regular dynamics in a delayed network of two neurons with all or none activation functions we consider a delayed network of two neurons with both self feedback and interaction described by an all or none threshold function the model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity we show that the dynamics of the network can be understood in terms of the iteration of a one dimensional map and we obtain simple criteria for the convergence of solutions the existence multiplicity and attractivity of periodic solutions

a criterion for the existence of common invariant subspaces of matrices it is shown that square matrices a and b have a common invariant subspace w of dimension k 1 if and only if for some scalar s a c si and b c si are invertible and their kth compounds have a common eigenvector which is a grassmann representative for w t he applicability of this criterion and its ability to yield a basis for the common invariant subspace are investigated 2001 elsevier science inc all rights reserved

boundary integral equation method for linear porous elasticity with applications to soil consolidation for physical phenomena governed by the biot model of porous elasticity a reciprocal relation similar to the betti s recoprocal theorem in elasticity is constructed in laplace transformed space integrating the reciprocal relation enables one to formulate boundary integral equations the fundamental kernels for the integral equations are solved in closed forms for the case of isotropic material numerical implementation of two dimensional problems includes finite element ideas of discretization and polynomial interpolation and numerical inversion of a laplace transform practical applications of the method are found in consolidation problems in soils which contain compressible as well as incompressible pore fluids also as a numerical experiment consolidation of partially saturated soil is simulated and interesting phenomena are observed the currently developed boundary integral equation method biem for porous elasticity may be viewed as an efficient and accurate alternative of existing finite element and finite difference methods for linear consolidation problems application of biem is always preferred to the other numerical methods whenever possible

two classes of ode models with switch like behavior abstract in cases where the same real world system can be modeled both by an ode system d and a boolean system b it is of interest to identify conditions under which the two systems will be consistent that is will make qualitatively equivalent predictions in this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions in contrast to the widely known class of glass networks the right hand sides of our odes are lipschitz continuous we prove that if b has certain structures consistency between d and b is implied by sufficient separation of timescales in one class of our models namely if the trajectories of b are one stepping then we prove a strong form of consistency and if b has a certain monotonicity property then there is a weaker consistency between d and b these results appear to point to more general structure properties that favor consistency between ode and boolean models

two and three dimensional soil consolidation by bem via particular integral abstract particular integral formulations are presented for two and three dimensional soil consolidation poroelasticity analysis the steady state poroelasticity equation is used as the complementary solution and a set of global shape functions is considered to approximate the transient terms of the poroelasticity equation to obtain the required particular integrals some numerical results for four example problems are given and compared with their analytical solutions or the results from the convolution method developed earlier by dargush and banerjee int j numer meth engng 28 1989 2423 j appl mech 58 1991 28

ultrafast physical generation of random numbers using hybrid boolean networks we describe a high speed physical random number generator based on a hybrid boolean network with autonomous and clocked logic gates realized on a reconfigurable chip the autonomous logic gates are arranged in a bidirectional ring topology and generate broadband chaos the clocked logic gates receive input from the autonomous logic gates so that random numbers are generated physically that pass standard randomness tests without further postprocessing the large number of logic gates on reconfigurable chips allows for parallel generation of random numbers as demonstrated by our implementation of 128 physical random number generators that achieve a real time bit rate of 12 8gbits s

state transition graph and stability of singular equilibria for piecewise linear biological models abstract many biological networks such as gene regulatory networks and neural networks can be modeled by piecewise linear switching systems of differential equations for these biological network models it is very difficult to describe the dynamics of the governing system near singular equilibria as they belong to the discontinuity set of the system in this paper using the concept of filippov solutions with the help of the state transition graph and employing a generalized lyapunov method we establish some stability results for singular equilibria of a class of piecewise linear biological models our results generalize many existing ones in particular we present an affirmative answer to conjecture 5 4 proposed in r casey h de jong j l gouze piecewise linear models of genetic regulatory networks equilibria and their stability journal of mathematical biology 52 1 2006 27 56

functional differential inclusions generated by functional differential equations with discontinuities abstract given a functional differential equation with a discontinuity a construction of its extension in the shape of a functional differential inclusion is offered this construction can be regarded as a generalization of the famous filippov approach to study ordinary differential equations with discontinuities some basic properties of the solutions of the introduced functional differential inclusions are studied the developed approach is applied to analysis of gene regulatory networks with general delays

chaos in a neural network circuit abstract we have constructed a neural network circuit of four clipped high grain integrating operational amplifiers coupled to each other through an array of digitally programmable resistor ladders mdacs in addition to fixed point and cyclic behavior the circuit exhibits chaotic behavior with complex strange attractors which are approached through period doubling intermittent attractor expansion and or quasiperiodic pathways couplings between the nonlinear circuit elements are controlled by a computer which can automatically search through the space of couplings for interesting phenomena we report some initial statistical results relating the behavior of the network to properties of its coupling matrix through these results and further research the circuit should help resolve fundamental issues concerning chaos in neural networks

from mixture theory to biot s approach for porous media abstract two apparently different approaches are used in dealing with the mechanics of a deformable porous medium mixture theories on the one hand and purely macroscale theories which are mainly associated with the work of biot on the other hand in the mixture theories the porous medium is represented by spatially superposed interacting media while macroscale theories assume that standard concepts of continuum mechanics are still relevant at the macro level the aim of this paper is two fold first it is shown that the macroscale field equations derived from mixture theories can be reformulated in terms of the measurable quantities involved in the macroscale theories second it is demonstrated how these field equations including the fundamental inequality obtained from the second law entail the existence of a macroscale c potential upon which a thermo dynamically consistent formulation of the constitutive equations can be firmly founded

common eigenvectors of two matrices abstract a computable criterion is given for two square matrices to possess a common eigenvector as well as a criterion for one matrix to have an eigenvector lying in a given subspace some applications are discussed

variationally consistent derivation of the stress partitioning law in saturated porous media abstract the linearized version of a recently formulated variational macroscopic theory of biphasic isotropic porous media vmtpm is employed to derive a general stress partitioning law for media undergoing flow conditions under prevented fluid seepage and negligible inertia effects typically met in biphasic specimens subjected to jacketed tests the principle of virtual work relevant to the specialization of vmtpm to such characteristic flow conditions naturally yields a stress partitioning law between solid and fluid phase of a saturated medium that exactly matches with the celebrated terzaghi s principle it is also shown that the stress tensor of the solid phase work associated with the strain measure of the vmtpm naturally corresponds to the terzaghi s effective stress accordingly under undrained conditions terzaghi s law is proved to be a completely general stress partitioning law for a saturated biphasic medium irrespective of its constitutive and or microstructural features as well as of the compressibility of its constituent phases since the developments reported are obtained ruling out thermodynamic constraints and any assumption on the internal microstructure and on the compressibility of the phases the results obtained indicate that terzaghi s law could more generally apply to a broader class of biphasic media and be not restricted within the context of geomechanics

a computational strategy for thermo poroelastic structures with a time space interface coupling this paper deals with a computational strategy suitable for the simulation of multiphysics problems based on the large time increment latin method the simulation of such problems must encounter the possible different time and space scales which usually arise with the different physics herein we focus on using different time and space discretizations for each physics by introducing an interface with its own discretization the feasibility of both a time and space couplings is exemplified on a nonlinear 3 physics strongly coupled problem the thermal fluid structure interaction in a thermo poroelastic structure

boundary conditions at fluid permeable interfaces in porous media a variational approach abstract a general set of boundary conditions at fluid permeable interfaces between dissimilar fluid filled porous matrices is established starting from an extended hamilton rayleigh principle these conditions do include friction and inertial effects once linearized they encompass boundary conditions relative to volume darcy brinkman and to surface saffman beavers joseph dissipation effects

an efficient finite element procedure for analyzing three phase porous media based on the relaxed picard method summary effective simulation of the solid liquid gas coupling effect in unsaturated porous media is of great significance in many diverse areas because of the strongly nonlinear characteristics of the fully coupled formulations for the three phase porous media an effective numerical solution scheme such as the finite element method with an efficient iterative algorithm has to be employed in this paper an efficient finite element procedure based on the adaptive relaxed picard method is developed for analyzing the coupled solid liquid gas interactions in porous media the coupled model and the finite element analysis procedure are implemented into a computer code poroush2m and the proposed procedure is validated through comparing the numerical simulations with the experimental benchmarks it is shown that the adaptive relaxed picard method has salient advantage over the traditional one with respect to both the efficiency and the robustness especially for the case of relatively large time step sizes compared with the newton raphson scheme the picard method successfully avoids the unphysical spurious unloading phenomenon under the plastic deformation condition although the latter shows a better convergence rate the proposed procedure provides an important reference for analyzing the fully coupled problems related to the multi phase multi field coupling in porous media copyright 2014 john wiley sons ltd

new global bifurcation diagrams for piecewise smooth systems transversality of homoclinic points does not imply chaos abstract in this paper we consider some piecewise smooth 2 dimensional systems having a possibly non smooth homoclinic t we assume that the critical point 0 lies on the discontinuity surface 0 we consider 4 scenarios which differ for the presence or not of sliding close to 0 and for the possible presence of a transversal crossing between t and 0 we assume that the systems are subject to a small non autonomous perturbation and we obtain 4 new bifurcation diagrams in particular we show that in one of these scenarios the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics

a control problem for affine dynamical systems on a full dimensional polytope given an affine system on a full dimensional polytope the problem of reaching a particular facet of the polytope using continuous piecewise affine state feedback is studied necessary conditions and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the polytope special attention is paid to affine systems on full dimensional simplices in this case the necessary and sufficient conditions are equivalent and a constructive procedure yields an affine feedback control law that solves the reachability problem under consideration

controlling a class of nonlinear systems on rectangles in this paper we focus on a particular class of nonlinear affine control systems of the form xdot f x bu where the drift f is a multi affine vector field i e affine in each state component the control distribution b is constant and the control u is constrained to a convex set for such a system we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant we then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time if the control constraints are polyhedral we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle this work is motivated by the need to construct discrete abstractions for continuous and hybrid systems in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs we show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators

a boundary integral equation method for consolidation problems abstract a time domain direct boundary integral equation method biem for biot s linear theory of consolidation is considered the potential representation of the solution is used not only for obtaining numerical solutions but also for investigating the behavior of the solution the obtained information concerning the behavior of the solution is utilized to implement an accurate biem a numerical example confirms the applicability of the present formulation

pausing for arbitrarily long times in dynamical systems it is well known that continuity in dynamical systems is not sufficient to guarantee uniqueness of solutions but less obvious is that non uniqueness can carry internal structure useful to characte

global dynamics for switching systems and their extensions by linear differential equations abstract switching systems use piecewise constant nonlinearities to model gene regulatory networks this choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of morse graphs associated to nodes of a parameter graph the parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical morse graphs however there are many cellular processes that do not exhibit threshold like behavior and thus are not well described by a switching system we consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations we show that the parameter graphs associated to the switching system and any of its extensions are identical for each parameter graph node there is an order preserving map from the morse graph of the switching system to the morse graph of any of its extensions we provide counterexamples that show why possible stronger relationships between the morse graphs are not valid

consolidation of elastic plastic saturated porous media by the boundary element method abstract this paper presents a domain boundary element formulation for inelastic saturated porous media with rate independent behavior for the solid skeleton the formulation is then applied to elastic plastic behavior for the solid biot s consolidation theory extended to include irreversible phenomena is considered and the direct boundary element technique is used for the numerical solution after time discretization by the implicit euler backward algorithm the associated nonlinear algebraic problem is solved by the newton raphson procedure whereby the loading unloading conditions are fully taken into account and the consistent tangent operator defined only domain nodes nodes defined inside the domain are used to represent all domain values and the corresponding integrals are computed by using an accurate sub elementation scheme the developments are illustrated through the drucker prager elastic plastic model for the solid skeleton and various examples are analyzed with the proposed algorithms

global dynamics for steep nonlinearities in two dimensions abstract this paper discusses a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations we study switching models of regulatory networks to each switching network we associate a morse graph a computable object that describes a morse decomposition of the dynamics in this paper we show that all smooth perturbations of the switching system share the same morse graph and we compute explicit bounds on the size of the allowable perturbation this shows that computationally tractable switching systems can be used to characterize dynamics of smooth systems with steep nonlinearities

streets tree networks and urban growth optimal geometry for quickest access between a finite size volume and one point abstract the geometric form of the tree network is deduced from a single mechanism the discovery that the shape of a heat generating volume can be optimized to minimize the thermal resistance between the volume and a point heat sink is used to solve the kinematics problem of minimizing the time of travel between a volume or area and one point the optimal path is constructed by covering the volume with a sequence of volume sizes building blocks which starts with the smallest size and continues with stepwise larger sizes assemblies optimized in each building block is the overall shape and the angle between constituents the speed of travel may vary from one assembly size to the next however the lowest speed is used to reach the infinity of points located in the smallest volume elements the volume to point path that results is a tree network a single design principle the geometric optimization of volume to point access determines all the features of the tree network

dynamics of transcription translation networks abstract a theory for qualitative models of gene regulatory networks has been developed over several decades generally considering transcription factors to regulate directly the expression of other transcription factors without any intermediate variables here we explore a class of models that explicitly includes both transcription and translation keeping track of both mrna and protein concentrations we mainly deal with transcription regulation functions that are steep sigmoids or step functions as is often done in protein only models though translation is governed by a linear term we extend many aspects of the protein only theory to this new context including properties of fixed points description of trajectories by mappings between switching points qualitative analysis via a state transition diagram and a result on periodic orbits for negative feedback loops we find that while singular behaviour in switching domains is largely avoided non uniqueness of solutions can still occur in the step function limit

dynamics of actively regulated gene networks abstract a popular way of modelling gene regulatory networks is to design the models as if the genes were acting directly on each other genes are activated or inhibited by transcription factors which are direct gene products the action of a transcription factor on a gene is modelled as a binary on off dose response function or a sigmoid around a certain threshold concentration a transcription factor may act on several genes with equal or different thresholds the combined effect of several transcription factors on a gene is generally assumed to obey boolean like composition rules a mathematical challenge related to such network models is to analyse the behaviour in the narrow domains in phase space where at least one variable is close to one of its thresholds called switching domains in the present paper we analyse the motion in switching domains for models with steep sigmoidal response functions under the assumption that each transcription factor only regulates one gene at each of its thresholds this biologically reasonable assumption allows us to establish rules that determine the passage through any switching domain and the sequence of domains through which the system passes the rules can be effectively implemented in a sound automated analyser for such networks the assumption simplifies certain mathematical and computational issues but unfortunately may be at variance with one of the necessary conditions for the singular perturbation theory which our derivations are based on we give sufficient conditions ensuring that despite this the difficulties raised by our assumption can be evaded and singular perturbation approximation safely applied

nonlinear dynamics and symbolic dynamics of neural networks a piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks a symbolic representation of the dynamics in this equation is given as a directed graph on an n dimensional hypercube this provides a formal link with discrete neural networks such as the original hopfield models analytic criteria are given to establish steady states and limit cycle oscillations independent of network dimension model networks that display multiple stable limit cycles and chaotic dynamics are discussed the results show that such equations are a useful and efficient method of investigating the behavior of neural networks

switching relay and complementarity systems a tutorial on their well posedness and relationships this paper reviews the relationships between switching systems defined from a partition of the state space into convex cells and relay or complementarity dynamical systems which are other classes of discontinuous systems well posedness results i e results on the existence and the uniqueness of solutions for different classes of relay and complementarity systems which are also switching systems are presented the reverse issue when can a switching system be rewritten equivalently as a relay or a complementarity system is also tackled many examples from mechanics circuits and biology illustrate the developments throughout the paper the paper focuses on systems with continuous solutions i e with no state jumps convexity is the central property

symbolic reachability analysis of genetic regulatory networks using discrete abstractions we use hybrid systems techniques for the analysis of reachability properties of a class of piecewise affine pa differential equations that are particularly suitable for the modeling of genetic regulatory networks more specifically we introduce a hyperrectangular partition of the state space that forms the basis for a discrete abstraction preserving the sign of the derivatives of the state variables the resulting discrete transition system provides a qualitative description of the network dynamics that is well adapted to available experimental data and that can be efficiently computed in a symbolic manner from inequality constraints on the parameters

bifurcation analysis of a mathematical model for genetic regulatory network with time delays in this paper we aim to investigate the dynamics of a gene regulatory network which is a time delayed version of the model proposed by elowitz and leibler nature 403 2000 335 338 based on the normal form theory and center manifold reduction hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated we also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network it shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation more precisely i the amplitude increases with small time delay while the change of amplitude is not sensitive to relatively large time delay however the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter where amplitude always increases quickly and linearly with the transcriptional rate ii the period of oscillation increases as the time delay increases but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value which implies that the robustness of period to the transcriptional rate variations occurs our numerical simulations also support the theoretical conclusions namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks

a filippov sliding vector field on an attracting co dimension 2 discontinuity surface and a limited loss of attractivity analysis we consider sliding motion in the sense of filippov on a discontinuity surface of co dimension 2 we characterize and restrict to the case of being attractive through sliding in this situation we show that a certain filippov sliding vector field ff suggested in alexander and seidman 1998 2 di bernardo et al 2008 6 dieci and lopez 2011 10 exists and is unique we also propose a characterization of first order exit conditions clarify its relation to generic co dimension 1 losses of attractivity for and examine what happens to the dynamics on for the aforementioned vector field ff examples illustrate our results

inflation of the edge of chaos in a simple model of gene interaction networks we study a set of linearized catalytic reactions to model gene and protein interactions the model is based on experimentally motivated interaction network topologies and is designed to capture some key properties of gene expression statistics we impose a nonlinearity to the system by enforcing a boundary condition which guarantees non negative concentrations of chemical substances system stability is quantified by maximum lyapunov exponents we find that the non negativity constraint leads to a drastic inflation of those regions in parameter space where the lyapunov exponent exactly vanishes within the model this finding can be fully explained as a result of a symmetry breaking mechanism induced by the positivity constraint the robustness of this finding with respect to network topologies and the role of intrinsic molecular and external noise is discussed we argue that systems with inflated edges of chaos could be much more easily favored by natural selection than systems where the lyapunov exponent vanishes only on a parameter set of measure zero

global dynamics of neural nets with infinite gain abstract we consider a model of neural and gene networks where the nonlinearities in the system of differential equations are discontinuous and piecewise constant we develop a framework for study of such systems as a first step we associate to the system a graph g on a hypercube and show how the collection of strongly connected components of g relates to the dynamics of the flow on the set of rays through the origin in the second step we discuss the relationship between the invariant sets of the ray flow and the invariant sets of the original flow we provide a sufficient condition for a one to one correspondence between these sets finally we study the class of binary networks within this framework under certain conditions we can determine the structure of an invariant set corresponding to the lowest strongly connected component of the hypercube graph

dominant vertices in regulatory networks dynamics discrete time regulatory networks are dynamical systems on directed graphs with a structure that is inspired on natural systems of interacting units using a notion of determination between vertices we define sets of dominant vertices and we prove that in the asymptotic regime the projection of the dynamics on a dominant set allows us to determine the state of the whole system at all times we provide an algorithm to find sets of dominant vertices and we test its accuracy on several examples we also explore the possibility of using the dominant set characteristics as a measure of the structural complexity of networks

when do several linear operators share an invariant cone abstract we establish a criterion for a finite family of matrices to possess a common invariant cone the criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family in spite of theoretical simplicity the practical use of the criterion may be difficult we show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable corollaries of the criterion which give sufficient and necessary conditions are derived finally we introduce a co directional number of several matrices we prove that this parameter is close to zero iff there is a small perturbation of matrices after which they get an invariant cone an algorithm for its computation is presented

a time domain boundary element method for poroelasticity the development of a general boundary element method bem for two and three dimensional quasistatic poroelasticity is discussed in detail the new formulation for the complete biot consolidation theory operates directly in the time domain and requires only boundary discretization as a result the dimensionality of the problem is reduced by one and the method becomes quite attractive for geotechnical analyses particularly those which involve extensive or infinite domains the presentation includes the definition of the two key ingredients for the bem namely the fundamental solutions and a reciprocal theorem then once the boundary integral equations are derived the focus shifts to an overview of the general purpose numerical implementation this implementation includes higher order conforming elements self adaptive integration and multi region capability finally several detailed examples are presented to illustrate the accuracy and suitability of this boundary element approach for consolidation analysis

on matrices with common invariant cones with applications in neural and gene networks motivated by a differential continuous time switching model for gene and neural networks we investigate matrix theoretic problems regarding the relative location and topology of the dominant eigenvectors of words constructed multiplicatively from two matrices a and b these problems are naturally associated with the existence of common invariant subspaces and common invariant proper cones of a and b the commuting case and the two dimensional case are rich and considered analytically we also analyze and recast the problem of the existence of a common invariant polyhedral cone in a multilinear framework as well as present necessary conditions for the existence of low dimensional common invariant cones

periodic solutions of gene networks with steep sigmoidal regulatory functions abstract we address the question of existence and stability of periodic solutions in gene regulatory networks the threshold dependent network dynamics divides the phase space into domains and a qualitative description can be derived specifying which transitions between domains can occur any periodic solution must follow a cyclic sequence of domains but the problem of determining when such a cyclic sequence of domains contains a periodic solution and when it is stable has not been completely resolved though results have been obtained before for restricted classes of networks here we develop a method by which existence or non existence of such solutions can be demonstrated analytically in any given example of a general class of gene networks with steep sigmoidal interactions under the assumption that any gene product that regulates multiple other genes does so at distinct thresholds our method determines qualitative stability but we also give a procedure that where applicable allows determination of quantitative stability of a periodic solution this complements the previous development of a local analysis method for this class of systems which allows computation of trajectories through any sequence of domains together the previous and current work form the basis for rigorous computer aided assessment of qualitative dynamics of a very general class of gene network models the ability to handle periodic solutions will also increase the applicability of such a computational tool to the design of synthetic networks

on the mechanical energy and effective stress in saturated and unsaturated porous continua abstract this paper has two objectives a to formulate a mechanical theory of porous continua within the framework of strong discontinuity concept commonly employed for the analysis of strain localization in inelastic solids and b to introduce an effective stress tensor for three phase solid water air partially saturated porous continua emerging from principles of thermodynamics to achieve the first objective strong forms of the boundary value problems are compared between two formulations the first in which the velocity jump at the solid fluid interface is treated as a strong discontinuity problem and the second in which the strong discontinuity is smeared in the representative volume element as for the second objective an effective cauchy stress tensor of the form 1 k k s p 1 emerges from the formulation where is the total stress tensor k and ks are the bulk moduli of the solid matrix and solid phase respectively and p is the mean pore water and pore air pressures weighted according to the degree of saturation we show from the first and second laws of thermodynamics that this effective stress tensor is power conjugate to the solid rate of deformation tensor and that it includes the mechanical power required to compress the solid phase

a survey of numerical methods for ivps of odes with discontinuous right hand side this work is dedicated to the memory of donato trigiante who has been the first teacher of numerical analysis of the second author the authors remember donato as a generous teacher always ready to discuss with his students able to give them profound and interesting suggestions here we present a survey of numerical methods for differential systems with discontinuous right hand side in particular we will review methods where the discontinuities are detected by using an event function so called event driven methods and methods where the discontinuities are located by controlling the local errors so called time stepping methods particular attention will be devoted to discontinuous systems of filippov s type where sliding behavior on the discontinuity surface is allowed

on common invariant cones for families of matrices abstract the existence and construction of common invariant cones for families of real matrices is considered the complete results are obtained for 2 2 matrices with no additional restrictions and for families of simultaneously diagonalizable matrices of any size families of matrices with a shared dominant eigenvector are considered under some additional conditions

stability of stationary solutions of piecewise affine differential equations describing gene regulatory networks abstract we study stability properties of a class of piecewise affine systems of ordinary differential equations arising in the modeling of gene regulatory networks our method goes back to the concept of a filippov stationary solution in the narrow sense to a differential inclusion corresponding to the system in question the main result of the paper justifies a reduction principle in the stability analysis enabling to omit the variables that are not singular i e that stay away from the discontinuity set of the system we suggest also the first approximation method to study asymptotic stability of stationary solutions based on calculating the principal part of the system which is 0 homogeneous rather than linear this leads to an efficient algorithm of how to check asymptotic stability without calculating the eigenvalues of the system s jacobian in appendix a we discuss and compare two other concepts of stationary solutions to the system in question

filippov solutions in the analysis of piecewise linear models describing gene regulatory networks abstract we study some properties of piecewise linear differential systems describing gene regulatory networks where the dynamics are governed by sigmoid type nonlinearities which are close to or coincide with the step functions to overcome the difficulty of describing the dynamics of the system near singular stationary points belonging to the discontinuity set of the system we use the concept of filippov solutions it consists in replacing discontinuous differential equations with differential inclusions the global existence and some other basic properties of the filippov solutions such as continuous dependence on parameters are studied we also study the uniqueness and non uniqueness of the filippov solutions in singular domains the concept of filippov stationary point is extensively exploited in the paper we compare two ways of defining the singular stationary points one is based on the filippov theory and the other consists in replacing step functions with steep sigmoids and investigating the smooth systems thus obtained the results are illustrated by a number of examples

dynamical stability and finite amplitude perturbations in coupled genetic networks abstract in this paper we show that coupled map lattice cml formalisms can be applied to study some general features of biological systems from a molecular point of view in particular we analyze the stability of such systems with respect to perturbations specifically we show that the response of the model of coupled formal genetic networks to finite amplitude perturbations may strongly depend on the level of the spatial coupling which determines different dynamical transient regimes before a periodic evolution is eventually attained in particular we observe transients that grow exponentially with the system size while the exact computation of the spectrum of the lyapunov exponents indicates that the maximum is always negative independent of the coupling strength a complete characterization of the correspondence between stability properties and dynamical features of the model has been obtained by extended dynamical simulations an interpretation of these results in a biological perspective indicates that signal transduction should be considered as a possible mechanism of modification of gene expression by transient perturbations on cell populations

sliding motion in filippov differential systems theoretical results and a computational approach in this work we discuss some theoretical and numerical aspects of solving differential equations with discontinuous right hand sides of filippov type in particular i we propose second order corrections to the theory of filippov ii we provide a systematic and nonambiguous way to define the vector field on the intersection of several surfaces of discontinuity and iii we propose and implement a numerical method to approximate a trajectory of systems with discontinuous right hand sides and illustrate its performance on a few examples

mathematical theory and simulations of thermoporoelasticity abstract in this paper we study the equations of semi linear thermoporoelasticity starting point is the dimensionless formulation in van duijn et al 2019 which was obtained by a formal two scale expansion nonlinearities in the equations arise through the fluid viscosity and the thermal conductivity both may depend on temperature and through the coupling in the heat convection by the darcy discharge in the energy equation the coupled system of equations involves as unknowns the skeleton displacement darcy discharge fluid pressure and temperature we treat the system in its incremental i e time discrete form we prove existence by applying a fundamental theorem of brezis on pseudo monotone operators moreover we show that the free energy of the system acts as a lyapunov functional this yields global stability in the time stepping process our theoretical results are substantiated with two dimensional numerical tests using a monolithic formulation temporal discretization is based on the backward euler scheme and finite elements are employed for the spatial discretization the semi linear discrete system is solved with newton s method in the proposed numerical examples different source terms are employed and spatial mesh refinement studies show computational convergence

analysis and generic properties of gene regulatory networks with graded response functions the mass of new genomic data has lead to a growing interest in gene regulatory models describing the regulatory aspects of gene activity in these models the rates of change of gene product concentrations are expressed as a sum of regulatory switches in terms of sums of products of sigmoid functions mimicking complex logical on off functions turning gene activity on and off the customary way to analyse such equations is to replace the sigmoid functions by step functions and disregard models with autoregulation this leads to discontinuous equations of motion but in models without autoregulation continuous solutions can easily be obtained with effective autoregulation this simple approach breaks down as real regulators have finite gain and autoregulation is ubiquitous in biological systems we propose a generalised gene regulatory model framework admitting autoregulation and graded sigmoid functions with different steepnesses using singular perturbation analysis methods we derive a set of reformulated continuous equations for the limit solution when the sigmoids approach step functions and show that the solution for steep sigmoids approaches this limit solution uniformly in a finite time interval a conspicuous feature of the limit solution is so called sliding motionduring which the solution slides along a threshold hyperplane or the intersection of such hyperplanes a particular mapping leads to a dual picture of the flow where the fast parts are magnified at the expense of the slow parts combining the two dual pictures a simple and powerful method to analyse the flow is obtained we show that for steep sigmoid functions the flow may be highly sensitive to the relative steepnesses of the sigmoids this has important consequences when stochastic effects are taken into account also it implies that the observation by glass and

chaos in high dimensional neural and gene networks abstract neural and gene networks are often modeled by differential equations if the continuous threshold functions in the differential equations are replaced by step functions the equations become piecewise linear pl equations the flow through the state space is represented schematically by paths and directed graphs on am n dimensional hypercube closed pathways called cycles may reflect periodic orbits with associated fixed points in a chosen poincare section a return map in the poincare section can be constructed by the composition of fractional linear maps the stable and unstable manifolds of the fixed points can be determined analytically these methods allow us to analyze the dynamics in higher dimensional networks as exemplified by a four dimensional network that displays chaotic behavior the three dimensional poincare map is projected to a two dimensional plane this much simpler piecewise linear two dimensional map conserves the important qualitative features of the flow

effect of time delay on pattern formation competition between homogenisation and patterning abstract we study a mathematical model for juxtacrine signalling in a discrete lattice of cells as the signalling is assumed to be under transcriptional control and transcription is a time consuming process we incorporate time delays in the equations and study the effect of this on the pattern forming potential of the model previous models without time delays have shown that the mechanism is able to generate spatial patterns the analysis of the delay model reveals a transient competition between patterning and homogeneous oscillations a fine grained pattern eventually appears over the whole lattice but the duration of the oscillatory behaviour increases as the time delay increases the results illustrate the importance of including the known delays in a model and of studying transients as these may not be favourable to the system in addition the results may suggest that there are other mechanisms regulating the signalling than transcription for example protein protein interactions which would render the patterning process much faster

on singular integral equations and fundamental solutions of poroelasticity abstract this paper presents a unified formulation of the various singular integral equations used in the boundary element methods bem for the solution of linear quasi static anisotropic poroelasticity in particular a derivation is provided that connects the direct method with the indirect methods the presentation begins with an alternative derivative of the time and space dependent reciprocal integral the somigliana type integral equations for the direct bem are first constructed by summing integral equations representing an interior and an exterior domain problem the somigliana displacement discontinuity and volterra stress discontinuity type dislocation equations for indirect bem are obtained an extension to the edge dislocation method is discussed these stress and displacement discontinuity equations are then combined to construct a symmetric galerkin integral equation system through such construction many intriguing connections among green s functions of fluid source dipole dilatation fluid body force total body force and displacement discontinuity are revealed finally a complete compilation of fundamental solutions for the isotropic case is provided

parameter space structure of continuous time recurrent neural networks a fundamental challenge for any general theory of neural circuits is how to characterize the structure of the space of all possible circuits over a given model neuron as a first step in this direction this letter begins a systematic study of the global parameter space structure of continuous time recurrent neural networks ctrnns a class of neural models that is simple but dynamically universal first we explicitly compute the local bifurcation manifolds of ctrnns we then visualize the structure of these manifolds in net input space for small circuits these visualizations reveal a set of extremal saddle node bifurcation manifolds that divide ctrnn parameter space into regions of dynamics with different effective dimensionality next we completely characterize the combinatorics and geometry of an asymptotically exact approximation to these regions for circuits of arbitrary size finally we show how these regions can be used to calculate estimates of the probability of encountering different kinds of dynamics in ctrnn parameter space

analysis of continuous time switching networks abstract models of a number of biological systems including gene regulation and neural networks can be formulated as switching networks in which the interactions between the variables depend strongly on thresholds an idealized class of such networks in which the switching takes the form of heaviside step functions but variables still change continuously in time has been proposed as a useful simplification to gain analytic insight these networks called here glass networks after their originator are simple enough mathematically to allow significant analysis without restricting the range of dynamics found in analogous smooth systems a number of results have been obtained before particularly regarding existence and stability of periodic orbits in such networks but important cases were not considered here we present a coherent method of analysis that summarizes previous work and fills in some of the gaps as well as including some new results furthermore we apply this analysis to a number of examples including surprising long and complex limit cycles involving sequences of hundreds of threshold transitions finally we show how the above methods can be extended to investigate aperiodic behaviour in specific networks though a complete analysis will have to await new results in matrix theory and symbolic dynamics

singular perturbation analysis and gene regulatory networks with delay abstract the problem of how the dynamics of the smooth gene networks is related to the simplified dynamics of the boolean networks is studied the emphasis is put on the gene regulatory networks with delay asymptotic analysis which is applied in the paper goes back to tikhonov s theory of singular perturbed differential equations and a modified algorithm of reducing delay equations to ordinary differential equations a number of illustrative examples complements the theory which is offered in the paper

a bifurcation analysis of the four dimensional generalized hopfield neural network abstract a four neuron hopfield neural network with asymmetric weights and self connection is analyzed its stable steady state and periodic attractors are identified and a complete bifurcation diagram is constructed a center manifold reduction is undertaken and by means of normal form theory the characteristics of the limit cycles are obtained the network is seen to have a large memory storage capacity with both fixed point and periodic memories coexisting the onset of chaos as reported previously is considered

optimal geometry of transportation networks motivated by the shape of transportation networks such as subways we consider a distribution of points in the plane and ask for the network g of given length l that is optimal in a certain sense in the general model the optimality criterion is to minimize the average over pairs of points chosen independently from the distribution time to travel between the points where a travel path consists of any line segments in the plane traversed at slow speed and any route within the subway network traversed at a faster speed of major interest is how the shape of the optimal network changes as l increases we first study the simplest variant of this problem where the optimization criterion is to minimize the average distance from a point to the network and we provide some general arguments about the optimal networks as a second variant we consider the optimal network that minimizes the average travel time to a central destination and discuss both analytically and numerically some simple shapes such as the star network the ring or combinations of both these elements finally we discuss numerically the general model where the network minimizes the average time between all pairs of points for this case we propose a scaling form for the average time that we verify numerically we also show that in the medium length regime as l increases resources go preferentially to radial branches and that there is a sharp transition at a value l c where a loop appears

common non trivial invariant closed cones for commuting contractions abstract let t t 1 t n be a system of n commuting contractions defined on a infinite dimensional separable hilbert space h in this article we will prove that if 1 1 he t t n where he t denotes the essential harte spectrum of t and t n the unit politorus respectively then there exists a non trivial cone c invariant for each contraction t j j 1 n this result complements recent results of tsatsomeros and co workers roderick edwards judith j mcdonald michael j tsatsomeros on matrices with common invariant cones with applications in neural and gene networks linear algebra appl 398 2005 37 67 michael tsatsomeros a criterion for the existence of common invariant subspaces of matrices linear algebra appl 322 1 3 2001 51 59

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

smooth dynamics becomes hybrid in the limit we show the appearance of an essentially nonlocal dynamics describing the limit behavior of trajectories of a class of dynamical systems defined by classical autonomous odes with smooth right hand sides containing a small parameter and becoming discontinuous in the formal limit the limit dynamics is shown to be described by an explicitly constructed nerode kohn hybrid dynamical system consisting of a continuous plant an ode and a finite state machine which are interacting and producing hybrid dynamics with possible memory effects we remark however that from the statistical point of view the limit behavior of an ensemble of trajectories can still be described by an ode with possibly time dependent and discontinuous right hand side depending on the chosen ensemble

analytical and variational numerical methods for unstable miscible displacement flows in porous media the miscible displacement of one fluid by another in a porous medium has received considerable attention in sub surface environmental and petroleum engineering applications when a fluid of higher mobility displaces another of lower mobility unstable patterns referred to as viscous fingering may arise their physical and mathematical study has been the object of numerous investigations over the past century the objective of this paper is to present a review of these contributions with particular emphasis on variational methods these algorithms are tailored to real field applications thanks to their advanced features handling of general complex geometries robustness in the presence of rough tensor coefficients low sensitivity to mesh orientation in advection dominated scenarios and provable convergence with fully unstructured grids this paper is dedicated to the memory of dr jim douglas jr for his seminal contributions to miscible displacement and variational numerical methods

chaotic dynamics in an electronic model of a genetic network we consider dynamics in a class of piecewise linear ordinary differential equations and in an electronic circuit that model genetic networks in these models gene activity varies continuously in time however as in boolean or discrete time switching networks gene activity is driven high or low based only on whether the activities of the regulating genes are high or low i e above or below certain thresholds depending on the regulatory logic these models can exhibit simple dynamics like stable fixed points or oscillation or chaotic dynamics the observed qualitative and quantitative differences between the dynamics in the idealized equations and the dynamics in the electronic circuit lead us to focus attention on the analysis of the dynamics as a function of parameter values we propose new techniques for solving the inverse problem the problem of inferring the regulatory logic and parameters from time series data we also give new symbolic and statistical methods for characterizing dynamics in these networks