

analysis of autocatalytic networks in biology autocatalytic networks in particular the glycolytic pathway constitute an important part of the cell metabolism changes in the concentration of metabolites and catalyzing enzymes during the lifetime of the cell can lead to perturbations from its nominal operating condition we investigate the effects of such perturbations on stability properties e g the extent of regions of attraction of a particular family of autocatalytic network models numerical experiments demonstrate that systems that are robust with respect to perturbations in the parameter space have an easily verifiable in terms of proof complexity region of attraction properties motivated by the computational complexity of optimization based formulations we take a compositional approach and exploit a natural decomposition of the system induced by the underlying biological structure into a feedback interconnection of two input output subsystems a small subsystem with complicating nonlinearities and a large subsystem with simple dynamics this decomposition simplifies the analysis of large pathways by assembling region of attraction certificates based on the input output properties of the subsystems it enables numerical as well as analytical construction of block diagonal lyapunov functions for a large family of autocatalytic pathways