a direct approach to the exit problem this paper considers the problem of exit for a dynamical system driven by small white noise from the domain of attraction of a stable state a direct singular perturbation analysis of the forward equation is presented based on kramers approach in which the solution to the stationary fokker planck equation is constructed for a process with absorption at the boundary and a source at the attractor in this formulation the boundary and matching conditions fully determine the uniform expansion of the solution without resorting to external selection criteria for the expansion coefficients such as variational principles or the lagrange identity as in our previous theory the exit density and the mean first passage time to the boundary are calculated from the solution of the stationary fokker planck equation as the probability current density and as the inverse of the total flux on the boundary respectively as an application a uniform expansion is constructed for the escape rate in kramers problem o

lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

consistent relations in the method of reducibility in the cluster variation method a redundancy is noted in the set of lagrange multipliers effective fields and effective interactions used in the expressions of the reduced density matrices given by morita 4 in some problems with less symmetry 6 the expressions of the reduced density matrices without this redundancy are given an argument showing that a certain expression of the free energy in terms of the lagrange multipliers is stationary with respect to the variations of these is presented

upper bounds for survival probability of the contact process a precise description of the nontrivial upper invariant measure for c is still an open problem for the basic contact process which is a self dual attractive but nonreversible markov process of an interacting particle system by its self duality to identify the invariant measure is equivalent to determining the initial state dependence of the survival probability of the process a procedure to give rigorous upper bounds for the survival probability is presented based on a lemma given by harris two new bounds are given improving the simple branching process bound in the one dimensional case the present procedure can be viewed as a trial to make approximate measures by generalized markov extensions

on some variational approximations in two dimensional classical lattice systems a new derivation is presented of some variational approximations for classical lattice systems that belong to the class of cluster variation methods among them the well known bethe peierls and kramers wannier approximations the limiting behavior of a hierarchical sequence of cluster variation approximations the so calledc hierarchy is discussed it is shown that this hierarchy provides a monotonically decreasing sequence of upper boundsfn on the free energy per lattice sitef and thatfn f asn our results are based on extension theorems for states given on subsets of the lattice which might be of some independent interest and on an application of transfer matrix concepts to the variational characterization of translation invariant equilibrium states

entropy and global markov properties we extend refine and give simple proofs of some recent results on the validity of global markov properties for classical spin systems one of the new results is that there is a global markov property that is satisfied by equilibrium states in general the proof of this establishes formulas for the entropy and free energy that show that these quantities are ford dimensional systems given in terms of d 1 dimensional systems furthermore we show that global markov properties imply the absence of some types of symmetry breaking

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

disconnected rational homotopy theory we construct two algebraic versions of homotopy theory of rational disconnected topological spaces one based on differential graded commutative associative algebras and the other one on complete differential graded lie algebras as an application of the developed technology we obtain results on the structure of maurer cartan spaces of complete differential graded lie algebras

an ordered line integral method for computing the quasi potential in the case of variable anisotropic diffusion abstract nongradient stochastic differential equations sdes with position dependent and anisotropic diffusion are often used in biological modeling the quasi potential is a crucial function in the large deviation theory that allows one to estimate transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths unfortunately the quasi potential can rarely be found analytically it is defined as the solution to a certain action minimization problem in this work the recently introduced ordered line integral method olim is extended for computing the quasi potential for 2d sdes with anisotropic and position dependent diffusion scaled by a small parameter on a regular rectangular mesh the presented solver employs the dynamical programming principle at each step a local action minimization problem is solved using straight line path segments and the midpoint quadrature rule the solver is tested on two examples where analytic formulas for the quasi potential are available the dependence of the computational error on the mesh size the update factor k a key parameter of olims as well as the degree and the orientation of anisotropy is established the effect of anisotropy on the quasi potential and the maximum likelihood paths is demonstrated on the maier stein model the proposed solver is applied to find the quasi potential and the maximum likelihood transition paths in a model of the genetic switch in lambda phage between the lysogenic state where the phage reproduces inside the infected cell without killing it and the lytic state where the phage destroys the infected cell

singularities of fluctuational paths for an overdamped two well system driven by white noise abstract noise induced escape from a metastable state in an overdamped double well system is investigated with the wkb approximation employed the pattern of fluctuational paths is given by using the action plot which gives rise to mpeps the pattern is divided into various regions where fluctuational paths inside show qualitatively distinguished characteristics the global quasi potential is calculated and its value at the saddle on the boundary agrees with the one obtained by action plot we show a contour and 3d plot of the quasi potential to illustrate the relative stabilities of metastable states the bifurcations of singularities as the system parameter varies are analyzed by the second order term of the wkb expansion in the vicinity of x axis its asymptotic behavior determines the positions of cusps and furthermore its evolution turns out to delineate the twisting structure of the lagrangian manifold

very long transients irregular firing and chaotic dynamics in networks of randomly connected inhibitory integrate and fire neurons we present results of an extensive numerical study of the dynamics of networks of integrate and fire neurons connected randomly through inhibitory interactions we first consider delayed interactions with infinitely fast rise and decay depending on the parameters the network displays transients which are short or exponentially long in the network size at the end of these transients the dynamics settle on a periodic attractor if the number of connections per neuron is large approximately 1000 this attractor is a cluster state with a short period in contrast if the number of connections per neuron is small approximately 100 the attractor has complex dynamics and very long period during the long transients the neurons fire in a highly irregular manner they can be viewed as quasistationary states in which depending on the coupling strength the pattern of activity is asynchronous or displays population oscillations in the first case the average firing rates and the variability of the single neuron activity are well described by a mean field theory valid in the thermodynamic limit bifurcations of the long transient dynamics from asynchronous to synchronous activity are also well predicted by this theory the transient dynamics display features reminiscent of stable chaos in particular despite being linearly stable the trajectories of the transient dynamics are destabilized by finite perturbations as small as o 1 n we further show that stable chaos is also observed for postsynaptic currents with finite decay time however we report in this type of network that chaotic dynamics characterized by positive lyapunov exponents can also be observed we show in fact that chaos occurs when the decay time of the synaptic currents is long compared to the synaptic delay provided that the network is sufficiently large

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

fluctuational transitions across different kinds of fractal basin boundaries we study fluctuational transitions in discrete and continuous dynamical systems that have two coexisting attractors in phase space separated by a fractal basin boundary which may be either locally disconnected or locally connected theoretical and numerical evidence is given to show that in each case the transition occurs via a unique accessible point on the boundary both in discrete systems and in flows the complicated structure of the escape paths inside the locally disconnected fractal basin boundary is determined by a hierarchy of homoclinic points the interrelation between the mechanism of transitions and the hierarchy is illustrated by consideration of fluctuational transitions in dynamical systems demonstrating fractal fractal basin boundary metamorphosis at some value of a control parameter the most probable escape path from an attractor which can be either regular or chaotic is found for each type of boundary using both statistical analysis of fluctuational trajectories and the hamiltonian theory of fluctuations

crossing the quasi threshold manifold of a noise driven excitable system we consider the noise induced escapes in an excitable system possessing a quasi threshold manifold along which there exists a certain point of minimal quasi potential in the weak noise limit the optimal escaping path turns out to approach this particular point asymptotically making it analogous to an ordinary saddle numerical simulations are performed and an elaboration on the effect of small but finite noise is given which shows that the ridges where the prehistory probability distribution peaks are located mainly within the region where the quasi potential increases gently the cases allowing anisotropic noise are discussed and we found that varying the noise term in the slow variable would dramatically raise the whole level of quasi potentials leading to significant changes in both patterns of optimal paths and exit locations

topological features of large fluctuations to the interior of a limit cycle received 29 october 1996 we investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves with overwhelming probability when it fluctuates far away from a stable state our emphasis is on systems that perform self sustained periodic vibrations and have an unstable focus inside a stable limit cycle we show that in the vicinity of the unstable focus the flow field of optimal paths generically displays a pattern of singularities in particular it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types the switching line spirals into the focus and has a self similar structure depending on the behavior of the system near the focus it may be smooth or have finite length branches our results are based on an analysis of the topology of the lagrangian manifold for an auxiliary purely dynamical problem that determines the optimal paths we illustrate our theory by studying both theoretically and numerically a van der pol oscillator driven by weak white noise s1063 651x 97 02703 7 many nonlinear dissipative systems display stable periodic oscillations i e their state spaces contain stable limit cycles 1 examples include lasers 2 radiofrequency generators 1 a chemical 3 a and biological 3 b systems in many cases the vibrations are nonsinusoidal as in a multimode laser with strong mode coupling 4 passive optically bistable elements 5 and various sorts of engineering structure 6 fluctuations in periodically oscillating systems play an extremely important role they eventually destroy periodicity of the self sustained oscillations so that the oscillation phase becomes random and a stationary probability distribution in the state space of the system is formed 7 8 this distribution reaches a maximum at the cycle does not vary much along the cycle and falls off rapidly in directions transverse to the cycle the shape of the maximum depends on the character of small transverse fluctuations away from the cycle if the fluctuations are due to the system being perturbed by weak noise of intensity d the transverse falloff will usually be approximately gaussian 7 9 with standard deviation proportional to d 1 2 as d 0

escape from attracting sets in randomly perturbed systems the dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science previous studies of escape in chaotic systems have rather focused on the case of unbounded noise usually assumed to have gaussian distribution in this paper we address the problem of escape induced by bounded noise we show that the dynamics of escape from an attractor s basin is equivalent to that of a closed system with an appropriately chosen hole using this equivalence we show that there is a minimum noise amplitude above which escape takes place and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition we verify our analytical predictions through numerical simulations of two well known two dimensional maps with noise

numerical limit of the spinodal point abstract the relation is studied between the phase boundary c pb and the spinodal point c s of the ferromagnetic ising model in the two dimensional triangular lattice using a series of approximations in the cvm cluster variation method hierarchy we keep the magnetization m fixed and extrapolate the temperature difference t pb m n t s m n for n 1 3 5 7 and 9 triangle approximations of the cvm it is found that the limiting value of the difference is practically zero our analysis indicates i the spinodal decomposition is to be treated as a system changing from the quenched initial state and the free energy curve used in the spinodal analysis is not a fixed curve but is changing in time and ii cases can occur in which a quenched system is initially decomposing in the nucleation mode but switches to the spinodal decomposition mode in time

mean field renormalization group for the boundary magnetization of strip clusters we analyse in some detail a recently proposed transfer matrix mean field approximation which yields the exact critical point for several two dimensional nearest neighbour ising models for the square lattice model we show explicitly that this approximation yields not only the exact critical point but also the exact boundary magnetization of a semi infinite ising model independent of the size of the strips used then we develop a new mean field renormalization group strategy based on this approximation and make connections with finite size scaling applying our strategy to the quadratic ising and three state potts models we obtain results for the critical exponents which are in very good agreement with the exact ones in this way we also clarify some advantages and limitations of the mean field renormalization group approach

weak noise approach to the logistic map using a nonperturbative weak noise approach we investigate the interference of noise and chaos in simple 1d maps we replace the noise driven 1d map by an area preserving 2d map modelling the poincare sections of a conserved dynamical system with unbounded energy manifolds we analyze the properties of the 2d map and draw conclusions concerning the interference of noise on the nonlinear time evolution we apply this technique to the standard period doubling sequence in the logistic map from the 2d area preserving analogue we in addition to the usual period doubling sequence obtain a series of period doubled cycles which are elliptic in nature these cycles are spinning off the real axis at parameters values corresponding to the standard period doubling events

reentrant transitions in the van hemmen model of a spin glass the van hemmen model of a spin glass which is an ising model with random couplings jij between sites i and j equal to j0 j i j j i where i i are independent identically distributed random variables is studied in the pair approximation of the cluster variation method for the family of probability distributions 12 1 p i a p i 12 1 p i a where p is varied phase diagrams are constructed they are qualitatively different from the mean field phase diagrams and display a competition between tendencies towards spin glass and towards ferromagnetic ordering which results in reentrant transitions it is argued that the observed effects are not accidental but are borne by the competition of bonds of the underlying lattice system

quasipotentials for simple noisy maps with complicated dynamics the theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems leading to exponential estimates of invariant probabilities and mean first exit times in the present article we develop the mathematical foundation of this theory for discrete time systems following and extending the work of freidlin and wentzell and kifer we discuss strategies for calculating and estimating quasipotentials and show their application to one dimensionals unimodal maps the method proves to be especially suited for describing the noise scaling behavior of invariant probabilities e g for the map occurring as the limit of the feigenbaum period doubling sequence we show that the method allows statements about the scaling behavior in the case of localized noise too which does not originally lie within the scope of the quasipotential formalism

noise induced escape through a fractal basin boundary we study noise induced escape within a discrete dynamical system that has two co existing chaotic attractors in phase space separated by a locally disconnected fractal basin boundary it is shown that escape occurs via a unique accessible point on the fractal boundary the structure of escape paths is determined by the original saddles forming the homoclinic structure of the system and by their hierarchical interrelations

use of hamiltonian mechanics in systems driven by colored noise the evaluation of the path integral representation for stochastic processes in the weak noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics we show that even when the noise is colored these may be put into a hamiltonian form which leads to better insights and improved numerical treatments we concentrate on solving hamilton s equations over an infinite time interval in order to determine the leading order contribution to the mean escape time for a bistable potential the paths may be oscillatory and inherently unstable in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy we look at two systems in some detail the underdamped langevin equation driven by external exponentially correlated noise and the overdamped langevin equation driven by external quasi monochromatic noise we deduce that caustics focusing and bifurcation of the optimal path are general features of all but the simplest stochastic processes

the cumulant approach for investigating the noise influence on mode locking bifurcations the influence of noise on mode locking bifurcations is investigated for the circle map and for coupled logistic maps the technique based on the cumulant expansion is used for the bifurcation analysis of these systems it is shown that this cumulant analysis in gaussian approximation provides a suitable description of the influence of weak noise we find that the universal scaling properties for the circle map in the critical point for the golden mean sequence are the same as those obtained analytically using the path integral technique we find the same scaling behaviour in the case of weak multiplicative noise

enhancement of noise induced escape through the existence of a chaotic saddle we study the noise induced escape process in a prototype dissipative nonequilibrium system the ikeda map in the presence of a chaotic saddle embedded in the basin of attraction of the metastable state we find the novel phenomenon of a strong enhancement of noise induced escape this result is established by employing the theory of quasipotentials our finding is of general validity and should be experimentally observable

quasipotential approach to critical scaling in noise induced chaos when a dynamical system exhibits transient chaos and a nonchaotic attractor as in a periodic window noise can induce a chaotic attractor in particular when the noise amplitude exceeds a critical value the largest lyapunov exponent of the attractor of the system starts to increase from zero while a scaling law for the variation of the lyapunov exponent with noise was uncovered previously it is mostly based on numerical evidence and a heuristic analysis this paper presents a more general approach to the scaling law one based on the concept of quasipotentials besides providing deeper insights into the problem of noise induced chaos the quasipotential approach enables previously unresolved issues to be addressed the fractal properties of noise induced chaotic attractors and applications to biological systems are also discussed

andre quillen cohomology and rational homotopy of function spaces we develop a simple theory of andre quillen cohomology for commutative differential graded algebras over a field of characteristic zero we then relate it to the homotopy groups of function spaces and spaces of homotopy self equivalences of rational nilpotent cw complexes this puts certain results of sullivan in a more conceptual framework

maurer cartan moduli and models for function spaces we set up a formalism of maurer cartan moduli sets for l algebras and associated twistings based on the closed model category structure on formal differential graded algebras a k a differential graded coalgebras among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of chevalley eilenberg and harrison cohomology we apply the developed technology to construct rational homotopy models for function spaces

algebraic models of non connected spaces and homotopy theory of l algebras we develop a homotopy theory of l algebras based on the lawrence sullivan construction a complete differential graded lie algebra which as we show satisfies the necessary properties to become the right cylinder in this category as a result we obtain a general procedure to algebraically model the rational homotopy type of non connected spaces 2012 elsevier ltd

pair correlations in the cluster variation approximation abstract the cluster variation method is used to obtain approximate free energy functions for a binary system in an inhomogeneous field in order to calculate the fourier transform of the pair correlation functions in the two dimensional square lattice using the nearest neighbors square as the basic cluster the correlation functions are correctly given up to the order 7 in the inverse temperature the effect of second neighbor pairs and many body interactions is investigated

exact variational methods and cluster variation approximations it is shown that for classical d dimensional lattice models with finite range interactions the translation invariant equilibrium states have the property that their mean entropy is completely determined by their restriction to a subset of the lattice that is infinite in d 1 dimensions and has a width equal to the range of the interaction in the dth dimension this property is used to show proper convergence toward the exact result for a hierarchy of approximations of the cluster variation method that uses one dimensionally increasing basis clusters in a two dimensional lattice

convergence of the cluster variation method for a system on a triangular lattice abstract the paper studies the convergence of the cluster variation method to the rigorous result as the cluster size increases the calculation is done on the phase boundary at t 0 between the a 2 b type ordered phase and the disordered phase on a two dimensional triangular lattice with nearest neighbor interaction it is shown that the phase boundary at t 0 is obtained by maximizing the entropy under the constraint that only a limited number of atomic configurations are allowed formulations are developed for clusters of n 3 5 7 9 11 and 13 points when thermodynamic quantities which are calculated using these clusters are plotted against 1 n they approach the known rigorous n results more or less linearly but with a pseudo period of n 6 an exception is the square of the long range order which bends down as 1 n tends to zero

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

noise induced failures of chaos stabilization large fluctuations and their control noise induced failures in the stabilization of an unstable orbit in the one dimensional logistic map are considered as large fluctuations from a stable state the properties of the large fluctuations are examined by determination and analysis of the optimal path and the optimal fluctuational force corresponding to the stabilization failure the problem of controlling noise induced large fluctuations is discussed and methods of control have been developed

escape problem for irreversible systems the problem of noise induced escape from a metastable state arises in physics chemistry biology systems engineering and other areas the problem is well understood when the underlying dynamics of the system obey detailed balance when this assumption fails many of the results of classical transition rate theory no longer apply and no general method exists for computing the weak noise asymptotic behavior of fundamental quantities such as the mean escape time in this paper we present a general technique for analyzing the weak noise limit of a wide range of stochastically perturbed continuous time nonlinear dynamical systems we simplify the original problem which involves solving a partial differential equation into one in which only ordinary differential equations need be solved this allows us to resolve some old issues for the case when detailed balance holds when it does not hold we show how the formula for the asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path we also present results on short time behavior and discuss the possibility of ital focusing along the escape path

minimal models of quantum homotopy lie algebras via the bv formalism using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral

tiling problems and undecidability in the cluster variation method in cluster approximations for lattice systems the thermodynamic behavior of the infinite system is inferred from that of a relatively small finite subsystem cluster approximations being made for the influence of the surrounding system in this context we study for translation invariant classical lattice systems the conditions under which a state for a cluster admits an extension to a global translation invariant state this extension problem is related to undecidable tiling problems the implication is that restrictions of global translation invariant states cannot be characterized purely locally in general this means that there is an unavoidable element of uncertainty in the application of a cluster approximation

stochastic bifurcation in a driven laser system experiment and theory we analyze the effects of stochastic perturbations in a physical example occurring as a higher dimensional dynamical system the physical model is that of a class b laser which is perturbed stochastically with finite noise the effect of the noise perturbations on the dynamics is shown to change the qualitative nature of the dynamics experimentally from a stochastic periodic attractor to one of chaoslike behavior or noise induced chaos to analyze the qualitative change we apply the technique of the stochastic frobenius perron operator l billings et al phys rev lett 88 234101 2002 to a model of the experimental system our main result is the identification of a global mechanism to induce chaoslike behavior by adding stochastic perturbations in a realistic model system of an optics experiment in quantifying the stochastic bifurcation we have computed a transition matrix describing the probability of transport from one region of phase space to another which approximates the stochastic frobenius perron operator this mechanism depends on both the standard deviation of the noise and the global topology of the system our result pinpoints regions of stochastic transport whereby topological deterministic dynamics subjected to sufficient noise results in noise induced chaos in both theory and experiment

cluster variation method in statistical physics and probabilistic graphical models the cluster variation method cvm is a hierarchy of approximate variational techniques for discrete ising like models in equilibrium statistical mechanics improving on the mean field approximation and the bethe peierls approximation which can be regarded as the lowest level of the cvm in recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models the foundations of the cvm are briefly reviewed and the relations with similar techniques are discussed the main properties of the method are considered with emphasis on its exactness for particular models and on its asymptotic properties the problem of the minimization of the variational free energy which arises in the cvm is also addressed and recent results about both provably convergent and message passing algorithms are discussed

a scaling theory of bifurcations in the symmetric weak noise escape problem we consider two dimensional overdamped double well systems perturbed by white noise in the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix the most probable escape path or mpep must terminate on the saddle between the two wells however as the parameters of a symmetric double well system are varied a unique mpep may bifurcate into two equally likely mpeps at the bifurcation point in parameter space the activation kinetics of the system become non arrhenius we quantify the non arrhenius behavior of a system at the bifurcation point by using the maslov wkb method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix the approximation is a formal asymptotic solution of the smoluchowski equation our construction relies on a new scaling theory which yields critical exponents describing weak noise behavior at the bifurcation point near the saddle

noise induced escape from attractors the author deals with several aspects of kautz 1987 1988 minimum principle for the noise induced escape from attractors he presents a slightly simpler derivation points out a connection with lyapunov exponents and invariant manifolds and generalises to chaotic maps

finding the quasipotential for nongradient sdes abstract the quasipotential with respect to a given attractor of a system evolving according to a stochastic differential equation sde with a small noise is a function that allows us to find the expected exit time from its basin of attraction and to estimate the equilibrium probability density in its neighborhood this work is devoted to the study of the quasipotential of nongradient sdes with isotropic diffusion the goals are i to understand the theoretical behavior of the quasipotential and ii to develop tools for its numerical study this work is focused on 2d equations with asymptotically stable equilibria and limit cycles in the theoretical part we establish a number of properties of the quasipotential for 2d linear equations we derive an exact formula for the quasipotential for 2d nonlinear equations we establish how the quasipotential grows near the stable equilibria and the stable limit cycles in the numerical part of this work we adjust the oum for computing the quasipotential on a regular mesh we show that the quasipotential satisfies a hamilton jacobi bellman equation with an anisotropic and unbounded speed function the unbounded speed function may lead to an additional consistency error in specific areas of the phase space however we conduct an error analysis and show that even if this error appears it decays quadratically as we refine the mesh we apply the proposed numerical algorithm to a number of examples linear systems the maier stein model a system with stochastic resonance and a system with two stable limit cycles whenever it is appropriate we compute the minimum action paths by shooting characteristics the minimum action paths found in this way are necessarily the global minimizers of the freidlin wentzell action

collective chaos and noise in the globally coupled complex ginzburg landau equation abstract we study a globally coupled version of the complex ginzburg landau equation gc cgle which consists of a large number n of identical two dimensional oscillators coupled through their mean amplitude depending on parameter values different dynamical regimes are attained we focus particularly on an interesting regime where the individual oscillators follow erratic motion but in a sufficiently coherent way so that the average motion does not vanish when n becomes large and is also chaotic a simple description of this state is proposed by considering the motion of a single forced two dimensional system which has both a limit cycle and a fixed point as stable attractors determining which of these two deterministic attractors is selected by a weak noise and how this depends on the parameter of the reduced system allows us to determine self consistently the average amplitude and dominant frequency of the collective behaviour of the full system finally we show that adding a small noise to the gc cgle transforms the chaotic collective behaviour into a purely periodic one

noise induced escape through a chaotic saddle lowering of the activation energy the noise induced escape mechanism is investigated for a prototype time discrete nonequilibrium system describing a laser in a ring cavity with optical feedback the ikeda map for certain parameter values a chaotic saddle is embedded in the basin of attraction of the metastable state this results in a lowering of the escape threshold for the activation energy which is established by employing the theory of quasipotentials our findings are explained by the computation of the most probable exit path mpep which lies partly on the chaotic saddle the enhancement of escape turns out to be maximal if the mpep lies entirely on the chaotic saddle

algorithms for controlling chaotic motion application for the bvp oscillator the bvp oscillator is a biologically and physically important dynamical system exhibiting chaotic motion phaselocking phenomenon and so on in this paper we investigate the controlling of chaos in this system both when periodic and constant currents are present we use the various control algorithms and investigate their effectiveness in particular we study the controlling by the following mechanisms i adaptive control algorithm ii weak periodic perturbations iii ott grebogi yorke method iv weak feedback control and v addition of noise also we show the suppression of chaos by numerical models

limiting exit location distributions in the stochastic exit problem consider a two dimensional continuous time dynamical system with an attracting fixed point s if the deterministic dynamics are perturbed by white noise random perturbations of strength epsilon the system state will eventually leave the domain of attraction omega of s we analyze the case when as epsilon to0 the exit location on the boundary partial omega is increasingly concentrated near a saddle point h of the deterministic dynamics we show using formal methods that the asymptotic form of the exit location distribution on partial omega is generically non gaussian and asymmetric and classify the possible limiting distributions a key role is played by a parameter mu equal to the ratio lambda s h lambda u h of the stable and unstable eigenvalues of the linearized deterministic flow at h if mu 1 then the exit location distribution is generically asymptotic as epsilon to0 to a weibull distribution with shape parameter 2 mu on the cal o epsilon mu 2

generalization of the kramers langer theory decay of the metastable state in the case of strongly anisotropic friction the problem of escape of a classical particle from a multidimensional potential well due to the influence of random forces is studied a new solution to the problem is obtained which shows that with fairly large friction anisotropy the qualitative picture of the process principally differs from that underlying the conventional multidimensional kramers langer theory according to the new process picture particles escape the well before reaching the saddle point the limiting stage of the process is the particle attainment of the transition region i e well dynamics not during passage through the barrier regions as the conventional theory suggests the new expression for the rate constant predicts a slower process rate in comparison with the kramers langer theory with decreasing friction anisotropy the new expression coincides with the conventional one

two dimensional translation invariant probability distributions approximations characterizations and no go theorems we study the properties of the set of marginal distributions of infinite translation invariant systems in the two dimensional square lattice in cases where the local variables can only take a smal

master equation simulations of a model of a thermochemical system master equation approach is used to study the influence of fluctuations on the dynamics of a model thermochemical system for appropriate values of parameters the deterministic description of the system gives the subcritical or supercritical hopf bifurcations for small systems containing 100 000 particles close to the supercritical hopf bifurcation the stochastic trajectories obtained from numerical simulations do not allow to distinguish between damped oscillations around a stable focus and sustained oscillations around a small stable limit cycle this uncertainty disappears if the number of particles in the system is increased up to 1 000 000 close to subcritical hopf bifurcation the stochastic trajectory of the system jumps from the basin of attraction of a stable focus to the basin of attraction of a stable limit cycle in this case the time dependencies of temperature and concentration of reactant in the system are apparently similar to intermittent chaotic oscillations the mean first passage time for the transitions from the stable focus to the stable limit cycle show the characteristic exponential dependence on the number of particles this passage time depends very strongly on the bifurcation parameter reaction heat which determines the distance between the stable focus and an unstable limit cycle

a version of the goldman millson theorem for filtered l algebras abstract in this paper we consider l algebras equipped with complete descending filtrations we prove that under some mild conditions an l quasi isomorphism u l l induces a weak equivalence between the deligne getzler hinich dgh groupoids corresponding to l and l respectively this paper may be considered as a modest addition to foundational paper 10 by ezra getzler

the global markov property for equilibrium states which are determined by correlations in a strip consider an interaction round face potential on the lattice d which may include nearest neighbor and next nearest neighbor pair interactions as well as more general plaquette terms assuming some periodicity of the potential it is shown that under the condition that equilibrium states can be distinguished by expectation values of sufficiently local observables equilibrium states possess a global markov property the condition under which this markov property is shown to hold is met in particular in case the equilibrium state is unique or determined by the magnetization the proof is based on an application of the variational principle to states which are constructed like markov chains