restricted ornstein uhlenbeck process and applications in neuronal models with periodic input signals restricted gauss markov processes are used to construct inhomogeneous leaky integrate and fire stochastic models for single neurons activity in the presence of a lower reflecting boundary and periodic input signals the first passage time problem through a time dependent threshold is explicitly developed numerical simulation and asymptotic results for firing densities are provided

an infinity norm bound for the inverse of dashnic zusmanovich type matrices with applications abstract an upper bound for the infinity norm for the inverse of dashnic zusmanovich type matrices is given it is proved that the upper bound is sharper than the well known varah s bound for strictly diagonally dominant matrices by introducing a new subclass of p matrices dashnic zusmanovich type b matrices dz type b matrices and using the proposed infinity norm bound an error bound is given for the linear complementarity problems of dz type b matrices we also give a new pseudospectra localization to measure the distance to instability

mode interaction in rayleigh be nard convection abstract a long wave model for two dimensional thermal convection is examined this model contains both quadratic and cubic nonlinearity unless the problem is symmetrical about the horizontal mid plane in which case only the cubic nonlinearity is present in the asymmetrical case a strong interaction between modes with wavenumbers k and 2k is responsible for wavenumber gaps regions in parameter space where steady states of a given spatial period cease to exist in both the symmetrical and asymmetrical cases a 1 3 resonance generates multiple steady solutions of a given spatial period we calculate steady states and determine their stability for the asymmetrical case we also solve the initial value problem and find stable travelling waves stable modulated travelling waves and stable heteroclinic cycles as predicted by an analysis of the 1 2 resonance

numerical methods for time periodic solutions of nonlinear parabolic boundary value problems this paper is devoted to a numerical analysis of periodic solutions of a finite difference system which is a discrete version of a class of nonlinear reaction diffusion convection equations under nonlinear boundary conditions three monotone iterative schemes for the finite difference system are presented and it is shown by the method of upper and lower solutions that the sequence of iterations from each of these iterative schemes converges monotonically to either a maximal periodic solution or a minimal periodic solution depending on whether the initial iteration is an upper solution or a lower solution a comparison theorem for the various monotone sequences is given it is also shown that the maximal and minimal periodic solutions of the finite difference system converge to the corresponding maximal and minimal periodic solutions of the reaction diffusion convection equation as the mesh size decreases to zero some error estimates between the theoretical and the computed iterations for each of the three iterative schemes are obtained and a discussion on the numerical stability of these schemes is given also given are some numerical results of a logistic reaction diffusion problem

computation of error bounds for p matrix linear complementarity problems we give new error bounds for the linear complementarity problem where the involved matrix is a p matrix computation of rigorous error bounds can be turned into a p matrix linear interval system moreover for the involved matrix being an h matrix with positive diagonals an error bound can be found by solving a linear system of equations which is sharper than the mathias pang error bound preliminary numerical results show that the proposed error bound is efficient for verifying accuracy of approximate solutions

on variational iterative methods for semilinear problems abstract in this article an iterative method suitable for inverting semilinear problems is presented the method employs a two way parametrization that is able to reduce semilinear boundary value problems into simpler and linear systems governed by the laplacian in contrast to newton picard iterations the method prescribes a well defined procedure for constructing initial iterates and thus does not suffer from lack of global convergence with rectangular domains and periodic boundary conditions the technique also allows fast poisson solvers to be engaged theoretical justifications are provided and supported by several experiments numerical results show that the method yields approximations which are comparable to reference solutions

error bounds for linear complementarity problems of db matrices doubly b matrices db matrices which properly contain b matrices are introduced by pena 2003 2 in this paper we present error bounds for the linear complementarity problem when the matrix involved is a db matrix and a new bound for linear complementarity problem of a b matrix the numerical examples show that the bounds are sharp

control of noisy heteroclinic cycles abstract we develop a control strategy for strongly nonlinear dynamical systems possessing heteroclinic cycles connecting saddle points and subject to small random perturbations in the absence of control attracting cycles lead to solutions which exhibit intermittent behavior consisting of quiescent periods interrupted by rapid bursts we show that suitable feedback control can retain solutions near the saddle points and hence reduce the typical bursting rate the model studied and the form of control are motivated by our interest in reducing turbulence production in the boundary layer

a numerical bifurcation function for homoclinic orbits we present a numerical method to locate periodic orbits near homoclinic orbits using a method of x b lin proc roy soc edinburgh 116a 1990 pp 295 325 and solutions of the adjoint variational equation we get a bifurcation function for periodic orbits whose periods are asymptotic to infinity on approaching a homoclinic orbit as a bonus a linear predictor for continuation of the homoclinic orbit is easily available numerical approximation of the homoclinic orbit and the solution of the adjoint variational equation are discussed we consider a class of methods for approximating the latter equation such that a scalar quantity is preserved we also consider a context where the effects of continuous symmetries of equations can be incorporated applying the method to an ordinary differential equation on r3 studied by e freire a rodriguez luis and e ponce phys d 62 1993 pp 230 253 we show the bifurcation function gives good agreement with path followed solutions even down to low period as an example application to a parabolic partial differential equation pde we examine the bifurcation function for a homoclinic orbit in the kuramoto sivashinsky equation

error bounds for the linear complementarity problem with a p matrix abstract we introduce a fundamental quantity associated with a p matrix and show how this quantity is useful in deriving error bounds for the linear complementarity problem of the p type we also obtain upper and lower bounds for the quantity introduced

secondary turing type instabilities due to strong spatial resonance we investigate the dynamics of pattern forming systems in large domains near a codimension two point corresponding to a strong spatial resonance where competing instabilities with wavenumbers in th

complex dynamics in the 1 3 spatial resonance abstract the interaction between two steady state bifurcations with spatial wave numbers in the ratio 1 3 is considered periodic boundary conditions are assumed the resulting o 2 equivariant normal form truncated at third order exhibits a number of global bifurcations that may result in complex dynamics the origin of this behavior is elucidated with the help of careful numerical simulations and analysis of appropriate return maps the results generalize to other 1 n resonances

forced symmetry breaking of homoclinic cycles in a pde with o 2 symmetry we perform a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a pde with o 2 symmetry to one with so 2 symmetry taking particular care of the consequences of the continuous group action we concentrate on the kuramoto sivashinsky equation with spatially periodic boundary conditions the breakup of structurally stable homoclinic cycles is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry in particular we note that although chossat 1993 has proved that generic perturbations cause the appearance of quasiperiodic orbits for the simplest possible flux terms this is not the case we compare these results with numerical simulations of a galerkin approximation of the equations

solving large nonlinear systems of equations by an adaptive condensation process we present an algorithm which efficiently solves large nonlinear systems of the form au f u u in mathbb r n whenever an iterative solver a 1 for the symmetric positive definite matrixa is available andf u is symmetric such problems arise from the discretization of nonlinear elliptic partial differential equations by means of an adaptive decomposition process we split the original system into a low dimensional system to be treated by any sophisticated solver and a remaining high dimensional system which can easily be solved by fixed point iteration specifically we choose a newton type trust region algorithm for the treatment of the small system we show global convergence under natural assumptions on the nonlinearity the convergence results typical for trust region algorithms carry over to the full iteration process the only large systems to be solved are linear ones with the fixed matrixa thus existing software for positive definite sparse linear systems can be used

bifurcation with memory a model equation containing a memory integral is posed the extent of the memory the relaxation time lambda controls the bifurcation behavior as the control parameter r is increased small large lambda gives steady periodic bifurcation there is a double eigenvalue at lambda lambda 1 separating purely steady lambda lambda 1 states with t to infty as lambda to lambda 1 analysis leads to the co existence of stable steady periodic states and as r is increased the periodic states give way to the steady states numerical solutions show that this behavior persists away from lambda lambda 1

global bifurcations close to symmetry abstract heteroclinic cycles involving two saddle foci where the saddle foci share both invariant manifolds occur persistently in some symmetric differential equations on the 3 dimensional sphere we analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1 dimensional connection the saddle foci have the same chirality when part of the symmetry is broken the 2 dimensional invariant manifolds intersect transversely creating a heteroclinic network of bykov cycles we show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics there are n pulse heteroclinic tangencies trajectories that follow the original cycle n times around before they arrive at the other node each n pulse heteroclinic tangency is accumulated by a sequence of n 1 pulse ones this coexists with the suspension of horseshoes defined on an infinite set of disjoint strips where the first return map is hyperbolic we also show how as the system approaches full symmetry the suspended horseshoes are destroyed creating regions with infinitely many attracting periodic solutions

nonsmooth dynamics in spiking neuron models large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue one approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models an alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies in this latter case the single neuron model of choice is often a variant of the classic integrate and fire model which is described by a nonsmooth dynamical system in this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate ranging from tonic to burst firing we show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis particularly those for treating grazing bifurcations importantly we highlight one particular single neuron model capable of generating realistic spike trains that is both computationally cheap and analytically tractable this is a planar nonlinear integrate and fire model with a piecewise linear vector field and a state dependent reset upon spiking we call this the pwl if model and analyse it at both the single neuron and network level the techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model as well as to develop the notion of liapunov exponents we also show how to construct the phase response curve for this system emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems the stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times at the network level we consider linear coupling between voltage variables as would occur in neurobiological networks with gap junction coupling and show how to analyse the properties existence and stability of both the asynchronous and synchronous states in the former case we use a phase density technique that is valid for any large system of globally coupled limit cycle oscillators whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics

first passage times of two dimensional correlated processes given a two dimensional correlated diffusion process we determine the joint density of the first passage times of the process to some constant boundaries this quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time first we show that these densities are solutions of a system of volterra fredholm first kind integral equations then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times the convergence of the method is theoretically proved for bivariate diffusion processes we derive explicit expressions for these and other quantities of interest in the case of a bivariate wiener process correcting previous misprints appearing in the literature finally we illustrate the application of the method through a set of examples

bifurcations with imperfect so 2 symmetry and pinning of rotating waves rotating waves are periodic solutions in so 2 equivariant dynamical systems their precession frequency changes with parameters and it may change sign passing through zero when this happens the

infimum of error bounds for linear complementarity problems of σ sdd and σ1 ssd matrices abstract in 2013 18 garcia esnaola and pena presented error bounds for the linear complementarity problems of sdd and 1 ssd matrices respectively these bounds all involve a parameter on an open interval it is shown in this paper that the infimum of these error bounds as a function of the parameter exists and then is determined numerical examples are given to verify the theoretical results

a parallel algorithm for the computation of invariant tori in large scale dissipative systems a parallelizable algorithm to compute invariant tori of high dimensional dissipative systems obtained upon discretization of pdes is presented the size of the set of equations to be solved is only a small multiple of the dimension of the original system the sequential and parallel implementations are compared with a previous method sanchez et al 2010 11 showing that important savings in wall clock time can be achieved in order to test it a thermal convection problem of a binary mixture of fluids has been used the new method can also be applied to problems with very low rotation numbers for which the previous is not suitable this is tested in two examples of two dimensional maps

nonlinear interactions of tearing modes in the vicinity of a bifurcation point of codimension two abstract the interaction of two different tearing modes in slab geometry is examined for certain physical parameter values the linearized problem has a four dimensional nullspace corresponding to two different marginal tearing modes with the center manifold theory the original partial differential equations could be reduced to a four dimensional system of ordinary ones these amplitude equations are equivalent under o 2 actions due to the symmetries of the physical problem because of these symmetries there exist many interesting spatial and spatio temporal solutions in the neighbourhood of the critical parameters such as standing travelling and modulated waves and a structural stable heteroclinic orbit see ambruster guckenheimer and holmes physica d 29 1988 257 282

the 1 2 hopf steady state mode interaction in three dimensional magnetoconvection abstract the first analysis of a properly three dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented by varying the fluid parameters for boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously we then construct and analyse the normal form the complex transition between steady and oscillatory convection near onset can be explained this extends and completes the work of clune and knobloch pattern selection in three dimensional magnetoconvection physica d 74 1994 151 176 selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer the symmetries of the resulting d 4 t 2 equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions a m rucklidge m silber bifurcations of periodic orbits with spatio temporal symmetries nonlinearity 11 1998 1435 1455 we also find new phenomena in the normal form for a hopf bifurcation with d 4 t 2 symmetry m silber e knobloch hopf bifurcation on a square lattice nonlinearity 4 1991 1063 1106 the introduction of weakly non boussinesq effects leads to qualitative changes in the dynamics near onset different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed

robust heteroclinic cycles in the one dimensional complex ginzburg landau equation numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one dimensional complex ginzburg landau equation cgl on the unit spatially periodic domain these cycles connect difierent spatially and temporally inhomogeneous time periodic solutions as t 1 a careful analysis of the connections is made using a projection onto 5 complex fourier modes it is shown flrst that the time periodic solutions can be treated as relative equilibria after consideration of the symmetries of the cgl second the cycles are shown to be robust since the individual heteroclinic connections exist in invariant

back in the saddle again a computer assisted study of the kuramoto sivashinsky equation a numerical and analytical study of the kuramoto sivashinsky partial differential equation pde in one spatial dimension with periodic boundary conditions is presented the structure stability and bifurcation characteristics of steady state and time dependent solutions of the pde for values of the parameter alpha less than 40 are examined the numerically observed primary and secondary bifurcations of steady states as well as bifurcations to constant speed traveling waves limit cycles are analytically verified persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of o 2 their effect on the system dynamics is discussed and several tertiary bifurcations observed numerically are presented

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

heteroclinic cycles and periodic orbits for the o 2 equivariant 0 1 2 mode interaction we study the quadratic normal form describing the generic interaction of fourier modes of wavenumbers 0 1 and 2 under the symmetry group o 2 of rotations and reflections in the case that the homogeneous quadratic terms preserve energy the sum of the squares of absolute values of the complex variables this system is a generalization of the 1 2 mode interaction studied by dangelmayr g dangelmayr steady state mode interactions in the presence of o 2 symmetry dyn stab syst 1 2 1986 159 185 armbruster et al d armbruster j guckenheimer p holmes heteroclinic cycles and modulated travelling waves in systems with o 2 symmetry physica d 29 1988 257 282 and others and its restriction to the 1 2 subspace is a degenerate case of that system it displays all the classes of fixed points periodic orbits standing and travelling waves invariant tori modulated travelling waves and heteroclinic cycles found in the 1 2 interaction as well as new heteroclinic cycles connecting pure and mixed modes chaotic cycles and strange periodic orbits we describe the key dynamical features show that the degenerate 1 2 case possesses a second organizing center at which bifurcation curves coalesce provide representative bifurcation sets and diagrams for the 1 2 and 0 1 2 systems and use a conservative limit to understand the periodic orbits in the latter system 2005 elsevier b v all rights reserved

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

dynamics in the 1 2 spatial resonance with broken reflection symmetry the dynamics near the 1 2 spatial resonance in systems with broken o 2 symmetry are described the system is assumed to remain so 2 invariant and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the o 2 symmetric system the symmetry breaking destroys these cycles and introduces a variety of new bifurcations into the system some which are associated with complex behavior the possible scenarios with increasing symmetry breaking are described and possible applications of the results are discussed

error bounds for the linear complementarity problem with a σ sdd matrix abstract we give new error bounds for the linear complementarity problem when the involved matrix is a sdd matrix with positive diagonals the new bounds can improve considerably other previous bounds we also find a subclass of sdd matrices for which the new bound is very simple

on an alternative to gerschgorin circles and ovals of cassini an alternative to gerschgorin circles for the localization of the real eigenvalues of a real matrix was studied in 8 in this paper we present a similar alternative to the brauer s theorem on ovals of cassini

local models of spatio temporally complex fields abstract we investigate the ability of local models of the one space dimensional kuramoto sivashinsky partial differential equation with periodic boundary conditions obtained by projection on a small set of fourier modes on a short subinterval to reproduce coherent events typical of solutions of the same equation on a much longer interval we find that systems containing as few as two linearly unstable modes can produce realistic local events in the short term but that for more reliable short time tracking and long term statistics three or four interacting modes are required and that the length of the short interval plays a subtle role certain resonant lengths giving superior results

modelling instability of abc flow using a mode interaction between steady and hopf bifurcations with rotational symmetries of the cube abstract we consider hydrodynamic stability of symmetric abc flow with a b c 1 regarded as a steady state of the three dimensional navier stokes equation with appropriate forcing numerical investigations have shown that its first instability on increasing the reynolds number is a hopf bifurcation at r 13 044 and that at this bifurcation the second dominant eigenvalue is real motivated by this we study generic interaction of steady state and hopf bifurcations in systems with rotational symmetry of the cube o with an eight dimensional normal form the generic branching and bifurcation behavior of the third order truncated normal form is investigated this is used to analyse a range of the bifurcations for particular values of the coefficients obtained by center manifold reduction from the hydrodynamic system the normal form system shows a sequence of bifurcations and attractors that closely follows the sequence observed for the original hydrodynamic system up to about r 13 91 this includes a torus breakdown to a chaotic attractor and a crisis of attractors that leads to a change in symmetry we discuss numerical simulations of the hydrodynamic system for larger values of r finally we present evidence that the system has robust heteroclinic cycles between fully symmetric abc flow and six steady states with broken symmetry for a range of parameter values near r 15

a broyden rank p 1 update continuation method with subspace iteration in this paper we present an efficient branch following procedure that can be used not only to compute branches of periodic solutions of periodically forced dynamical systems but also to determine the stability of the periodic solutions the procedure combines broyden s method with a subspace iteration method to determine the dominant eigenvalues the method has connections with the hybrid newton picard methods developed by lust et al in siam j sci comput 19 1998 pp 1188 1209 a convergence analysis of the procedure is presented the method is applied to the computation of periodic states of a reverse flow reactor and its performance is compared with two variants of the hybrid newton picard method

wavelet projections of the kuramoto sivashinsky equation i heteroclinic cycles and modulated traveling waves for short systems abstract we study fairly low dimensional projections of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a short interval onto bases spanned by periodic wavelets such projections break the translation reflection symmetry o 2 replacing it by a finite dihedral group dk however we show that for the perrier basdevant wavelets used here the loss of symmetry is sufficiently mild that key global features of the dynamics are preserved in particular we observe heteroclinic cycles and modulated travelling waves arising from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets we use invariant manifold reductions in our analysis and pay particular attention to symmetries and the relation between periodic wavelets and fourier modes which preserve full o 2 symmetry and are also optimal in that fourier truncations maximise the energy l2 norm among all finite dimensional models this study provides a foundation for current and future work in which we use wavelet bases to extract local models of evolution equations in large space domains

reduction of the hodgkin huxley equations to a single variable threshold model it is generally believed that a neuron is a threshold element that fires when some variable u reaches a threshold here we pursue the question of whether this picture can be justified and study the four dimensional neuron model of hodgkin and huxley as a concrete example the model is approximated by a response kernel expansion in terms of a single variable the membrane voltage the first order term is linear in the input and its kernel has the typical form of an elementary postsynaptic potential higher order kernels take care of nonlinear interactions between input spikes in contrast to the standard volterra expansion the kernels depend on the firing time of the most recent output spike in particular a zero order kernel that describes the shape of the spike and the typical after potential is included our model neuron fires if the membrane voltage given by the truncated response kernel expansion crosses a threshold the threshold model is tested on a spike train generated by the hodgkin huxley model with a stochastic input current we find that the threshold model predicts 90 percent of the spikes correctly our results show that to good approximation the description of a neuron as a threshold element can indeed be justified

instabilities induced by a weak breaking of a strong spatial resonance abstract through multiple scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady state pattern forming instabilities in the case that the wavelengths of the instabilities are nearly in the ratio 1 2 in the case of exact 1 2 resonance the amplitude equations are odes here they are pdes we discuss the stability of spatially periodic solutions to long wavelength disturbances by including these modulational effects we are able to explore the relevance of the exact 1 2 results to spatially extended physical systems for parameter values near to this codimension two bifurcation point these new instabilities can be described in terms of reduced normal form pdes near various secondary codimension two points the robust heteroclinic cycle in the odes is destabilised by long wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle an analytic expression giving the approximate period of this orbit is derived

noise induced intermittency in a model of a turbulent boundary layer abstract the presence of attracting heteroclinic cycles in a set of truncated ordinary differential equations for the modal amplitudes in a proper orthogonal decomposition of the navier stokes equations leads to burst like intermittent phenomenon in the reconstructed velocity field the time between bursts or the cycle time is effectively randomized by the addition of small random perturbations in this case taking the form of irregular pressure fluctuations from the outer flow this suggests that interaction of the inner and outer flow is important in the generation and triggering of these burst events a scaling law is developed for the time between burst events as well as an expression for the probability distribution of these time intervals

structure and dynamics of modulated traveling waves in cellular flames we describe the formation and evolution of spatial and temporal patterns in cylindrical premixed flames we consider the cellular regime le 1 where the lewis number le is the ratio of thermal to mass diffusivity of a deficient component of the combustible mixture a transition from stationary axisymmetric flames to stationary cellular flames is predicted analytically if le is decreased below a critical value we present the results of numerical computations to show that as le is further decreased with all other parameters fixed traveling waves tws along the flame front arise via an infinite period bifurcation which breaks the reflection symmetry of the cellular array upon further decreasing le we find the development of different kinds of periodically modulated traveling waves mtws as well as a branch of quasiperiodically modulated traveling waves qpmtws these transitions are accompanied by the development of different spatial and temporal symmetries including period doublings and period halvings in appropriate coordinate systems we also observe the apparently chaotic temporal behavior of a disordered cellular pattern involving creation and annihilation of cells we analytically describe the stability of the tw solution near its onset using suitable phase amplitude equations within this framework one of the mtws can be identified as a localized wave traveling through an underlying stationary spatially periodic structure

the reliability of a stochastically complex dynamical system abstract the first passage problem is an important issue in the theory of dynamical systems it determines whether the system is safe and reliable or destroyed therefore taking a suitable force to control first passage failure is very significant for such systems in this paper we study the reliability of a stochastically complex dynamical system by using the stochastic averaging method the reliability results show that the reliability function is monotonically decreasing with respect to time and that the reliability of the system is relatively worse with external and parametric white noise excitations a bang bang feedback control is used to improve the reliability for the original system numerical results display that even a small control force can greatly enhance the reliability of the original system

weak convergence of marked point processes generated by crossings of multivariate jump processes applications to neural network modeling abstract we consider the multivariate point process determined by the crossing times of the components of a multivariate jump process through a multivariate boundary assuming to reset each component to an initial value after its boundary crossing we prove that this point process converges weakly to the point process determined by the crossing times of the limit process this holds for both diffusion and deterministic limit processes the almost sure convergence of the first passage times under the almost sure convergence of the processes is also proved the particular case of a multivariate stein process converging to a multivariate ornstein uhlenbeck process is discussed as a guideline for applying diffusion limits for jump processes we apply our theoretical findings to neural network modeling the proposed model gives a mathematical foundation to the generalization of the class of leaky integrate and fire models for single neural dynamics to the case of a firing network of neurons this will help future study of dependent spike trains

application of dynamical system theory to coherent structures in the wall region abstract in an attempt to apply dynamical system approaches to turbulence we derive closed sets of nonlinear ordinary differential equations for the wall region this is achieved by galerkin projection onto statistical coherent structures using the proper orthogonal decomposition lumley 1 which converges optimally fast in the quadratic mean sense application of this definition to the instantaneous velocity field of the wall region shows the presence of streamwise rolls a severe truncation of the dynamical system 10 dimensions is studied to investigate the time behavior of these rolls the energy transfer to unresolved modes is adjusted by a heisenberg parameter for large values of the parameter large loss the solution is steady and represents streamwise rolls having the experimentally observed cross stream spacing for lower values of the parameter intermittency appears due to the presence of a heteroclinic attracting orbit in the phase space the behavior of the streamwise rolls consists in a long time quasi steady state followed by a growing oscillation and a sudden burst during which the rolls lose their identify subsequently the rolls reform and the process is repeated these dynamics appear to minic the bursting behavior observed in the wall region for lower parameter values the regime is much more complex apparently chaotic the fluctuating pressure term appears as a forcing term at the upper boundary of the integration domain although its amplitude is very small it acts as a trigger for bursts and equilibrates the mean time between the events which would otherwise increase with time as the trajectory is attracted closer and closer to the heteroclinic cycle the whole mechanism is qualitatively the same when the wall region is artificially thickened which is done by applying stretching transformations this is in agreement with experimental results concerning drag reduced flows

error bounds for linear complementarity problems of s nekrasov matrices and b s nekrasov matrices abstract an error bound involving a parameter which does not always work for the linear complementarity problem lcp when the involved matrices are s nekrasov matrices is provided by m garcia esnaola and j m pena 2014 in this paper a new error bound for the lcp of s nekrasov matrices is presented which depends only on the entries of the involved s nekrasov matrices based on the obtained results we also give an error bound for the lcp of b s nekrasov matrices

classification of robust heteroclinic cycles for vector fields in with symmetry we consider a classification of robust heteroclinic cycles in the positive octant of under the action of the symmetry group we introduce a coding system to represent different classes up to a topological equivalence and produce a characterization of all types of robust heteroclinic cycle that can arise in this situation these cycles may or may not contain the origin within the cycle we proceed to find a connection between our problem and meandric numbers we find a direct correlation between the number of classes of robust heteroclinic cycle that do not include the origin and the mercedes benz sequence of integers characterizing meanders through a y shaped configuration we investigate upper and lower bounds for the number of classes possible for robust cycles between n equilibria one of which may be the origin

resonant triad interactions in symmetric systems abstract we analyse resonant triad interactions in symmetric systems the case of coupling at quadratic order in amplitudes and the case of coupling at cubic order are considered we study modulated travelling waves and heteroclinic cycles in amplitude equations and we discuss their stability types we describe possible dynamical regimes and spatial patterns arising from triad interactions of long waves in rapidly rotating hagen poiseuille flow

analytic approximations of statistical quantities and response of noisy oscillators abstract nonlinear oscillators have been utilized in many contexts because they encompass a large class of phenomena for a reduced phase oscillator model with weak noise forcing that is necessarily multiplicative we provide analytic formulas for the stationary statistical quantities of the random period this is an important quantity which we term response i e the spike times instantaneous frequency in neuroscience the cycle time in chemical reactions etc that is often analytically intractable in noisy oscillator systems the analytic formulas are accurate in the weak noise limit so that one does not have to numerically solve a time varying fokker planck equation the steady state and dynamic responses are also analyzed with deterministic forcing a second order analytic formula is derived for the steady state response whereas the dynamic response with time varying forcing is more complicated we focus on the specific case where the forcing is sinusoidal and accurately capture the frequency response with an analytic approximation that is obtained with a rescaling of the equation by utilizing various techniques in the weak noise regime this work leads to a better understanding of how the random period of nonlinear oscillators are affected by multiplicative noise and external forcing comparisons of the asymptotic formulas with a full oscillator system confirm the qualitative accurateness of the theory

numerical computation of nonlinear normal modes in mechanical engineering this paper reviews the recent advances in computational methods for nonlinear normal modes nnms different algorithms for the computation of undamped and damped nnms are presented and their respective advantages and limitations are discussed the methods are illustrated using various applications ranging from low dimensional weakly nonlinear systems to strongly nonlinear industrial structures

peer methods with improved embedded sensitivities for parameter dependent odes recently an extension of peer two step methods to parameter dependent ivps has been proposed by the author and e kostina where additional embedded stages allow cheap computation of solution derivatives sensitivities with respect to problem parameters for each parameter only one additional satellite stage was used which provides a high order approximation to a neighboring solution however the accuracy of the sensitivities was only first order in the time stepsize h in the present paper we derive implicit peer methods where the sensitivity accuracy is of second order with the same number of stages approximate implementations yield efficient linearly implicit methods for stiff problems and explicit predictor corrector type versions for nonstiff problems tests confirm improved and more robust convergence of inexact newton iterations in shooting for different applications with nonstiff and stiff ivps

continuation techniques and interactive software for bifurcation analysis of odes and iterated maps physics abstract we present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase parameter space the invariant sets involved in our study are equilibrium points and limit cycles of autonomous odes periodic solutions of time periodic nonautonomous odes fixed points and periodic orbits of iterated maps the more general concept of a continuation strategy is also discussed it allows the analysis of various singularities of generic systems and of their mutual relationships the approach is extended to codimension three singularities we introduce several bifurcation functions and show how to use them to construct well posed continuation problems the described continuation technique is supported by an interactive graphical program called locbif we discuss briefly the concepts of the locbif interface and give some examples of typical applications

the 2 1 steady hopf mode interaction in the two layer be nard problem abstract the interaction of hopf and steady modes with wavenumber ratio 2 1 is investigated for a critical situation two different immiscible liquids lie in layers between horizontal walls and are heated from below a situation with a pair of complex conjugate eigenvalues at wavenumber and a real eigenvalue at wavenumber 2 is at criticality weakly nonlinear amplitude equations are derived for the interaction of these oscillatory and steady modes the two modes generate a two parameter bifurcation the coefficients involved in the equations are determined numerically based on the physical parameters of the system at criticality three obvious equilibrium solutions of the amplitude equations are the steady solution the traveling waves and the mixed standing waves the eigenvalues governing the stability of these solutions are found explicitly numerical results and bifurcation diagrams are given for the critical situation the steady solution and the traveling wave solution are unstable there is a region of stability for the standing wave solution a new equilibrium solution the asymmetric mixed mode is found to be stable in a parameter range bifurcations from the standing wave solution and the asymmetric mixed mode are described

hyperbolic periodic orbits in nongradient systems and small noise induced metastable transitions abstract small noise can induce rare transitions between metastable states which can be characterized by maximum likelihood paths mlps nongradient systems contrast gradient systems in that mlp does not have to cross the separatrix at a saddle point but instead possibly at a point on a hyperbolic periodic orbit a numerical approach for identifying such unstable periodic orbits is proposed based on string method in a special class of nongradient systems orthogonal type there are provably local mlps that cross such saddle point or hyperbolic periodic orbit and the separatrix crossing location determines the associated local maximum of transition rate in general cases however the separatrix crossing may not determine a unique local maximum of the rate as we numerically observed a counter example in a sheared 2d space allen cahn spde it is a reasonable conjecture that there are always local mlps associated with each attractor on the separatrix such as saddle point or hyperbolic periodic orbit our numerical experiments did not disprove so

heteroclinic cycles involving periodic solutions in mode interactions with o 2 symmetry in this paper we show that in o 2 symmetric systems structurally stable asymptoticallystable heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions these cycles are found in the third order truncated normal forms of specific codimension two steady state hopf and hopf hopf mode interactions we find these cycles using group theoretic techniques in particular we look for certainpatterns in the lattice of isotropy subgroups once the pattern has been identified the heteroclinic cycle can be constructed by decomposing the vector field on fixed point subspaces into phase amplitude equations it is here that we use the assumption of normal form the final proof of existence and stability relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied

new type of complex dynamics in the 1 2 spatial resonance abstract the amplitude equations describing the 1 2 spatial resonance in o 2 symmetric systems are re examined complex dynamical behavior associated with a new class of global connections is identified and analyzed these connections form further from the mode interaction point where the structurally stable heteroclinic cycles described by armbruster et al physica d 29 1988 257 no longer exist the resulting behavior is characteristic of systems with o 2 z 2 symmetry in which the symmetry z 2 is weakly broken

an analysis of equivalent operator preconditioning for equation free newton krylov methods we consider the computation of a fixed point of a time stepper using newton krylov methods and propose and analyze equivalent operator preconditioning for the resulting linear systems for a linear scalar advection reaction diffusion equation we investigate in detail how the convergence rate depends upon the choice of preconditioner parameters and upon the time discretization the results are especially valuable when computing fixed points of a coarse time stepper in the equation free multiscale framework in which one simulates an unavailable coarse scale model by wrapping a set of computational routines around appropriately initialized fine scale simulations both analytical results and numerical experiments are presented showing that one can speed up the convergence of iterative methods significantly for a wide range of parameter values in the preconditioner

suppression of bursting abstract we investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear o 2 symmetric dynamical systems possessing structurally stable heteroclinic cycles the leading order behavior near the equilibria is bilinear and homogeneous in the state variables while nonlinearities representing the symmetry breaking effect of the controller are crucial after a series of simplifying transformations we use ideas from optimal control theory to construct a stabilizing controller this study is motivated by the desire to stabilize the burst sweep cycle in low dimensional models of a turbulent boundary layer in the last two sections we apply the techniques to the 10 dimensional system of aubry et al 1988 kw bilinear systems dynamical systems heteroclinic cycles normal forms optimal control symmetry

an example of symmetry breaking to heteroclinic cycles is orbitally asymptotically stable suppose now that we consider a small system symmetry breakingperturbation of 1 1 z f z g z 1 2 where is small and g is only 2 equivariant where 2 1 is a lie subgroup when is sufficiently small normal hyperbolicity guarantees that there isa perturbed flow invariant manifold x

doubly diagonally dominant matrices we consider the class of doubly diagonally dominant matrices a ajj e c la 1 l jjl ck i laiklck jlajkl i j and its subclasses we give necessary and sufficient conditions in terms of the directed graph for an irreducibly doubly diagonally dominant matrix to be a singular matrix or to be an h matrix as in the case of diagonal dominance we show that the schur complements of doubly diagonally dominant matrices inherit this property moreover we describe when a schur complement of a strictly doubly diagonally dominant matrix is strictly diagonally dominant 0 elsevier science inc 1997 1 preliminaries

a linear complementarity problem with a p matrix in this paper we present an application of a linear complementarity problem where m is a p matrix but in general is neither an h matrix nor a positive definite matrix this application occurs originally in j rohn linear algebra appl 126 1989 pp 39 78 which is less known to the lcp community its focus is in computing the exact interval enclosures of the components of the solution set of an interval linear system we extend the idea such that the calculations can be done by a computer with rigorous error control

a comparison of error bounds for linear complementarity problems of h matrices we give new error bounds for the linear complementarity problem when the involved matrix is an h matrix with positive diagonals we find classes of h matrices for which the new bounds improve considerably other previous bounds we also show advantages of these new bounds with respect the computational cost a new perturbation bound of h matrices linear complementarity problems is also presented

a new error bound for linear complementarity problems with weakly chained diagonally dominant b matrices abstract recently some error bounds for the linear complementarity problem with a weakly chained diagonally dominant b matrix have been given under certain conditions in this paper we provide a new and even optimal error bound for such linear complementarity problems under weaker conditions than those there greatly improving the existing ones some numerical examples are performed to illustrate the sharpness and optimality of our new bound

dynamics near a periodically perturbed robust heteroclinic cycle abstract robust heteroclinic cycles rhcs arise naturally in collections of symmetric differential equations derived as dynamical models in many fields including fluid mechanics game theory and population dynamics in this paper we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle rhc we give a detailed derivation of the poincare map for trajectories near the rhc asymptotically correct in the limit of small amplitude perturbations this reduces the nonautonomous system in r 3 to a 2d map we identify three distinct dynamical regimes the distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem the first regime corresponds to the rhc being only weakly attracting here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque in the second regime the periodically perturbed rhc is more strongly attracting and the system dynamics corresponds to that of a non invertible or invertible circle map in the third regime of yet stronger attraction the dynamics of the return map is chaotic and no longer reducible to a one dimensional map this third regime has been noted previously our analysis in this paper focusses on providing quantitative results in the first two regimes

an adaptive newton picard algorithm with subspace iteration for computing periodic solutions this paper is concerned with the efficient computation of periodic orbits in large scale dynamical systems that arise after spatial discretization of partial differential equations pdes a hybrid newton picard scheme based on the shooting method is derived which in its simplest form is the recursive projection method rpm of shroff and keller siam j numer anal 30 1993 pp 1099 1120 and is used to compute and determine the stability of both stable and unstable periodic orbits the number of time integrations needed to obtain a solution is shown to be determined only by the system s dynamics this contrasts with traditional approaches based on newton s method for which the number of time integrations grows with the order of the spatial discretization two test examples are given to show the performance of the methods and to illustrate various theoretical points

heteroclinic cycles in rings of coupled cells abstract symmetry is used to investigate the existence and stability of heteroclinic cycles involving steady state and periodic solutions in coupled cell systems with d n symmetry using the lattice of isotropy subgroups we study the normal form equations restricted to invariant fixed point subspaces and prove that it is possible for the normal form equations to have robust asymptotically stable heteroclinic cycles connecting periodic solutions with steady states and periodic solutions with periodic solutions a center manifold reduction from the ring of cells to the normal form equations is then performed using this reduction we find parameter values of the cell system where asymptotically stable cycles exist simulations of the cycles show trajectories visiting steady states and periodic solutions and reveal interesting spatio temporal patterns in the dynamics of individual cells we discuss how these patterns are forced by normal form symmetries

a numerical liapunov schmidt method with applications to hopf bifurcation on a square we discuss an iterative method for calculating the reduced bifurcation equation of the liapunov schmidt method and its numerical approximation using appropriate genericity assumptions with symmetry we derive a taylor series for the reduced equation where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation this method is used to calculate reduced equations at hopf bifurcation of the two dimensional brusselator equations on a square with neumann and dirichlet boundary conditions we examine several hopf bifurcations within the three parameter space there are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry

low dimensional chaotic dynamics versus intrinsic stochastic noise a paradigm model several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables these models are built by coupling a four dimensional ode with known analytical properties including heteroclinic cycles with a suitable deterministic bath of fast variables a systematic strategy for stochastic mode reduction is applied to these models with 104 degrees of freedom to derive four dimensional reduced stochastic equations for the slow variables due to the internal chaotic dynamics of the slow variables the stochastic mode reduction strategy is very robust in this case and yields reduced models which accurately capture the statistical behavior of the original deterministic system furthermore it is also shown here that even in the regime of a weak coupling between the slow variables and the fast heat bath the detailed structure of the stochastic terms derived through the mode elimination procedure is essential for reproducing the statistical behavior of the slow dynamics

transverse bifurcations of homoclinic cycles abstract homoclinic cycles exist robustly in dynamical systems with symmetry and may undergo various bifurcations not all of which have an analog in the absence of symmetry we analyze such a bifurcation the transverse bifurcation and uncover a variety of phenomena that can be distinguished representation theoretically for example exponentially flat branches of periodic solutions a typical feature of bifurcation from homoclinic cycles occur for some but not all representations of the symmetry group our study of transverse bifurcations is motivated by the problem of intermittent dynamos in rotating convection